Bitcoin's Issuance Schedule
https://blog.lopp.net/how-is-the-21-million-bitcoin-cap-defined-and-enforced/
What is a Geometric Series
https://en.wikipedia.org/wiki/Geometric_series
Calculating the Sum of a Geometric Series
https://www.youtube.com/watch?v=PqXAjCXYbNk
Fundamentals
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AverageGary
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In this episode, we delve into the complexities of mathematics and its profound impact on our understanding of the world, particularly in the realm of cryptography. We discuss the challenges of making complex mathematical concepts accessible and why it's crucial not to blindly trust mathematical protocols without understanding their foundations. Our conversation explores the role of mathematics as a language that explains the workings of the world, emphasizing its importance beyond mere numbers.
We also touch on the human brain's ability to recognize patterns and how this relates to our perception of reality and survival instincts. The discussion extends to neuroplasticity and the potential to "rewire" our brains through practice and repetition, drawing parallels between physical and mental fitness.
Our exploration includes a deep dive into the concept of geometric series, particularly in the context of Bitcoin's block subsidy and its mathematical underpinnings. We explain how numbers can be represented as polynomials and the significance of fields and rings in mathematics, highlighting the unique properties of binary systems.
Throughout the episode, we emphasize the importance of teaching and sharing knowledge to deepen understanding, and we encourage listeners to engage with mathematical concepts actively. We also reflect on the challenges and rewards of discussing complex topics and the personal growth that comes from pushing intellectual boundaries.
[00:00:00]
Unknown:
Because it Yeah. This is, like, not an easy topic to talk about. And so I think it's prudent.
[00:00:08] Unknown:
If it if it was, right, there'd be a lot better people than you and I doing it professionally.
[00:00:14] Unknown:
Right? Well, hold on. Wait. You're saying that if it was easy, there would be better people doing what we're doing. There would be, like, if it was easy, right Yeah. It would be scaled.
[00:00:24] Unknown:
Like, the influencer class would already figure out how to productionalize these conversations. Right? I mean, you see a lot of math channels. You see, you know, you'd see it. There's a few cryptography channels that I found now. You do see them. They try. Right? But it's hard. And the hardest thing is it's it may be easy to teach it may be easy to make videos of a math subject. Right? Right. But what's difficult to get across I I don't I'm what I'm more interested in doing here, right, is in keeping alive this notion that we should not be just trusting math. Right? And how do you do that is by giving examples as to why we should be, why should we worry that we've trusted, like, we have many of us here have trusted our life savings on our protocol, right, that has a lot of reliances on things that we might want to make sure in our own minds we're very comfortable with.
[00:01:29] Unknown:
Well, and it's not even it's not even just, you know, Bitcoiners or or or or anything like that. Right? Like, the entire modern infrastructure of the web is, like, fund like, the the fun one of the fundamental pillars is this mathematics called cryptography. And then even further than that, though, I think it's math is is is an explanation or is is the closest approximation of using human language. Right? Because math is essentially just a language of explaining things
[00:02:03] Unknown:
to, like, how the world works. Right? Because it's like you're measuring all the any science has, like, numbers in it. Like, any science has numbers. And then then there's other words to science, but there's always numbers. And as we know we now know after three or four episodes, math isn't necessarily just numbers. Right? Right. Like, we had we did two episodes about, like, what a group is and without really mentioning any numbers except for zero and one maybe. Right? And you can go through the entire textbook, a lot of math textbooks, not see any numbers. Right?
So, like, it's what's going on. Right? It's not just number crunching. Right?
[00:02:38] Unknown:
Yeah. It's like pattern pattern recognition. Right? Like, humans were great at, like, pattern recognition and and deducing patterns in relationships.
[00:02:46] Unknown:
Some of us. Some of us. Quite good at it. Quite good at it. Right. Right? So, you know, so what you're getting at here is that is the second thing I think is really important. It's probably more important, but you don't get there without, I think, the fear. It's like the fear of yielding your power is, I think, the motive the real motivator to then this notion that math is access to power. You know? Like, so once you get over, okay, I don't want to yield my power to somebody else, that's really important. Mhmm. Then you say, well, okay, what about this source of power? Can I claim it? Can I claim this power for myself, and what might that make possible for myself?
Right?
[00:03:30] Unