Wrath of Math: https://www.youtube.com/watch?v=VzsAehzmjrU&list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN
A Book of Abstract Algebra: https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/
In this second episode of "Motivate the Math," we dive into the foundational concepts of group theory, a crucial element in understanding cryptography and mathematics. We start by addressing feedback from our first episode, emphasizing the importance of making math accessible and correcting any inaccuracies. We introduce the concept of an errata to keep ourselves accountable and transparent.
We then explore the definition of a group in mathematics, discussing its four key axioms: closure, associativity, identity, and inverse. We explain how these properties are essential for a set of elements to be considered a group and why this matters in the broader context of math and cryptography. We also touch on the concept of commutativity, or Abelian groups, and introduce the idea of cyclic groups.
Throughout the episode, we reflect on the feedback and support from our listeners, sharing some of the boosts and messages we've received. We discuss the motivation behind the podcast and the desire to make complex mathematical concepts more digestible and engaging.
As we wrap up, we hint at future topics, including rings and fields, and the importance of understanding these concepts for cryptography. We also discuss the need for additional resources and problem-solving sessions to support the podcast's content, emphasizing the collaborative nature of learning and the journey we're on together.
[00:00:04]
Unknown:
I wanna set the right expectation here. Right? So, like, we you and I like to just rip. So we don't have pretox. We don't do much. Like, I have an idea and you're like, let's save it. Let's save it for the show. And like, you know, I think you have the same thinking. You're like, let's save everything for live. K? So let me just let me let me get a couple of things out of the way before we dig in. So, this is our 2nd episode. First things first, we talked about our backgrounds. Neither of us are professionals at what we do. Well, you're more of a professional in cryptography than I am in math. And there was some feedback and I do appreciate it and I encourage it and I do encourage it. There was some feedback that just some technical things that I misspoke on and what I did was I promised I would send this feedback back to the show without necessarily, you know, giving an opinion on the assertions just to let people know that some of the things have been challenged.
Like they don't, like I think everybody who's interacting with me, so some of this happens on Noster, some of it happens just on text, but like they understand what I'm trying to do, right? So they understand that we're trying to make this math accessible to people, we're trying to demystify this sort of abyss of unknown, right? And the more we can kind of just shine light on these things, the smaller the world gets and you're no longer drinking out of a fire hose. And so, there were 2 things that happened last week that I was
[00:01:54] Unknown:
I received feedback on. So what so what I ask before you before you get into this, let me ask, is this feedback on the wording, how you frame something, or was it feedback on, like, something you said was technically inaccurate due to the way you framed it? Technically inaccurate,
[00:02:11] Unknown:
I. E. Didn't and, you know, it's I think we just want to be we wanna keep ourselves honest. Right? But we also wanna keep our eye on the ball of what we're doing. So what I what I thought would be a good idea would be to just publish an errata, which is just us which would just be me spending 3 minutes telling all of you what I was told last week about, you know, what might have been incorrect. Right? And that's like to me the best mechanic and so I'm not gonna worry about it when I'm trying to explain something because I'm still gonna tell, I'm gonna do my best to reach the most limited person.
Right?
[00:02:55] Unknown:
And we're gonna get things wrong. Like, we will get things wrong. And and if nothing else, you know, this thing the show is titled Motivate the Math. It's not because we're gonna get everything right. Yes. We're There's some math savants. It's Right. To get you to dig into these things. And ideally, we get fact checked on every step of this way. Now ideally, we're never wrong. Right? Like that's what I think hopefully
[00:03:17] Unknown:
most of you guys are for. I actually like I don't let's put it this way. I don't mind being wrong because here's something that I found in math at least it's like if there's no struggle you tend not to grow, you tend not to actually you actually have to be wrong and be told you're wrong for certain concepts to really pop out. So I'm like, I think it's part of the process and I'm good with it, I embrace it, right? It doesn't embarrass me. I'm an amateur yet I'm the only one who has applied for this job which is to be the one to explain to all you guys, what I think is going on and what I think will help us.
We talked about it last week, none of the professional mathematicians are here volunteering to do this. They're all working for Shitcoins, some of them work for Bitcoin companies. Right. But they're not doing podcast to try, they're not trying to explain to you. It's like when I said, well, guys we consider the good guys, they're still not, they're not telling me why I saw a wall of numbers on a SecP website. Right? They're just they're doing their thing. So Yeah. I'm the guy that volunteered for the job and you'll have to just accept my limitations. Alright. So we have some errata. Let's get to the link and go right to the baby. Now, so yeah. So, yeah. It was 2 things. 1, I knew it. I knew it the second it left my mouth and it wasn't like I was wrong. I just didn't know who was right or how to talk about it. And it was this thing with when we talked about finite field and weather 0 was part of it and I had that confused just with another concept and it was the other concept that I really wanted to that I thought was part of it was the other concept to call the unit, like a unit field not a finite field.
The finite field clearly has 0 in it. Okay? Finite field modulo n has 0. Right. Someone's gonna tell me I'm wrong about this next week I know it. But it's this is what we're going with right now, okay? I just wanted to clear that confusion because I knew like when you and I were sort of disagreeing on that I knew it was gonna be an issue. But it's not like, it's like the detail is not pertinent to me to what we're trying to explain and that's why I sort of just, it's not in my view, but I want to communicate that this was pointed out. And the other thing I just want to communicate was pointed out was, I'm laughing because it's something I will never know in my life, but I talked about rational numbers and real numbers and we talked about the number systems. Yeah. And I said that real numbers kind of give you continuity whereas the rational numbers are like the known numbers that you can make out of fractions, but like the irrationals which are in between irrationals like pi and e and the square root of 2 which are in between rationals and the reals. They complete the real, they complete that number system.
The detail I was told was that actually the and I'm sorry guys if this is so fucking obtuse, just this will be over in a second. The detail is that, you could have continuity in the irrationals. Sorry, in the rationals. The rationals actually can have continuity, and it was explained to me great detail by our friend, Alan Farrington. And so, wanted to put that out there. The errata is complete, we are accountable and, say, let's start the show.
[00:06:56] Unknown:
All right.
[00:06:59] Unknown:
The one thing I did not expect to be doing in episode 2 at all was to read Boost.
[00:07:07] Unknown:
Why did you not expect that? I had
[00:07:10] Unknown:
I did not expect our first episode to get for anyone to even find because I've launched a few podcasts before and usually they don't have any traction yet.
[00:07:20] Unknown:
So Well, the the one thing I saw in Nasser, and I I don't necessarily see in the boost here, but, a great artist friend of of ours, I think. I think you know him as well. Was like, wow. I had no idea I needed this in my life. But, like, people out there have this desire to learn this. So I think Yes. This this is something that that was a huge motivation for me. Right? We're here to motivate everybody else to learn, but getting the feedback immediately from somebody that I know in meet space that I've met in real life saying, like, wow. This is awesome. Like, I wish I had this before. And then not only that, on on Noster, just chatting with some random NIM. And they're like, oh, yeah. I'm also, like, looking into Schnorr signatures and, like, learning mathematics and everything like that. So, the the meme is out there,
[00:08:06] Unknown:
and we're here to just squeeze the every living life out of it. What it tells me is that we're like, like, most of the things I seem to be involved in and talk about are just so early. They're so early, like, I'm the only one there. Right? Tells me this is we're closer to on time, maybe, than we think on this subject. It tells me this subject is really relevant to a lot of people right now. Right. That's I know that's corny to point out. It's just that I did not I I never expect to get boosts like this. Right? And it's just so early. And Boosts like what? Why don't you read the first one? So let's do it. Okay? We're gonna read boosts here because because we love we love the audience who boosts us.
So, the first boost is from Borst, who I didn't know but now started following. He boosted 21,000 sats. Just says, really looking forward to more of these. Pretty cool. Wanna hit the
[00:09:04] Unknown:
10,000 sats. Great idea for a podcast. Yep. Love Stackitoshi,
[00:09:08] Unknown:
our guy. Another guy, Jason See, who, I started a podcast with last week, another podcast about fish. So he was very gracious to come and boost the podcast and, he says looking forward to more great content related to the overlap of math and Bitcoin.
[00:09:28] Unknown:
Cheers to Nasser Npub. That's your Npub and then my Npub. Oh, god. Okay.
[00:09:34] Unknown:
Yeah. I mean, you know, he he and I are doing the overlap of fish and Bitcoin and now with you, we're doing the overlap of Yeah. Math. Next boo is 5,000 stats from John. Consider me motivated. Interesting stuff. Shaka emoji. That's j a w n. May or may not know him from the Bitcoin, John, but thanks, man. And then we got Dan who you mentioned, the artist. It's as if I've been asking God to teach me his universal language, and these fine gentlemen received the call to teach it to me. Thank you, Average Gary and Fundamentals. And I just wanna say that, yeah, I think it's like yeah. We have we're 2 human beings once again. We're just 2 humans that are hearing the call, and we're gonna we are going to do our best here.
Yep. Thanks, Dan.
[00:10:27] Unknown:
Some more more shout outs and booths from God Seth. Thank you, gentlemen. Montani, math is not my strong suit, but this is an engaging approach. It makes a complex subject digestible. Some anonymous booths and, JMCL. It's good to motivate. Keep going.
[00:10:41] Unknown:
Cool. So that was great. Thank you guys.
[00:10:48] Unknown:
So I had a Go ahead. I had a I had I I've been going through this understanding cryptography book and it it it's just in the first couple chapters, and it's talking about the advanced encryption standard. Right? Which is like first of all, it's it's a it's a block cipher, but it's used within anyway, it got into the math behind, like, how, one it got into why it does things a certain way, and then it got into math using some of these finite fields or a Gallois field is a number another name for finite field I found out. Yeah. Or And I was embarrassed
[00:11:32] Unknown:
by, I was running around calling it Gallois because I read it in that book, and I had a guy when this is back when I was working. Yeah. And I had a guy in my team who was a math major, and he just, like, pulls me aside. He's like, I just need you to know for the thing you've been saying for the last 2 weeks. Like, I was didn't wanna say anything, but you keep going with it. It's Galois. There's a fascinating and Galois is, like, maybe the most interesting biography of any of these guys, but that's, I thought I'd interlude that. Interesting in what way? He died at the age of 19 in a duel.
[00:12:07] Unknown:
Oh, wow.
[00:12:08] Unknown:
But he has a he has a mathematical field He's he's he's named after him. He is one of the top probably 5 most important mathematicians that's ever lived. And before he died, he sent he sent us just his stack, his ream of everything he had discovered to, some famous mathematician whose name escapes me right now, but like and then it took a 100 years because this guy was a lunatic and they they ignored him. And then at some point somebody picked it up and figured out, holy shit. This, like, explains all of algebra. This is ridiculous. This is where groups, rings, fields, and, you know, this is where this is where a lot of it I'm gonna get corrected on this too. Like, this is Well, no. That's it. You're leading into the next thing. Like, what the next definition I encountered in here was, like, what is a group?
[00:12:57] Unknown:
Right? And it it's a set of elements. Right? When you think of an element, you think of, like, a thing within a set. Right? Like, when, you think of a a deck of cards, the elements would be the individual cards. Right? There's 52 of them. Yep. But a group has these these properties that it has to follow.
[00:13:17] Unknown:
They're axioms. They're called that like, basically, they're called axioms. Okay. Right? Which is like, axioms are things in like, I think they're things in logic that we can take for granted that we don't have to prove. So, you know, and I did I'm still, like, I'm already thinking about, getting corrected on all this, but I'm gonna embrace this. I'm gonna lean into it, and I'm just gonna let the audience know at some point, Alan Farrington's gonna insist on doing this with us so that he doesn't have to keep yelling at his radio every time I say something like this. So, Axiom, and we're dealing with his baby Axiom is the name of this company. Axiom, so, but these are these things in logic, right, that we can take for granted. They're in there, it's almost like told to us by God, right? These axioms. And you don't have to prove these theorems every single time you want to prove something is true.
So, there are 4 axioms in a group. I didn't, you know, I mean, we could say what they are. Yeah, let's try it. Then talk about So, the 4 axioms of the group are 1 the first one's called closure. We talked about it last week. Yep. When you can say so there's 4 axioms and then there's something called an operation. Okay? So the operation, let's say, is addition. K? So now you have this axiom called closure where if you add any two numbers in the group, the result is in the group. That means you have closure. And the same applies for subtraction, multiplication, division, any operation? Not quite Well, it did not quite division and not quite you don't it's the same would apply, let's just say, the operation is very general. So you could have a group that has that closure on division, but but you, you know, you could have a set that does not, but just does not. Right?
Like it has to basically it has to satisfy the axiom to be called a group. And then people might ask who the fuck cares if you call something a group or not, and the answer is well, by by identifying something as a group you open up some very special properties that you can do, certain kinds of math on that actually allow all this to be possible. That's kind of the answer as to why does it matter. So, the first property is called closure. So, closure to the operation. The operation is very general. When we're in 7th grade, we get taught that algebra, you have addition and multiplication, right? But when we go talking about groups, we're saying the operation could be anything. It could be, a function of a function of another. It could be like what called it's called composition where I say one function is a function of another set.
I don't wanna be confusing here. I'm just saying that it could be ever it could be anything. It doesn't have to be limited to addition, multiplication. It could there there are many things that could be an operation.
[00:16:22] Unknown:
Yeah. The the closure the the operation the group operation is closed, but that operation is not just what we think about when we think about math. Right? Everybody, when you think about math, it's like multiplication, addition, and division. What you're saying is there are more complex more complex operations that can be inserted into that definition that still apply and bring closure to the group.
[00:16:46] Unknown:
Yes. Like so, you know, there could be an operation called, if an element is less than 5, the result is 1, if the element is greater than 5 the result is 0. That that like function can be an operation, right, to say, you like it you it's unlimited really what an operation can be. Oh, and it's oh, it's not unlimited. There has to be a binary operation. And what's binary operation? It means it's either like either has a defined result or it doesn't. So when we add two numbers we know there's a result, right? Even when we divide two numbers we know there's a result or there's not, right? Because we know if you're dividing by 0, there's no result, right? So, the operation is called binary.
The result is true or false. Okay. So, that's the only real restriction on what you call the operation. Okay. And so you only have closure when the result is true and the result is still is a member of that group. Right. Okay.
[00:17:50] Unknown:
And and this is where I'm learning in this text specifically. There's, like, specific mathematical notation, and I'm gonna do probably a horrible job explaining this, but it was basically showing there's, like, a symbol. It's like a backwards e, but it looks more like a pitchfork shaped symbol. Means there exists. It's include yeah. It includes within the group. So it's like a and b. Do you know what that symbol is called? Is that there's a name for that symbol?
[00:18:17] Unknown:
No. Alright. I'll have to look it up. The 2 the 2 symbols you see are, like, the upside down a, which means for any, and then the backwards e, which means there exists. But let's get let let I mean, let's try to get let's try to get through these 4 prop these 4 axioms. What's the next axiom? Okay. So number 1 was completeness. Closure, you mean? Sorry. Closure. Goddamn. This is gonna be so hard to do. We are really are gonna need to support this with, like, side you know, we're gonna need to support this with other, some other medium where we do this on the side too. Absolutely.
Okay. I'm try I I wanna get them in the right order, but, I'm Listen here I have associative. Yeah. Okay. Great. Great. Great. Okay. Number 2 is associative. What does that mean? It's tell you if you've if you learn 7th grade math it's trivial, but it basically just means that the order doesn't matter. Meaning, sorry, like so sorry. So, if you have like A plus B, that quantity, plus C, right? Mhmm. Like it doesn't matter if you add the A's and the B first or the B and the C first. That's just what associativity means, man. This is so esoteric and who gives a fuck? But these are little tiny, tiny details that just matter when you're trying to sew a sweater later on and you want to make sure you got that first thing right. This is a very esoteric thing called this. So in like formula terms, you'd have a, so here's a symbol you'd need to know. Instead of plus, minus, multiplication, the symbol for, like, just the generic operation is a little tiny circle.
Like an open circle, not a not a filled in circle. It's right. An open circle. So you would have, like, a circle b would be a using the operation on b. And then Right. So if your operation was addition, that would mean a plus b. Okay? So associativity is where, in a circle b is like done first and then you and then you operate on c, Then you'd say that's the same as having a operate on the total of b operating on c.
[00:20:37] Unknown:
Yeah. The the outcome of the operation doesn't matter which order. If you have multiple, I don't know if it's elements. But if if you have multiple elements that are all having the same operation done with each other,
[00:20:50] Unknown:
the ordering doesn't matter. So let's just take 1, 3, and 5 as an exam. Right? Our our group, our set, is 1, 3, and 5. So if I have on my left hand side, I do the 1 and the 3 first. So I can say 4 plus 5 is the same as 1 +8. That's all associativity means is that if that turns out to be true, right, which we would expect it to be true, it basically means that these operations are preserved within themselves, And if that turns out to be true, then you can be called a group. Right. Again, very esoteric, but it should remember It must be true to be a group. Yes. So just remember that it's just important it's one of those things. It's it's important to be called a group. Okay?
Even though we don't know why right now. Just it's important that that is, something we can call a set if we can because it match it opens up magical magical properties. Okay? So those are you know, the closure thing makes sense and is intuitive. You know, I always liked it. Right? Mhmm. The associative, I hate having to prove that. It's like, you know, yes, who cares? Right? But it's just one of those little details that if it doesn't, if you miss it, then you're screwed and you think you can do math in a certain way and you can't. So it's just this it's important to know it. And then the next 2 are both also seemingly really trivial but they're not as well. Number 3 is there has to be something called an identity. There exists something called an identity. Now that sounds, I don't know.
The key example is if you're talking about addition, the identity would be 0 because your element plus 0 is itself. So in other words, the identity is the thing that when you run the operation on it, you get yourself. So in multiplication, the identity would be 1.
[00:22:50] Unknown:
Well, for the group for for defining a group, according to the text here, is saying there is an element that is 1 that is in the group, and that is the that is the identity element.
[00:23:03] Unknown:
That is the neutral. 1 is the neutral identity element from if the operation is multiplication. And if the operation is addition, that identity is 0. You operate on 0 and you may the a operating on 0 is a if the operator is plus.
[00:23:25] Unknown:
And that's just you need something to get back to whatever element you're starting with, you need something to get back to that original element.
[00:23:33] Unknown:
Yeah. A group wrote the the this thing called the group for whatever magical things it's gonna do, right, needs to know that it can operate on itself. Very important. And I think the 4th property is probably the reason the most really the real reason why that's the case. The 4th property is that it also contains an element called the inverse. Now what's the inverse? So the inverse is the thing that you can, essentially multiply an element by and the result will be the identity. So if we're adding, you have an element and you wanna add something to it and get 0.
[00:24:15] Unknown:
Right? You have to add the negative of whatever that is. Negative of that is the inverse in addition.
[00:24:20] Unknown:
And in multiplication, you wanna multiply by something and get the inverse. You wanna multiply by something, you get 1, then you multiply by the inverse is called the reciprocal. 1 o it's 1 divided
[00:24:34] Unknown:
by Yeah. So 2 times half or one half is 1.
[00:24:41] Unknown:
Yes. So yeah. So if you take the set of integers, right, 0, 1, 2, 3, 4, negative 1, negative 2, negative 3, negative 4. Right? That is, that can be a group under addition but not multiplication because that, the integers under multiplication has no inverse. You need so you would need, the fraction. In other words right. In in 0, 1, 2, 3, 4, I pick element 3, the element inverse of 3 is 1 third, 1 third is not in the group. Therefore, when, the group of the set of integers is not a group when the operation is multiplication, but I could say it is a group when the operation is addition because I could pick any I can pick any element and find its inverse.
[00:25:34] Unknown:
Interesting. Okay. Yeah. I guess that that that might have been a key thing that I just sort of, tuned into as far as my brain goes is is the the operation is another key defining point of the group. Yes. Right? You're you keep saying, like, it it's a group with addition. It's a group with multiplication. It's a group with this. That's right. It's almost like
[00:25:55] Unknown:
it's like the operation is the thing that you're testing all the axioms on.
[00:26:01] Unknown:
Right. Okay. To determine if it's a group for that.
[00:26:05] Unknown:
Yes. So in other words, that the axioms don't really make sense without specifying the operation. Right? But, also, know that the operation can be general. Right? Mhmm. In theory, you know, like, so in theory of a group, when we talk about what groups can do, right, all the magical math you can do with groups, we don't care what the operation is. Right? But when we're testing a specific set, then, yes, we wanna make sure we understand what operate under what operations this set becomes a group. Right? The set of numbers.
[00:26:45] Unknown:
Yeah. Okay. It I mean, it's almost like it's like a dance. Like, if you're not if you don't have the right steps to follow the dance, doesn't matter what that dance is. Right? Like, you need to I'm trying to think. Maybe that's a bad analogy.
[00:27:02] Unknown:
You can't And the Go ahead. I I guess maybe the question is, like, why why would we even wanna define a set of elements as a group? Right? Like, how Exactly. Who gives a shit? Right. Right? Pardon my language here. Right? But who cares if we call something a group or not? At this stage of the podcast, that's a valid question. Right. And the the answer will be when we find out what it is, like, what becomes possible When you have a group. Algebra when you have when you actually have groups. So there's some everything we said about finite fields last week, you know, not 100% of everything, but a lot of it a lot of it relates and is is is applicable to what's possible with groups. Groups are groups can be well so in abstract algebra, just just to contrast a group and a field, you know, they can be the same. For example, the real numbers is a field
[00:28:09] Unknown:
and, you know We have field definition coming up next, but there was one more, It can't qualify as a property. Of a definition of group that that was here in the text that I wanted to bring up, and it it's called is Abelian or Camino?
[00:28:26] Unknown:
Abelian. Abelian? Yeah. So and which okay. So but let's go back and review real quick. Okay? The so just review a group. We got operation, and we have 4 axioms.
[00:28:36] Unknown:
Yep.
[00:28:38] Unknown:
Closure, associativity, identity an identity element, sorry, and inverse. Right? Closure, sensitivity, identity element, inverse element. K? Okay. I feel like it's a lot to try to cover more more than that, but the thing is so the the the logical extension is a ring which basically has 2 operations and they're specifically addition and multiplication, and then you have the distributed property. Okay. So, like, that's The what property? The distributed property. So this is just, like, things that I think that people learned in 7th grade for the most part, but don't know that there's any greater structure to them. Right? Distributor property meaning, like, if you take a times this quantity b plus c, you get a b plus a c.
Right. Okay. Yeah. You can, like, factor it out. Right? Is that Yeah. By using that word. Right? Yeah. And in order to be able to do that, you need both you need both operations. You need both operations to be, preserved. And, by the way, this is another thing. What groups do is they preserve the operations. That's so when the operation gets preserved, no matter what you're doing with the group, and, again, people ask why, who cares, we'll find out later, it's a very key it's just a really important property so that this operation is preserved no matter really no matter what you're doing, no matter how many times you're using this group and transforming it, you preserve this basic operation.
You know, I think we this is not a terrible place to start. Okay? You know, that book Understanding Cryptography, I think has 2 really, 2 lanes and we have to just focus 1 at a time on each lane. First lane's abstract algebra and the second lane, which is what we're talking about right now. Right? Okay. The abstract and and which is like the rules. Right? Abstract algebra is like the rules. It's it's like struct it's structures and rules. Right? And then you have number theory, which is more of like, how do I do actual numbers and stuff? And these two lanes connect, they intersect, right? So it's like we have one of these highways going from California to Minnesota and we have another one going from Florida to Minnesota. We're all gonna meet in Minnesota and freeze our asses off up there.
But, we're all gonna meet yeah, we're eventually gonna these two concepts are gonna converge, These two things are gonna converge, and then we're gonna figure out really what cryptography you know, why how does this then enable, you know Magic Internet numbers. Yeah. Yeah. Yeah. So the abstract, you know, I think it's tough, but I think we can get through it. Yeah. I think the abstract algebra needs probably some support, and I think I'll I'll put in the show notes some recommendations for YouTube videos, YouTube video series that I think are good. The one that comes to mind right now is called Wrath of Math.
Okay. Dude is like looks like he's 10 years old, but he knows a shit and he's really good at explaining explaining stuff to normies.
[00:32:21] Unknown:
W r a t h, wrath. Yeah. Yeah. Yeah. I got that written down. Yeah.
[00:32:26] Unknown:
He's really actually, I subscribed to his Patreon, and I think he's I think he's really good.
[00:32:31] Unknown:
Value for value. Gotta get him on Nastir.
[00:32:34] Unknown:
Yeah. Maybe, like, I you know, I do a lot of Patreon stuff, but I don't communicate with any of them. I don't try to, like, you know, try to communicate with them. You guys should. In the future. Get these guys on Nostra. But, anyway, like, you really wanna go through, like and the the thing about videos is that, you know, they will support, like, they'll support some base baseline knowledge. You go through a book and you're like, what the why do I care about this? And the videos I think will help continue sort of the thinking that you would need, right, as to why you should care and then what are some of the, you know, like the videos motivate the math as well, you know, as well, probably in a more constructive way than we are, right?
This podcast is important so that you know to watch the video. Right? We're really pointing you into, like, where do where do you get started. Right?
[00:33:29] Unknown:
Well, and just trying to lay out some of these core fundamental definitions. Right? Because that's the whole thing. I think I we talked about this first episode is without me sitting here going through what a group is, and I'd highlighted this whole definition, it it wouldn't it wouldn't click. Right? It it would not click as to, these individual properties. I can sit here and read it all day long. But Yeah. Going through and dissecting it like this,
[00:33:58] Unknown:
I think, is gonna help us. It's gonna be even more like, it's gonna be even more, like, litigious when we talk about rings and how you, like because there's so okay. So let's talk about groups real quick. And you mentioned you mentioned abelian. So now there's property.
[00:34:14] Unknown:
Or commutative.
[00:34:16] Unknown:
Yes. So commutative, I e, that's really so when we said associative, we said the order of operations doesn't matter. Meaning, like, if you do if you have a bunch of, operations on that the order in which you'd commit those operations don't matter. Right? Right. We said 1 +8 was the same as, 4, you know, 4+5. Right. Right? In com with commutative I hate the word. Commutativity. Abelian. That's why we like Abelian. So the mathematician whose name was Abel, and he may have been again, he might have actually been studying this before Galois. But Delois.
That's right. He had a specific type of a special type of group, a group with all of the axioms on an operation but with a very very special property, which is that the order of the variables doesn't matter. So in other words, like 1 plus 5 would equal 5 plus 1, right? A operates b equals b operates a. Right? 2 times 7 equals 7 times 2. You see like that that they could be essentially Well, the crazy thing is I can't
[00:35:37] Unknown:
think of numbers
[00:35:40] Unknown:
where, like, that isn't That's not true. Property. Right? Right. Because that's in, like, the real numbers that, you know, and the stuff that we grew up with, it was that was an axiom. Commutativity was an axiom because, because in fields that's true. So this is like what you you know, the beauty of this, the first time I encountered this was when I took linear algebra in college, and I was like, why does it matter whether I multiply a times b or b times a? And I think when I start doing some problem solving sessions, I spent almost like 6 hours on Sunday like, on one problem and it was like a cryptography problem where, you have these matrices that represent the plaintext and the ciphertext.
And the reason I spent 6 hours on it is because I actually made the mistake of screwing up the order of the multiplication because I was dealing with something that was not commutative and I forgot. And I was pulling my freaking eyelashes out trying to figure out what I was doing wrong. So it's just, I guess, again, know that it the reason you brought this property up is more of a mind blowing, it's like a it should be more of a mind blowing, discovery that the order sometimes does matter. Meaning, I think the I need a better term than order, and I think the one I like is left rightness. Meaning, like, which is left and which is right doesn't matter.
Because I like the order talking about associativity, meaning the order in which you do the operations.
[00:37:33] Unknown:
And if you're and if you're picturing in your mind, right, we talked about the circle being representation for an operation. It's like a circle b is equal to b circle a. Meaning left multiplication
[00:37:44] Unknown:
and right multiplication are the same. Right. That's probably the most artistically accurate way to to say that. Yeah. Right? And, again, it's like something you probably took for granted your whole life and maybe right now is like, what the hell? They're always the same. Well, they're not. I could I could tell you, like, some very ease I could in in a problem solving session, which we I will create and we'll start doing. Mhmm. This podcast will never stand alone. I mean, and it's coming very clear to me in episode 2, 38 minutes, Ed, like, yeah, this is not enough. Right? We just can't sit here and talk about axioms. Like, so there will be there definitely will be problem solving.
[00:38:29] Unknown:
Well, if nothing else, we can pull out what these sessions are gonna be. Right? Like, what is it that needs to be written down and demonstrated to really make the concrete points and and really hammer home, like, what is this group thing? And then what it what is not a group? Right? Like, how do you get to these, like, when do these axioms break? How would they break? Right? These are all questions that, I think are important because if you're am I correct in understanding is, like, to define the group, you have to make sure that all of these things are applicable within that group. And so doing the opposite is also a useful endeavor.
[00:39:08] Unknown:
So in any proof you'll ever see in a group, right, what you'll see is the 4 tests.
[00:39:16] Unknown:
Okay. You just okay. Proof. I have always wondered, like, what mathematical proofs meant Mhmm. In the in like, oh, yeah. We have, like, this proof for whatever. It's like, I don't know what that means to prove something. Mhmm. But it sounds like with proving a group, you just go through and demonstrate that each of these properties are applicable to every element in the group.
[00:39:44] Unknown:
That's right. So you can ask the question, is the integer, you know, is the set of integers a group? And then a astute student would say, well, what do you mean with respect to what operation? And then you'll say, okay, is the set of integers a group with respect to addition? And then you would say, yes, and then you would you would demonstrate closure. Right? The any integer plus any other integer is an integer. Right? Mhmm. You would demonstrate associativity as we said 1 +8 equals 4+5. Right? 13 with 13 and 5, you would show that. And then you would show that 0 is in the integers and 0 plus any integer is that integer and then you would show that the inverse exists, right? Because you could add any integer to its negative because it's negative. Get back to 0, the identity element. You could go through that and then then, you know, your teacher might say, okay. Well, what about multiplication?
And then you would go through the same exercise again and you'd say, oh, well, I got closure. But the multiplication group has a different identity element. That's right. Well, that's interesting. But you go through you would go through and you'd say closure check. Absolutely. Right? Right. You would look at associativity. Let's do it real quick with 13 and 5. Right? So if I do the 13 first, I get 3 times 5, which is 15.
[00:41:05] Unknown:
Yep.
[00:41:05] Unknown:
If I do the 5 and 3 first, I get 15 times 1. So, yeah, that's looks pretty good. Right? Yep. Social activity looks pretty good. Maybe I'll do an example without 1, if you, you know, you guys could do that yourself. Left, leave that to the reader. So then let's check for an identity element. Well, clearly, one would be an identity under multiplication, and that is an integer. So all goes Checks out. Yep. It's in the group. And now we get to an inverse. And if I look at any number other than 1, right, I cannot determine the inverse. I cannot get back to 1, and so, therefore, I say there's an inverse element does not exist generically.
Right? And therefore The axiom does not exist, right? Well, there's no element. The axiom does not hold because the element does not exist.
[00:41:55] Unknown:
Ah, okay.
[00:41:56] Unknown:
I cannot say there exists generically an inverse element in these set of integers for multiple
[00:42:05] Unknown:
for the operational multiplication. So integers are not a group when it comes to the operational multiplication?
[00:42:11] Unknown:
That's right. They got 3 out of the 4, and that's don't that's no good. So the So does that just mean that so okay. It's just an example of how you can't how you it's just this is a good example of how to contrast
[00:42:29] Unknown:
Yeah. But defining a group. To take it a step further, like, what does that mean for integers? That means that integers are not useful for certain types of mass that we're trying to accomplish. Right? That's right. And that's
[00:42:42] Unknown:
I would say that's why probably we invented the rationals, the group, the the set of rationals. So because rational numbers now have the fractions
[00:42:52] Unknown:
and now Which give you the inverse.
[00:42:55] Unknown:
Now you got a group under multiplication.
[00:42:58] Unknown:
But that just expanded the group. The number of that just expanded the number of elements in the group. Like Oh, no. I'm already hearing my friend Alan screaming
[00:43:07] Unknown:
bad mouth alert on that one. Yeah. That okay. But, like, not to get into it, but there the number of elements in a group when it's in well, like, that's a subject in and of in and of itself, let's just say. Because we're talking about infinity when we're talking about integers. And, you know, intuitively, I'll just say for me, I would guess that the odd integers have half the elements of the all the integers, but I'd be wrong. And there's read there's there's a proof for that, but, like, that the odd integers and the integers have the same amount of there's the same infinity, and the same infinity is the natural numbers.
And, you know, it is what it is. So those things break intuition a little bit but, like, it's not that it's, I think the Wrath of Math video series does by, like, the 5th video gets to this question, by the way? And there's a because it's a there's a very important result called Lagrange. I think it's Lagrange's Lagrange theorem and it's about if you know the number of elements in 1 group, if you and it's equal to another group that they would have to have the same number of elements. There's it's actually look. Just Wait. Say say that one more time. If you know of algebra. So no. So one of the magical things you can do with groups Mhmm.
Is you can establish mapping relationships. You can establish, like, what's called equivalents, like, relationships between 2 groups that don't seem like they're equivalent at all. And when you do that, you can basically figure out that they have the same number of elements. You can work with them now in different ways. You can work with 1 group in a way that you couldn't in the other same math, basically. You're the the the group theory tells you they're the same thing even though your eyes say they're totally different and I could never learn. When you say the same math, you mean the same operation? Well, what what remember when I said there's this magical math that you can do after you've established things or groups?
Yeah. It's that. It's like it's you can make you can draw conclusions about one system versus another. I think the best way I could just say it is that the the the algebra tells you they're the same even though your eyes and your brain tell you otherwise. You can have 2 sets and 2 group you can have 2 things that don't look the same at all, don't appear to be the same, and you would you would have zero intuition that they would help you answer a question. Right? But there's a world in which they are equivalent. Right?
And in that world you can answer a lot of you can actually answer a lot of questions. And that world is the one where they're both they're both groups and they're both they're, like, equivalent to each other based on laws of math in the group. Now this is, again, very esoteric, but Yeah. I was trying to tease out why it matters because you could take 2 things that are seemingly unrelated and then you can basically learn you know, it's almost like now you have a map between them and they become decoder rings, I. E. This is cryptography. Right? They do become decoder rings for each other.
That is why it matters. Right? You will want like, that's why we really like to focus on things, blobs, that possess this property. That are that are yeah. Structure because we could work with them in a very powerful way that allow it to, you know, draw relationships with other things that are seemingly unrelated.
[00:47:02] Unknown:
Okay. Do do we wanna continue and talk about the definition of a field, which is like I wouldn't that's too much of a leap. Too much of a leap. Okay. Let like, I would pre see
[00:47:14] Unknown:
I mean, you know, in reality, this takes a long time. I had to go through about 3 different textbooks just to understand what the difference between a ring and a field. And the reason why is because it's all these different stages. So, like, you know, a group is absolute. Right? A group is absolute. Like, it either follows these 4 axioms with the operation or it doesn't. So, textbooks usually start with groups and they go just absolutely hog wild on all the theorems of groups, okay? Okay. Because there's not a lot of distinction. It either follows the axioms.
[00:47:55] Unknown:
Or it doesn't.
[00:47:57] Unknown:
Right? And so it's sort of simple it's simple on that way. Right? Now you got fun little groups like abelian groups. There's another type of group called the cyclic group, which is one where you have where you can generate the entire group from one element. Just taking repeated powers of the same element. So Hold on. Say that a cyclical group is when you can take one element and create raising it to you you just keep raising it. So in other words, if you took the group, we haven't talked about congruence and modulos. We talked about it a little last week. We haven't, like, introduced that topic. Mhmm. We were if this podcast were like a textbook, right, it wouldn't know what congruence it really is. Right? But let's let's say for the sake you got, like, you understand this idea of modulo. Right? Where we if you take a number, say, 5, and you wanna have everything modulo 5, and we talked last week about a field with the elements 0, 1, 2, 3, and 4. Right? And just a reminder, modulo means you you divide by a number and then remainder is the answer. The remainder is the answer. Right? The remainder is the answer. So we talked about we had the finite field. We said it was 0, 1, 2, 3, and 4, and this is this field modulo 5. Right? So any number 11 modulo 5 is 1. Right? Because 5 times 2 is 10 and the remainder is 1.
[00:49:23] Unknown:
Right.
[00:49:24] Unknown:
724 modulo 5 is 4 because because Oh, because 720 is yeah. That's right. I have 5 into it 4 is my remainder. So my elements of remainders module of 5 or the the formal term I think is residues. But my my my my quote unquote group is 1, 2, 3, and 4. K? So let's just I'm gonna create an example just to show that if I took the element 2. Right? Yeah. And I kept taking it to higher powers. So 2 to the for 2 to the 0th power is 1. 1. 2 to the first power is 2. Yep. So I've already got I've already you see the group forming. Right? I've got 12. Yep. 2 squared is 4. 4. That's also in the group. I've got 3 of my 5 elements already. Right? Right.
Okay. Let's go to to the 3rd. 8. 8 and 8 modulo 5 is?
[00:50:26] Unknown:
3.
[00:50:27] Unknown:
Oh, look at this. I got I got 1, 2, 3, and 4. Okay. You see that? Yeah. Okay.
[00:50:35] Unknown:
What is and then what's 2 to 4th is 16. Modulo 5 is 1. Yes. Also.
[00:50:43] Unknown:
So now you basically when I say the the the group in this case so this is where I got a little confused last week and thought it shouldn't include 0. The cyclic group would not include this. Like, you know, you never you you're never gonna get to 0 and that's because you're dividing by prime number. So that's the whole point. Right? You're dividing by you're dividing by a prime number and you can't by definition, you're not gonna get the you're not gonna get that number or a number a multiple of that number by definition. So you're gonna have so you just generated this group 1, 2, 3, and 4. Right? By taking excessive powers of 2. And if you just take 2 arbitrarily, 2 to the 5th is 32. Right? That's 2.
Then you get You just see down the line. 64, okay, that's your 4. Okay. 128, that's 3 again. 256, okay, that's 1 again. And it's just down on line, a line, a line you go, you're gonna get 1, 2, 3, 4. You're gonna get them in different orders, right? Which is also an interesting and powerful property but that's called a cyclic group and the whole point of that of the last 5 minutes was to say there's another variation of a group that's that's interesting and actually important. So you can have an abelian or non abelian group, you can have a cyclic group or I don't know if there's called a non cyclic group, but, you know, those are really those are very important and powerful as well.
It's probably something. It's probably one I'm forgetting in groups. See now, see, when you get to rings just to preview. Right? Mhmm. The reason why it gets complicated is because what rings are is like it it has all of the you have 2 operations. So now it's twice as complicated a little bit. Right? But they are addition and multiplication. Okay? And by the way, when we do rings, it's probably gonna do be 3 episodes or just rings. Okay. But just to just to just to tease it out. Right? You've got 2 operations and they are multiplication and addition.
Right. And most of what the reason why those are interesting and you'll just see real quick. Right? Any, when you multiply numbers with like exponents like 2 to the 5th plus 2 to the 6th sorry, 2 to the 5th times 2 to the 6th, you know how you add, you end up adding the exponents, right? Mhmm. So in the in the you have a multiplication but in the exponent space you have addition. Those are the Is that the, like, distributive
[00:53:19] Unknown:
property
[00:53:19] Unknown:
again? Not quite yet. No. The distributive property is nothing more than a times b plus c is a b plus a c. A times the quantity b plus c. Meaning, if we take let's just take 2, 4 and 6, right? Mhmm. If I wanted to take 4+6 and multiply that by 2, that's 20, right?
[00:53:42] Unknown:
But I can get that by saying it's 2 times 4 which is 8. I guess it was more so like the 2 to the 5th plus 2 to the 6th. Times 2 to the 6th. Or 2 to 2 to 5th times 2 to the 6th, you're actually
[00:53:54] Unknown:
doing addition of the exponent. Right? That's right. So that equals 2 to the 11th. Right? So the operate but and it's also yeah. So the operation is of multiplication is preserved, but also you have a ring will allow you to basically have both and that's why it's it's powerful because having 2 operations is also pretty pretty common. Right? It's just calm it's very common to have math where you're trying to do addition and multiplication. The entire field of linear algebra with vector spaces
[00:54:31] Unknown:
is addition and scalar multiplication. Right? So like what And what you're describing, there's at least according to this the text, right, was, in order to have all 4 basic arithmetic operations, addition, subtraction, multiplication, division in one structure, we need a set which contains an additive and multiplicative group. This is what we call a field. So for the definition in this this understanding cryptography PDF that we're we've been I've been reviewing at least and you've reviewed, it's like the field means that you have both additive and multiplicative.
[00:55:03] Unknown:
That's because a field is a very specific version of a ring. Okay. So a a field to get to field, you got to start with a ring and then you have to do some basically, imagine a structure where we started with a group and said it has to be a billion and it has to be cyclic, and then that's a certain thing with a name. Mhmm. You take a ring. See, fields have commutivity in multiplication. Rings not don't necessarily. A ring the ring is the so the ring is the most general version of the field. I think that's the best way to say it. And, again, that's why I said I think it's gonna take, like, 3, like, 3 episodes and problem solving sessions to get through to get from what a ring is to a field is because it's a much more gen ring is so general. Right?
And a field is just a very, very specific version of a ring. For example, it has it has so a ring has all the group things for addition, but not for multiplication necessarily. Right? So you'd have to specify if it's commutative ring, then it's commutative for multiplication. K? It's already commutated for addition, but it can but then you might specify that it's commutated for multiplication. You might specify that, it has a unit. It has in other words, it has a one. Okay. The Again, this is all esoteric, but I'm what I'm saying is I think it's very hard to make this leap to a from a group to a field
[00:56:46] Unknown:
without dealing with this. Is hasn't all elements of a field form an additive group with a neutral element of 0. Yes. And we kinda talked about this earlier. It also said all elements of a field except 0 form a multiplicative group with a neutral element of 1.
[00:57:06] Unknown:
Yes. And 0 couldn't be part of that because you can't divide by 0. Right. You have to be able to divide by any element in the group to get right? So you would specify that like, you know, by the way, rationals are not inherently a group because you have to specify that your denominator can't equal 0. Your element that you're using in your denominator cannot be 0. You have to specify that. Mhmm. Alright.
[00:57:36] Unknown:
Okay. So we're gonna start talking about rings, like, we're talking about rings next week then?
[00:57:42] Unknown:
I suppose. I suppose we will. Let me, also just you know, if you ever, have you ever heard of the book Programming Bitcoin by Jimmy Song?
[00:57:51] Unknown:
I have. Yeah.
[00:57:52] Unknown:
So, like, in the first chapter of this book, it just so happens he goes through he defines a finite field in the first three pages. Mhmm. But he, you know, he starts creating his own objects for addition and multiplication and inverses and he yeah. So, like, if you actually go through the code and the thing the thing I give this book the most credit for, I've gotten a lot of programming books in my life and almost none of them work when you go to the GitHubs. Okay. Right? Very few, like, you go to the GitHub and the code just doesn't, you know, it's like it doesn't work in your version of Python or whatever you're using. Yeah. This thing works. You go to the GitHub, he's got the Jupyter Notebooks all set up and if you go through it now after sort of listening to this, I think that's a good thing, it's probably not a bad thing to do.
Also check out like the rat check out, you know, rathemath or other YouTube videos and bear with us. We're gonna supplement this somehow with, we're gonna supplement this with, problem solving session. Eyeballs. Yeah. It needs I mean, just talking about math is honestly stupid. Right?
[00:59:08] Unknown:
Certain at a certain level. It takes a special type of person to sit here, for an hour at a time and discuss this. Yes. It's well, again, this is that's why it's motivating
[00:59:18] Unknown:
the math. You still have to then get good at it. Well, go do the proofs. Right? So one of the things I wanted to say, I was thinking about just wanted to say on the podcast was, you know, I think this conversation is designed to be slow. Like I was thinking of somebody who's gonna go and listen to 10 of these episodes in a row, right? Like people tell, like I hear about people who find rock paper bitcoin and they say I just listened to the first ten episodes and I'm like Binge the whole thing. Yeah. It's amazing. Right? And I just imagine somebody doing that here and really really struggling. So I think so the structure of why we're doing I was thinking of, the structure of how to do this, I thought might follow a textbook where, like, what I'm worried about is people just drinking out of a fire hose, right? And that's like what we're I mean, in a certain level it's a little bit of reality until we get some very basic things across, right?
But, like, I guess what I'll say is communicate with us and let us know how this is going. Because there's no real template for how to do this. Not at all. We could go if you're crying uncle on the abstract algebra front, then we can just start playing with numbers because that's fun too. And we can start doing what I consider to be also super important which is prime numbers, congruence, Euclidean algorithm, and all of those tools as well. So, like, that's another lane if we start getting if we start puking on the algebra.
[01:00:58] Unknown:
Right? Well, understanding the math of cryptography, like, that's the the the north star as we laid out in the first episode. There there are many,
[01:01:06] Unknown:
paths. You need both. You need to To that. Yeah. And most most abstract algebra classes are essentially number theory as well. And they require they require it. Personally, for myself, when I finished the cryptography textbook, I went right into a number theory textbook. I just felt like that was the thing. I looked at the list of things from that book that I needed to learn and, if you look at Galois theory, that's the last chapter of every abstract algebra textbook is, and so I'm like, okay, I guess that's the goal, but then in order to get into the abstract algebra textbook, it I think it helps for people who are totally new maybe to also have some of the number theory.
I I I I can't figure out what We're gonna get into all of it. It's gonna be a random walk the whole way there. Yeah. And I guess I'll ask people to just stay in touch with us. You know? Yeah. Let us let us know and, yeah, let us know how it's going and what you think. Best way to reach out would be on Noster. Noster or Boost?
[01:02:18] Unknown:
Boost it. Yep. You know?
[01:02:20] Unknown:
We I I, we associate higher truthfulness to the size of the Boosts.
[01:02:27] Unknown:
Well, Fundy, thanks, thanks for spending your morning with me again,
[01:02:32] Unknown:
and and I look forward to next time. We're getting this plane is, like, in the air now, and now it's it's like, you know, we gotta now figure out how to stabilize it and get it, you know, start getting it to the direction we want it to go. Planes in the air, though. I feel good. We're headed there. I mean, we just spent an hour talking about groups.
[01:02:53] Unknown:
Beautiful. So thank you. And we'll see you next time on Motivate the Math.
[01:02:57] Unknown:
See you, guys.
I wanna set the right expectation here. Right? So, like, we you and I like to just rip. So we don't have pretox. We don't do much. Like, I have an idea and you're like, let's save it. Let's save it for the show. And like, you know, I think you have the same thinking. You're like, let's save everything for live. K? So let me just let me let me get a couple of things out of the way before we dig in. So, this is our 2nd episode. First things first, we talked about our backgrounds. Neither of us are professionals at what we do. Well, you're more of a professional in cryptography than I am in math. And there was some feedback and I do appreciate it and I encourage it and I do encourage it. There was some feedback that just some technical things that I misspoke on and what I did was I promised I would send this feedback back to the show without necessarily, you know, giving an opinion on the assertions just to let people know that some of the things have been challenged.
Like they don't, like I think everybody who's interacting with me, so some of this happens on Noster, some of it happens just on text, but like they understand what I'm trying to do, right? So they understand that we're trying to make this math accessible to people, we're trying to demystify this sort of abyss of unknown, right? And the more we can kind of just shine light on these things, the smaller the world gets and you're no longer drinking out of a fire hose. And so, there were 2 things that happened last week that I was
[00:01:54] Unknown:
I received feedback on. So what so what I ask before you before you get into this, let me ask, is this feedback on the wording, how you frame something, or was it feedback on, like, something you said was technically inaccurate due to the way you framed it? Technically inaccurate,
[00:02:11] Unknown:
I. E. Didn't and, you know, it's I think we just want to be we wanna keep ourselves honest. Right? But we also wanna keep our eye on the ball of what we're doing. So what I what I thought would be a good idea would be to just publish an errata, which is just us which would just be me spending 3 minutes telling all of you what I was told last week about, you know, what might have been incorrect. Right? And that's like to me the best mechanic and so I'm not gonna worry about it when I'm trying to explain something because I'm still gonna tell, I'm gonna do my best to reach the most limited person.
Right?
[00:02:55] Unknown:
And we're gonna get things wrong. Like, we will get things wrong. And and if nothing else, you know, this thing the show is titled Motivate the Math. It's not because we're gonna get everything right. Yes. We're There's some math savants. It's Right. To get you to dig into these things. And ideally, we get fact checked on every step of this way. Now ideally, we're never wrong. Right? Like that's what I think hopefully
[00:03:17] Unknown:
most of you guys are for. I actually like I don't let's put it this way. I don't mind being wrong because here's something that I found in math at least it's like if there's no struggle you tend not to grow, you tend not to actually you actually have to be wrong and be told you're wrong for certain concepts to really pop out. So I'm like, I think it's part of the process and I'm good with it, I embrace it, right? It doesn't embarrass me. I'm an amateur yet I'm the only one who has applied for this job which is to be the one to explain to all you guys, what I think is going on and what I think will help us.
We talked about it last week, none of the professional mathematicians are here volunteering to do this. They're all working for Shitcoins, some of them work for Bitcoin companies. Right. But they're not doing podcast to try, they're not trying to explain to you. It's like when I said, well, guys we consider the good guys, they're still not, they're not telling me why I saw a wall of numbers on a SecP website. Right? They're just they're doing their thing. So Yeah. I'm the guy that volunteered for the job and you'll have to just accept my limitations. Alright. So we have some errata. Let's get to the link and go right to the baby. Now, so yeah. So, yeah. It was 2 things. 1, I knew it. I knew it the second it left my mouth and it wasn't like I was wrong. I just didn't know who was right or how to talk about it. And it was this thing with when we talked about finite field and weather 0 was part of it and I had that confused just with another concept and it was the other concept that I really wanted to that I thought was part of it was the other concept to call the unit, like a unit field not a finite field.
The finite field clearly has 0 in it. Okay? Finite field modulo n has 0. Right. Someone's gonna tell me I'm wrong about this next week I know it. But it's this is what we're going with right now, okay? I just wanted to clear that confusion because I knew like when you and I were sort of disagreeing on that I knew it was gonna be an issue. But it's not like, it's like the detail is not pertinent to me to what we're trying to explain and that's why I sort of just, it's not in my view, but I want to communicate that this was pointed out. And the other thing I just want to communicate was pointed out was, I'm laughing because it's something I will never know in my life, but I talked about rational numbers and real numbers and we talked about the number systems. Yeah. And I said that real numbers kind of give you continuity whereas the rational numbers are like the known numbers that you can make out of fractions, but like the irrationals which are in between irrationals like pi and e and the square root of 2 which are in between rationals and the reals. They complete the real, they complete that number system.
The detail I was told was that actually the and I'm sorry guys if this is so fucking obtuse, just this will be over in a second. The detail is that, you could have continuity in the irrationals. Sorry, in the rationals. The rationals actually can have continuity, and it was explained to me great detail by our friend, Alan Farrington. And so, wanted to put that out there. The errata is complete, we are accountable and, say, let's start the show.
[00:06:56] Unknown:
All right.
[00:06:59] Unknown:
The one thing I did not expect to be doing in episode 2 at all was to read Boost.
[00:07:07] Unknown:
Why did you not expect that? I had
[00:07:10] Unknown:
I did not expect our first episode to get for anyone to even find because I've launched a few podcasts before and usually they don't have any traction yet.
[00:07:20] Unknown:
So Well, the the one thing I saw in Nasser, and I I don't necessarily see in the boost here, but, a great artist friend of of ours, I think. I think you know him as well. Was like, wow. I had no idea I needed this in my life. But, like, people out there have this desire to learn this. So I think Yes. This this is something that that was a huge motivation for me. Right? We're here to motivate everybody else to learn, but getting the feedback immediately from somebody that I know in meet space that I've met in real life saying, like, wow. This is awesome. Like, I wish I had this before. And then not only that, on on Noster, just chatting with some random NIM. And they're like, oh, yeah. I'm also, like, looking into Schnorr signatures and, like, learning mathematics and everything like that. So, the the meme is out there,
[00:08:06] Unknown:
and we're here to just squeeze the every living life out of it. What it tells me is that we're like, like, most of the things I seem to be involved in and talk about are just so early. They're so early, like, I'm the only one there. Right? Tells me this is we're closer to on time, maybe, than we think on this subject. It tells me this subject is really relevant to a lot of people right now. Right. That's I know that's corny to point out. It's just that I did not I I never expect to get boosts like this. Right? And it's just so early. And Boosts like what? Why don't you read the first one? So let's do it. Okay? We're gonna read boosts here because because we love we love the audience who boosts us.
So, the first boost is from Borst, who I didn't know but now started following. He boosted 21,000 sats. Just says, really looking forward to more of these. Pretty cool. Wanna hit the
[00:09:04] Unknown:
10,000 sats. Great idea for a podcast. Yep. Love Stackitoshi,
[00:09:08] Unknown:
our guy. Another guy, Jason See, who, I started a podcast with last week, another podcast about fish. So he was very gracious to come and boost the podcast and, he says looking forward to more great content related to the overlap of math and Bitcoin.
[00:09:28] Unknown:
Cheers to Nasser Npub. That's your Npub and then my Npub. Oh, god. Okay.
[00:09:34] Unknown:
Yeah. I mean, you know, he he and I are doing the overlap of fish and Bitcoin and now with you, we're doing the overlap of Yeah. Math. Next boo is 5,000 stats from John. Consider me motivated. Interesting stuff. Shaka emoji. That's j a w n. May or may not know him from the Bitcoin, John, but thanks, man. And then we got Dan who you mentioned, the artist. It's as if I've been asking God to teach me his universal language, and these fine gentlemen received the call to teach it to me. Thank you, Average Gary and Fundamentals. And I just wanna say that, yeah, I think it's like yeah. We have we're 2 human beings once again. We're just 2 humans that are hearing the call, and we're gonna we are going to do our best here.
Yep. Thanks, Dan.
[00:10:27] Unknown:
Some more more shout outs and booths from God Seth. Thank you, gentlemen. Montani, math is not my strong suit, but this is an engaging approach. It makes a complex subject digestible. Some anonymous booths and, JMCL. It's good to motivate. Keep going.
[00:10:41] Unknown:
Cool. So that was great. Thank you guys.
[00:10:48] Unknown:
So I had a Go ahead. I had a I had I I've been going through this understanding cryptography book and it it it's just in the first couple chapters, and it's talking about the advanced encryption standard. Right? Which is like first of all, it's it's a it's a block cipher, but it's used within anyway, it got into the math behind, like, how, one it got into why it does things a certain way, and then it got into math using some of these finite fields or a Gallois field is a number another name for finite field I found out. Yeah. Or And I was embarrassed
[00:11:32] Unknown:
by, I was running around calling it Gallois because I read it in that book, and I had a guy when this is back when I was working. Yeah. And I had a guy in my team who was a math major, and he just, like, pulls me aside. He's like, I just need you to know for the thing you've been saying for the last 2 weeks. Like, I was didn't wanna say anything, but you keep going with it. It's Galois. There's a fascinating and Galois is, like, maybe the most interesting biography of any of these guys, but that's, I thought I'd interlude that. Interesting in what way? He died at the age of 19 in a duel.
[00:12:07] Unknown:
Oh, wow.
[00:12:08] Unknown:
But he has a he has a mathematical field He's he's he's named after him. He is one of the top probably 5 most important mathematicians that's ever lived. And before he died, he sent he sent us just his stack, his ream of everything he had discovered to, some famous mathematician whose name escapes me right now, but like and then it took a 100 years because this guy was a lunatic and they they ignored him. And then at some point somebody picked it up and figured out, holy shit. This, like, explains all of algebra. This is ridiculous. This is where groups, rings, fields, and, you know, this is where this is where a lot of it I'm gonna get corrected on this too. Like, this is Well, no. That's it. You're leading into the next thing. Like, what the next definition I encountered in here was, like, what is a group?
[00:12:57] Unknown:
Right? And it it's a set of elements. Right? When you think of an element, you think of, like, a thing within a set. Right? Like, when, you think of a a deck of cards, the elements would be the individual cards. Right? There's 52 of them. Yep. But a group has these these properties that it has to follow.
[00:13:17] Unknown:
They're axioms. They're called that like, basically, they're called axioms. Okay. Right? Which is like, axioms are things in like, I think they're things in logic that we can take for granted that we don't have to prove. So, you know, and I did I'm still, like, I'm already thinking about, getting corrected on all this, but I'm gonna embrace this. I'm gonna lean into it, and I'm just gonna let the audience know at some point, Alan Farrington's gonna insist on doing this with us so that he doesn't have to keep yelling at his radio every time I say something like this. So, Axiom, and we're dealing with his baby Axiom is the name of this company. Axiom, so, but these are these things in logic, right, that we can take for granted. They're in there, it's almost like told to us by God, right? These axioms. And you don't have to prove these theorems every single time you want to prove something is true.
So, there are 4 axioms in a group. I didn't, you know, I mean, we could say what they are. Yeah, let's try it. Then talk about So, the 4 axioms of the group are 1 the first one's called closure. We talked about it last week. Yep. When you can say so there's 4 axioms and then there's something called an operation. Okay? So the operation, let's say, is addition. K? So now you have this axiom called closure where if you add any two numbers in the group, the result is in the group. That means you have closure. And the same applies for subtraction, multiplication, division, any operation? Not quite Well, it did not quite division and not quite you don't it's the same would apply, let's just say, the operation is very general. So you could have a group that has that closure on division, but but you, you know, you could have a set that does not, but just does not. Right?
Like it has to basically it has to satisfy the axiom to be called a group. And then people might ask who the fuck cares if you call something a group or not, and the answer is well, by by identifying something as a group you open up some very special properties that you can do, certain kinds of math on that actually allow all this to be possible. That's kind of the answer as to why does it matter. So, the first property is called closure. So, closure to the operation. The operation is very general. When we're in 7th grade, we get taught that algebra, you have addition and multiplication, right? But when we go talking about groups, we're saying the operation could be anything. It could be, a function of a function of another. It could be like what called it's called composition where I say one function is a function of another set.
I don't wanna be confusing here. I'm just saying that it could be ever it could be anything. It doesn't have to be limited to addition, multiplication. It could there there are many things that could be an operation.
[00:16:22] Unknown:
Yeah. The the closure the the operation the group operation is closed, but that operation is not just what we think about when we think about math. Right? Everybody, when you think about math, it's like multiplication, addition, and division. What you're saying is there are more complex more complex operations that can be inserted into that definition that still apply and bring closure to the group.
[00:16:46] Unknown:
Yes. Like so, you know, there could be an operation called, if an element is less than 5, the result is 1, if the element is greater than 5 the result is 0. That that like function can be an operation, right, to say, you like it you it's unlimited really what an operation can be. Oh, and it's oh, it's not unlimited. There has to be a binary operation. And what's binary operation? It means it's either like either has a defined result or it doesn't. So when we add two numbers we know there's a result, right? Even when we divide two numbers we know there's a result or there's not, right? Because we know if you're dividing by 0, there's no result, right? So, the operation is called binary.
The result is true or false. Okay. So, that's the only real restriction on what you call the operation. Okay. And so you only have closure when the result is true and the result is still is a member of that group. Right. Okay.
[00:17:50] Unknown:
And and this is where I'm learning in this text specifically. There's, like, specific mathematical notation, and I'm gonna do probably a horrible job explaining this, but it was basically showing there's, like, a symbol. It's like a backwards e, but it looks more like a pitchfork shaped symbol. Means there exists. It's include yeah. It includes within the group. So it's like a and b. Do you know what that symbol is called? Is that there's a name for that symbol?
[00:18:17] Unknown:
No. Alright. I'll have to look it up. The 2 the 2 symbols you see are, like, the upside down a, which means for any, and then the backwards e, which means there exists. But let's get let let I mean, let's try to get let's try to get through these 4 prop these 4 axioms. What's the next axiom? Okay. So number 1 was completeness. Closure, you mean? Sorry. Closure. Goddamn. This is gonna be so hard to do. We are really are gonna need to support this with, like, side you know, we're gonna need to support this with other, some other medium where we do this on the side too. Absolutely.
Okay. I'm try I I wanna get them in the right order, but, I'm Listen here I have associative. Yeah. Okay. Great. Great. Great. Okay. Number 2 is associative. What does that mean? It's tell you if you've if you learn 7th grade math it's trivial, but it basically just means that the order doesn't matter. Meaning, sorry, like so sorry. So, if you have like A plus B, that quantity, plus C, right? Mhmm. Like it doesn't matter if you add the A's and the B first or the B and the C first. That's just what associativity means, man. This is so esoteric and who gives a fuck? But these are little tiny, tiny details that just matter when you're trying to sew a sweater later on and you want to make sure you got that first thing right. This is a very esoteric thing called this. So in like formula terms, you'd have a, so here's a symbol you'd need to know. Instead of plus, minus, multiplication, the symbol for, like, just the generic operation is a little tiny circle.
Like an open circle, not a not a filled in circle. It's right. An open circle. So you would have, like, a circle b would be a using the operation on b. And then Right. So if your operation was addition, that would mean a plus b. Okay? So associativity is where, in a circle b is like done first and then you and then you operate on c, Then you'd say that's the same as having a operate on the total of b operating on c.
[00:20:37] Unknown:
Yeah. The the outcome of the operation doesn't matter which order. If you have multiple, I don't know if it's elements. But if if you have multiple elements that are all having the same operation done with each other,
[00:20:50] Unknown:
the ordering doesn't matter. So let's just take 1, 3, and 5 as an exam. Right? Our our group, our set, is 1, 3, and 5. So if I have on my left hand side, I do the 1 and the 3 first. So I can say 4 plus 5 is the same as 1 +8. That's all associativity means is that if that turns out to be true, right, which we would expect it to be true, it basically means that these operations are preserved within themselves, And if that turns out to be true, then you can be called a group. Right. Again, very esoteric, but it should remember It must be true to be a group. Yes. So just remember that it's just important it's one of those things. It's it's important to be called a group. Okay?
Even though we don't know why right now. Just it's important that that is, something we can call a set if we can because it match it opens up magical magical properties. Okay? So those are you know, the closure thing makes sense and is intuitive. You know, I always liked it. Right? Mhmm. The associative, I hate having to prove that. It's like, you know, yes, who cares? Right? But it's just one of those little details that if it doesn't, if you miss it, then you're screwed and you think you can do math in a certain way and you can't. So it's just this it's important to know it. And then the next 2 are both also seemingly really trivial but they're not as well. Number 3 is there has to be something called an identity. There exists something called an identity. Now that sounds, I don't know.
The key example is if you're talking about addition, the identity would be 0 because your element plus 0 is itself. So in other words, the identity is the thing that when you run the operation on it, you get yourself. So in multiplication, the identity would be 1.
[00:22:50] Unknown:
Well, for the group for for defining a group, according to the text here, is saying there is an element that is 1 that is in the group, and that is the that is the identity element.
[00:23:03] Unknown:
That is the neutral. 1 is the neutral identity element from if the operation is multiplication. And if the operation is addition, that identity is 0. You operate on 0 and you may the a operating on 0 is a if the operator is plus.
[00:23:25] Unknown:
And that's just you need something to get back to whatever element you're starting with, you need something to get back to that original element.
[00:23:33] Unknown:
Yeah. A group wrote the the this thing called the group for whatever magical things it's gonna do, right, needs to know that it can operate on itself. Very important. And I think the 4th property is probably the reason the most really the real reason why that's the case. The 4th property is that it also contains an element called the inverse. Now what's the inverse? So the inverse is the thing that you can, essentially multiply an element by and the result will be the identity. So if we're adding, you have an element and you wanna add something to it and get 0.
[00:24:15] Unknown:
Right? You have to add the negative of whatever that is. Negative of that is the inverse in addition.
[00:24:20] Unknown:
And in multiplication, you wanna multiply by something and get the inverse. You wanna multiply by something, you get 1, then you multiply by the inverse is called the reciprocal. 1 o it's 1 divided
[00:24:34] Unknown:
by Yeah. So 2 times half or one half is 1.
[00:24:41] Unknown:
Yes. So yeah. So if you take the set of integers, right, 0, 1, 2, 3, 4, negative 1, negative 2, negative 3, negative 4. Right? That is, that can be a group under addition but not multiplication because that, the integers under multiplication has no inverse. You need so you would need, the fraction. In other words right. In in 0, 1, 2, 3, 4, I pick element 3, the element inverse of 3 is 1 third, 1 third is not in the group. Therefore, when, the group of the set of integers is not a group when the operation is multiplication, but I could say it is a group when the operation is addition because I could pick any I can pick any element and find its inverse.
[00:25:34] Unknown:
Interesting. Okay. Yeah. I guess that that that might have been a key thing that I just sort of, tuned into as far as my brain goes is is the the operation is another key defining point of the group. Yes. Right? You're you keep saying, like, it it's a group with addition. It's a group with multiplication. It's a group with this. That's right. It's almost like
[00:25:55] Unknown:
it's like the operation is the thing that you're testing all the axioms on.
[00:26:01] Unknown:
Right. Okay. To determine if it's a group for that.
[00:26:05] Unknown:
Yes. So in other words, that the axioms don't really make sense without specifying the operation. Right? But, also, know that the operation can be general. Right? Mhmm. In theory, you know, like, so in theory of a group, when we talk about what groups can do, right, all the magical math you can do with groups, we don't care what the operation is. Right? But when we're testing a specific set, then, yes, we wanna make sure we understand what operate under what operations this set becomes a group. Right? The set of numbers.
[00:26:45] Unknown:
Yeah. Okay. It I mean, it's almost like it's like a dance. Like, if you're not if you don't have the right steps to follow the dance, doesn't matter what that dance is. Right? Like, you need to I'm trying to think. Maybe that's a bad analogy.
[00:27:02] Unknown:
You can't And the Go ahead. I I guess maybe the question is, like, why why would we even wanna define a set of elements as a group? Right? Like, how Exactly. Who gives a shit? Right. Right? Pardon my language here. Right? But who cares if we call something a group or not? At this stage of the podcast, that's a valid question. Right. And the the answer will be when we find out what it is, like, what becomes possible When you have a group. Algebra when you have when you actually have groups. So there's some everything we said about finite fields last week, you know, not 100% of everything, but a lot of it a lot of it relates and is is is applicable to what's possible with groups. Groups are groups can be well so in abstract algebra, just just to contrast a group and a field, you know, they can be the same. For example, the real numbers is a field
[00:28:09] Unknown:
and, you know We have field definition coming up next, but there was one more, It can't qualify as a property. Of a definition of group that that was here in the text that I wanted to bring up, and it it's called is Abelian or Camino?
[00:28:26] Unknown:
Abelian. Abelian? Yeah. So and which okay. So but let's go back and review real quick. Okay? The so just review a group. We got operation, and we have 4 axioms.
[00:28:36] Unknown:
Yep.
[00:28:38] Unknown:
Closure, associativity, identity an identity element, sorry, and inverse. Right? Closure, sensitivity, identity element, inverse element. K? Okay. I feel like it's a lot to try to cover more more than that, but the thing is so the the the logical extension is a ring which basically has 2 operations and they're specifically addition and multiplication, and then you have the distributed property. Okay. So, like, that's The what property? The distributed property. So this is just, like, things that I think that people learned in 7th grade for the most part, but don't know that there's any greater structure to them. Right? Distributor property meaning, like, if you take a times this quantity b plus c, you get a b plus a c.
Right. Okay. Yeah. You can, like, factor it out. Right? Is that Yeah. By using that word. Right? Yeah. And in order to be able to do that, you need both you need both operations. You need both operations to be, preserved. And, by the way, this is another thing. What groups do is they preserve the operations. That's so when the operation gets preserved, no matter what you're doing with the group, and, again, people ask why, who cares, we'll find out later, it's a very key it's just a really important property so that this operation is preserved no matter really no matter what you're doing, no matter how many times you're using this group and transforming it, you preserve this basic operation.
You know, I think we this is not a terrible place to start. Okay? You know, that book Understanding Cryptography, I think has 2 really, 2 lanes and we have to just focus 1 at a time on each lane. First lane's abstract algebra and the second lane, which is what we're talking about right now. Right? Okay. The abstract and and which is like the rules. Right? Abstract algebra is like the rules. It's it's like struct it's structures and rules. Right? And then you have number theory, which is more of like, how do I do actual numbers and stuff? And these two lanes connect, they intersect, right? So it's like we have one of these highways going from California to Minnesota and we have another one going from Florida to Minnesota. We're all gonna meet in Minnesota and freeze our asses off up there.
But, we're all gonna meet yeah, we're eventually gonna these two concepts are gonna converge, These two things are gonna converge, and then we're gonna figure out really what cryptography you know, why how does this then enable, you know Magic Internet numbers. Yeah. Yeah. Yeah. So the abstract, you know, I think it's tough, but I think we can get through it. Yeah. I think the abstract algebra needs probably some support, and I think I'll I'll put in the show notes some recommendations for YouTube videos, YouTube video series that I think are good. The one that comes to mind right now is called Wrath of Math.
Okay. Dude is like looks like he's 10 years old, but he knows a shit and he's really good at explaining explaining stuff to normies.
[00:32:21] Unknown:
W r a t h, wrath. Yeah. Yeah. Yeah. I got that written down. Yeah.
[00:32:26] Unknown:
He's really actually, I subscribed to his Patreon, and I think he's I think he's really good.
[00:32:31] Unknown:
Value for value. Gotta get him on Nastir.
[00:32:34] Unknown:
Yeah. Maybe, like, I you know, I do a lot of Patreon stuff, but I don't communicate with any of them. I don't try to, like, you know, try to communicate with them. You guys should. In the future. Get these guys on Nostra. But, anyway, like, you really wanna go through, like and the the thing about videos is that, you know, they will support, like, they'll support some base baseline knowledge. You go through a book and you're like, what the why do I care about this? And the videos I think will help continue sort of the thinking that you would need, right, as to why you should care and then what are some of the, you know, like the videos motivate the math as well, you know, as well, probably in a more constructive way than we are, right?
This podcast is important so that you know to watch the video. Right? We're really pointing you into, like, where do where do you get started. Right?
[00:33:29] Unknown:
Well, and just trying to lay out some of these core fundamental definitions. Right? Because that's the whole thing. I think I we talked about this first episode is without me sitting here going through what a group is, and I'd highlighted this whole definition, it it wouldn't it wouldn't click. Right? It it would not click as to, these individual properties. I can sit here and read it all day long. But Yeah. Going through and dissecting it like this,
[00:33:58] Unknown:
I think, is gonna help us. It's gonna be even more like, it's gonna be even more, like, litigious when we talk about rings and how you, like because there's so okay. So let's talk about groups real quick. And you mentioned you mentioned abelian. So now there's property.
[00:34:14] Unknown:
Or commutative.
[00:34:16] Unknown:
Yes. So commutative, I e, that's really so when we said associative, we said the order of operations doesn't matter. Meaning, like, if you do if you have a bunch of, operations on that the order in which you'd commit those operations don't matter. Right? Right. We said 1 +8 was the same as, 4, you know, 4+5. Right. Right? In com with commutative I hate the word. Commutativity. Abelian. That's why we like Abelian. So the mathematician whose name was Abel, and he may have been again, he might have actually been studying this before Galois. But Delois.
That's right. He had a specific type of a special type of group, a group with all of the axioms on an operation but with a very very special property, which is that the order of the variables doesn't matter. So in other words, like 1 plus 5 would equal 5 plus 1, right? A operates b equals b operates a. Right? 2 times 7 equals 7 times 2. You see like that that they could be essentially Well, the crazy thing is I can't
[00:35:37] Unknown:
think of numbers
[00:35:40] Unknown:
where, like, that isn't That's not true. Property. Right? Right. Because that's in, like, the real numbers that, you know, and the stuff that we grew up with, it was that was an axiom. Commutativity was an axiom because, because in fields that's true. So this is like what you you know, the beauty of this, the first time I encountered this was when I took linear algebra in college, and I was like, why does it matter whether I multiply a times b or b times a? And I think when I start doing some problem solving sessions, I spent almost like 6 hours on Sunday like, on one problem and it was like a cryptography problem where, you have these matrices that represent the plaintext and the ciphertext.
And the reason I spent 6 hours on it is because I actually made the mistake of screwing up the order of the multiplication because I was dealing with something that was not commutative and I forgot. And I was pulling my freaking eyelashes out trying to figure out what I was doing wrong. So it's just, I guess, again, know that it the reason you brought this property up is more of a mind blowing, it's like a it should be more of a mind blowing, discovery that the order sometimes does matter. Meaning, I think the I need a better term than order, and I think the one I like is left rightness. Meaning, like, which is left and which is right doesn't matter.
Because I like the order talking about associativity, meaning the order in which you do the operations.
[00:37:33] Unknown:
And if you're and if you're picturing in your mind, right, we talked about the circle being representation for an operation. It's like a circle b is equal to b circle a. Meaning left multiplication
[00:37:44] Unknown:
and right multiplication are the same. Right. That's probably the most artistically accurate way to to say that. Yeah. Right? And, again, it's like something you probably took for granted your whole life and maybe right now is like, what the hell? They're always the same. Well, they're not. I could I could tell you, like, some very ease I could in in a problem solving session, which we I will create and we'll start doing. Mhmm. This podcast will never stand alone. I mean, and it's coming very clear to me in episode 2, 38 minutes, Ed, like, yeah, this is not enough. Right? We just can't sit here and talk about axioms. Like, so there will be there definitely will be problem solving.
[00:38:29] Unknown:
Well, if nothing else, we can pull out what these sessions are gonna be. Right? Like, what is it that needs to be written down and demonstrated to really make the concrete points and and really hammer home, like, what is this group thing? And then what it what is not a group? Right? Like, how do you get to these, like, when do these axioms break? How would they break? Right? These are all questions that, I think are important because if you're am I correct in understanding is, like, to define the group, you have to make sure that all of these things are applicable within that group. And so doing the opposite is also a useful endeavor.
[00:39:08] Unknown:
So in any proof you'll ever see in a group, right, what you'll see is the 4 tests.
[00:39:16] Unknown:
Okay. You just okay. Proof. I have always wondered, like, what mathematical proofs meant Mhmm. In the in like, oh, yeah. We have, like, this proof for whatever. It's like, I don't know what that means to prove something. Mhmm. But it sounds like with proving a group, you just go through and demonstrate that each of these properties are applicable to every element in the group.
[00:39:44] Unknown:
That's right. So you can ask the question, is the integer, you know, is the set of integers a group? And then a astute student would say, well, what do you mean with respect to what operation? And then you'll say, okay, is the set of integers a group with respect to addition? And then you would say, yes, and then you would you would demonstrate closure. Right? The any integer plus any other integer is an integer. Right? Mhmm. You would demonstrate associativity as we said 1 +8 equals 4+5. Right? 13 with 13 and 5, you would show that. And then you would show that 0 is in the integers and 0 plus any integer is that integer and then you would show that the inverse exists, right? Because you could add any integer to its negative because it's negative. Get back to 0, the identity element. You could go through that and then then, you know, your teacher might say, okay. Well, what about multiplication?
And then you would go through the same exercise again and you'd say, oh, well, I got closure. But the multiplication group has a different identity element. That's right. Well, that's interesting. But you go through you would go through and you'd say closure check. Absolutely. Right? Right. You would look at associativity. Let's do it real quick with 13 and 5. Right? So if I do the 13 first, I get 3 times 5, which is 15.
[00:41:05] Unknown:
Yep.
[00:41:05] Unknown:
If I do the 5 and 3 first, I get 15 times 1. So, yeah, that's looks pretty good. Right? Yep. Social activity looks pretty good. Maybe I'll do an example without 1, if you, you know, you guys could do that yourself. Left, leave that to the reader. So then let's check for an identity element. Well, clearly, one would be an identity under multiplication, and that is an integer. So all goes Checks out. Yep. It's in the group. And now we get to an inverse. And if I look at any number other than 1, right, I cannot determine the inverse. I cannot get back to 1, and so, therefore, I say there's an inverse element does not exist generically.
Right? And therefore The axiom does not exist, right? Well, there's no element. The axiom does not hold because the element does not exist.
[00:41:55] Unknown:
Ah, okay.
[00:41:56] Unknown:
I cannot say there exists generically an inverse element in these set of integers for multiple
[00:42:05] Unknown:
for the operational multiplication. So integers are not a group when it comes to the operational multiplication?
[00:42:11] Unknown:
That's right. They got 3 out of the 4, and that's don't that's no good. So the So does that just mean that so okay. It's just an example of how you can't how you it's just this is a good example of how to contrast
[00:42:29] Unknown:
Yeah. But defining a group. To take it a step further, like, what does that mean for integers? That means that integers are not useful for certain types of mass that we're trying to accomplish. Right? That's right. And that's
[00:42:42] Unknown:
I would say that's why probably we invented the rationals, the group, the the set of rationals. So because rational numbers now have the fractions
[00:42:52] Unknown:
and now Which give you the inverse.
[00:42:55] Unknown:
Now you got a group under multiplication.
[00:42:58] Unknown:
But that just expanded the group. The number of that just expanded the number of elements in the group. Like Oh, no. I'm already hearing my friend Alan screaming
[00:43:07] Unknown:
bad mouth alert on that one. Yeah. That okay. But, like, not to get into it, but there the number of elements in a group when it's in well, like, that's a subject in and of in and of itself, let's just say. Because we're talking about infinity when we're talking about integers. And, you know, intuitively, I'll just say for me, I would guess that the odd integers have half the elements of the all the integers, but I'd be wrong. And there's read there's there's a proof for that, but, like, that the odd integers and the integers have the same amount of there's the same infinity, and the same infinity is the natural numbers.
And, you know, it is what it is. So those things break intuition a little bit but, like, it's not that it's, I think the Wrath of Math video series does by, like, the 5th video gets to this question, by the way? And there's a because it's a there's a very important result called Lagrange. I think it's Lagrange's Lagrange theorem and it's about if you know the number of elements in 1 group, if you and it's equal to another group that they would have to have the same number of elements. There's it's actually look. Just Wait. Say say that one more time. If you know of algebra. So no. So one of the magical things you can do with groups Mhmm.
Is you can establish mapping relationships. You can establish, like, what's called equivalents, like, relationships between 2 groups that don't seem like they're equivalent at all. And when you do that, you can basically figure out that they have the same number of elements. You can work with them now in different ways. You can work with 1 group in a way that you couldn't in the other same math, basically. You're the the the group theory tells you they're the same thing even though your eyes say they're totally different and I could never learn. When you say the same math, you mean the same operation? Well, what what remember when I said there's this magical math that you can do after you've established things or groups?
Yeah. It's that. It's like it's you can make you can draw conclusions about one system versus another. I think the best way I could just say it is that the the the algebra tells you they're the same even though your eyes and your brain tell you otherwise. You can have 2 sets and 2 group you can have 2 things that don't look the same at all, don't appear to be the same, and you would you would have zero intuition that they would help you answer a question. Right? But there's a world in which they are equivalent. Right?
And in that world you can answer a lot of you can actually answer a lot of questions. And that world is the one where they're both they're both groups and they're both they're, like, equivalent to each other based on laws of math in the group. Now this is, again, very esoteric, but Yeah. I was trying to tease out why it matters because you could take 2 things that are seemingly unrelated and then you can basically learn you know, it's almost like now you have a map between them and they become decoder rings, I. E. This is cryptography. Right? They do become decoder rings for each other.
That is why it matters. Right? You will want like, that's why we really like to focus on things, blobs, that possess this property. That are that are yeah. Structure because we could work with them in a very powerful way that allow it to, you know, draw relationships with other things that are seemingly unrelated.
[00:47:02] Unknown:
Okay. Do do we wanna continue and talk about the definition of a field, which is like I wouldn't that's too much of a leap. Too much of a leap. Okay. Let like, I would pre see
[00:47:14] Unknown:
I mean, you know, in reality, this takes a long time. I had to go through about 3 different textbooks just to understand what the difference between a ring and a field. And the reason why is because it's all these different stages. So, like, you know, a group is absolute. Right? A group is absolute. Like, it either follows these 4 axioms with the operation or it doesn't. So, textbooks usually start with groups and they go just absolutely hog wild on all the theorems of groups, okay? Okay. Because there's not a lot of distinction. It either follows the axioms.
[00:47:55] Unknown:
Or it doesn't.
[00:47:57] Unknown:
Right? And so it's sort of simple it's simple on that way. Right? Now you got fun little groups like abelian groups. There's another type of group called the cyclic group, which is one where you have where you can generate the entire group from one element. Just taking repeated powers of the same element. So Hold on. Say that a cyclical group is when you can take one element and create raising it to you you just keep raising it. So in other words, if you took the group, we haven't talked about congruence and modulos. We talked about it a little last week. We haven't, like, introduced that topic. Mhmm. We were if this podcast were like a textbook, right, it wouldn't know what congruence it really is. Right? But let's let's say for the sake you got, like, you understand this idea of modulo. Right? Where we if you take a number, say, 5, and you wanna have everything modulo 5, and we talked last week about a field with the elements 0, 1, 2, 3, and 4. Right? And just a reminder, modulo means you you divide by a number and then remainder is the answer. The remainder is the answer. Right? The remainder is the answer. So we talked about we had the finite field. We said it was 0, 1, 2, 3, and 4, and this is this field modulo 5. Right? So any number 11 modulo 5 is 1. Right? Because 5 times 2 is 10 and the remainder is 1.
[00:49:23] Unknown:
Right.
[00:49:24] Unknown:
724 modulo 5 is 4 because because Oh, because 720 is yeah. That's right. I have 5 into it 4 is my remainder. So my elements of remainders module of 5 or the the formal term I think is residues. But my my my my quote unquote group is 1, 2, 3, and 4. K? So let's just I'm gonna create an example just to show that if I took the element 2. Right? Yeah. And I kept taking it to higher powers. So 2 to the for 2 to the 0th power is 1. 1. 2 to the first power is 2. Yep. So I've already got I've already you see the group forming. Right? I've got 12. Yep. 2 squared is 4. 4. That's also in the group. I've got 3 of my 5 elements already. Right? Right.
Okay. Let's go to to the 3rd. 8. 8 and 8 modulo 5 is?
[00:50:26] Unknown:
3.
[00:50:27] Unknown:
Oh, look at this. I got I got 1, 2, 3, and 4. Okay. You see that? Yeah. Okay.
[00:50:35] Unknown:
What is and then what's 2 to 4th is 16. Modulo 5 is 1. Yes. Also.
[00:50:43] Unknown:
So now you basically when I say the the the group in this case so this is where I got a little confused last week and thought it shouldn't include 0. The cyclic group would not include this. Like, you know, you never you you're never gonna get to 0 and that's because you're dividing by prime number. So that's the whole point. Right? You're dividing by you're dividing by a prime number and you can't by definition, you're not gonna get the you're not gonna get that number or a number a multiple of that number by definition. So you're gonna have so you just generated this group 1, 2, 3, and 4. Right? By taking excessive powers of 2. And if you just take 2 arbitrarily, 2 to the 5th is 32. Right? That's 2.
Then you get You just see down the line. 64, okay, that's your 4. Okay. 128, that's 3 again. 256, okay, that's 1 again. And it's just down on line, a line, a line you go, you're gonna get 1, 2, 3, 4. You're gonna get them in different orders, right? Which is also an interesting and powerful property but that's called a cyclic group and the whole point of that of the last 5 minutes was to say there's another variation of a group that's that's interesting and actually important. So you can have an abelian or non abelian group, you can have a cyclic group or I don't know if there's called a non cyclic group, but, you know, those are really those are very important and powerful as well.
It's probably something. It's probably one I'm forgetting in groups. See now, see, when you get to rings just to preview. Right? Mhmm. The reason why it gets complicated is because what rings are is like it it has all of the you have 2 operations. So now it's twice as complicated a little bit. Right? But they are addition and multiplication. Okay? And by the way, when we do rings, it's probably gonna do be 3 episodes or just rings. Okay. But just to just to just to tease it out. Right? You've got 2 operations and they are multiplication and addition.
Right. And most of what the reason why those are interesting and you'll just see real quick. Right? Any, when you multiply numbers with like exponents like 2 to the 5th plus 2 to the 6th sorry, 2 to the 5th times 2 to the 6th, you know how you add, you end up adding the exponents, right? Mhmm. So in the in the you have a multiplication but in the exponent space you have addition. Those are the Is that the, like, distributive
[00:53:19] Unknown:
property
[00:53:19] Unknown:
again? Not quite yet. No. The distributive property is nothing more than a times b plus c is a b plus a c. A times the quantity b plus c. Meaning, if we take let's just take 2, 4 and 6, right? Mhmm. If I wanted to take 4+6 and multiply that by 2, that's 20, right?
[00:53:42] Unknown:
But I can get that by saying it's 2 times 4 which is 8. I guess it was more so like the 2 to the 5th plus 2 to the 6th. Times 2 to the 6th. Or 2 to 2 to 5th times 2 to the 6th, you're actually
[00:53:54] Unknown:
doing addition of the exponent. Right? That's right. So that equals 2 to the 11th. Right? So the operate but and it's also yeah. So the operation is of multiplication is preserved, but also you have a ring will allow you to basically have both and that's why it's it's powerful because having 2 operations is also pretty pretty common. Right? It's just calm it's very common to have math where you're trying to do addition and multiplication. The entire field of linear algebra with vector spaces
[00:54:31] Unknown:
is addition and scalar multiplication. Right? So like what And what you're describing, there's at least according to this the text, right, was, in order to have all 4 basic arithmetic operations, addition, subtraction, multiplication, division in one structure, we need a set which contains an additive and multiplicative group. This is what we call a field. So for the definition in this this understanding cryptography PDF that we're we've been I've been reviewing at least and you've reviewed, it's like the field means that you have both additive and multiplicative.
[00:55:03] Unknown:
That's because a field is a very specific version of a ring. Okay. So a a field to get to field, you got to start with a ring and then you have to do some basically, imagine a structure where we started with a group and said it has to be a billion and it has to be cyclic, and then that's a certain thing with a name. Mhmm. You take a ring. See, fields have commutivity in multiplication. Rings not don't necessarily. A ring the ring is the so the ring is the most general version of the field. I think that's the best way to say it. And, again, that's why I said I think it's gonna take, like, 3, like, 3 episodes and problem solving sessions to get through to get from what a ring is to a field is because it's a much more gen ring is so general. Right?
And a field is just a very, very specific version of a ring. For example, it has it has so a ring has all the group things for addition, but not for multiplication necessarily. Right? So you'd have to specify if it's commutative ring, then it's commutative for multiplication. K? It's already commutated for addition, but it can but then you might specify that it's commutated for multiplication. You might specify that, it has a unit. It has in other words, it has a one. Okay. The Again, this is all esoteric, but I'm what I'm saying is I think it's very hard to make this leap to a from a group to a field
[00:56:46] Unknown:
without dealing with this. Is hasn't all elements of a field form an additive group with a neutral element of 0. Yes. And we kinda talked about this earlier. It also said all elements of a field except 0 form a multiplicative group with a neutral element of 1.
[00:57:06] Unknown:
Yes. And 0 couldn't be part of that because you can't divide by 0. Right. You have to be able to divide by any element in the group to get right? So you would specify that like, you know, by the way, rationals are not inherently a group because you have to specify that your denominator can't equal 0. Your element that you're using in your denominator cannot be 0. You have to specify that. Mhmm. Alright.
[00:57:36] Unknown:
Okay. So we're gonna start talking about rings, like, we're talking about rings next week then?
[00:57:42] Unknown:
I suppose. I suppose we will. Let me, also just you know, if you ever, have you ever heard of the book Programming Bitcoin by Jimmy Song?
[00:57:51] Unknown:
I have. Yeah.
[00:57:52] Unknown:
So, like, in the first chapter of this book, it just so happens he goes through he defines a finite field in the first three pages. Mhmm. But he, you know, he starts creating his own objects for addition and multiplication and inverses and he yeah. So, like, if you actually go through the code and the thing the thing I give this book the most credit for, I've gotten a lot of programming books in my life and almost none of them work when you go to the GitHubs. Okay. Right? Very few, like, you go to the GitHub and the code just doesn't, you know, it's like it doesn't work in your version of Python or whatever you're using. Yeah. This thing works. You go to the GitHub, he's got the Jupyter Notebooks all set up and if you go through it now after sort of listening to this, I think that's a good thing, it's probably not a bad thing to do.
Also check out like the rat check out, you know, rathemath or other YouTube videos and bear with us. We're gonna supplement this somehow with, we're gonna supplement this with, problem solving session. Eyeballs. Yeah. It needs I mean, just talking about math is honestly stupid. Right?
[00:59:08] Unknown:
Certain at a certain level. It takes a special type of person to sit here, for an hour at a time and discuss this. Yes. It's well, again, this is that's why it's motivating
[00:59:18] Unknown:
the math. You still have to then get good at it. Well, go do the proofs. Right? So one of the things I wanted to say, I was thinking about just wanted to say on the podcast was, you know, I think this conversation is designed to be slow. Like I was thinking of somebody who's gonna go and listen to 10 of these episodes in a row, right? Like people tell, like I hear about people who find rock paper bitcoin and they say I just listened to the first ten episodes and I'm like Binge the whole thing. Yeah. It's amazing. Right? And I just imagine somebody doing that here and really really struggling. So I think so the structure of why we're doing I was thinking of, the structure of how to do this, I thought might follow a textbook where, like, what I'm worried about is people just drinking out of a fire hose, right? And that's like what we're I mean, in a certain level it's a little bit of reality until we get some very basic things across, right?
But, like, I guess what I'll say is communicate with us and let us know how this is going. Because there's no real template for how to do this. Not at all. We could go if you're crying uncle on the abstract algebra front, then we can just start playing with numbers because that's fun too. And we can start doing what I consider to be also super important which is prime numbers, congruence, Euclidean algorithm, and all of those tools as well. So, like, that's another lane if we start getting if we start puking on the algebra.
[01:00:58] Unknown:
Right? Well, understanding the math of cryptography, like, that's the the the north star as we laid out in the first episode. There there are many,
[01:01:06] Unknown:
paths. You need both. You need to To that. Yeah. And most most abstract algebra classes are essentially number theory as well. And they require they require it. Personally, for myself, when I finished the cryptography textbook, I went right into a number theory textbook. I just felt like that was the thing. I looked at the list of things from that book that I needed to learn and, if you look at Galois theory, that's the last chapter of every abstract algebra textbook is, and so I'm like, okay, I guess that's the goal, but then in order to get into the abstract algebra textbook, it I think it helps for people who are totally new maybe to also have some of the number theory.
I I I I can't figure out what We're gonna get into all of it. It's gonna be a random walk the whole way there. Yeah. And I guess I'll ask people to just stay in touch with us. You know? Yeah. Let us let us know and, yeah, let us know how it's going and what you think. Best way to reach out would be on Noster. Noster or Boost?
[01:02:18] Unknown:
Boost it. Yep. You know?
[01:02:20] Unknown:
We I I, we associate higher truthfulness to the size of the Boosts.
[01:02:27] Unknown:
Well, Fundy, thanks, thanks for spending your morning with me again,
[01:02:32] Unknown:
and and I look forward to next time. We're getting this plane is, like, in the air now, and now it's it's like, you know, we gotta now figure out how to stabilize it and get it, you know, start getting it to the direction we want it to go. Planes in the air, though. I feel good. We're headed there. I mean, we just spent an hour talking about groups.
[01:02:53] Unknown:
Beautiful. So thank you. And we'll see you next time on Motivate the Math.
[01:02:57] Unknown:
See you, guys.
Setting Expectations and Feedback
Unexpected Boosts and Audience Engagement
Understanding Cryptography and Finite Fields
Exploring Group Theory and Axioms
Rings and Fields: A Preview
The Importance of Group Properties in Math
Future Directions and Audience Interaction
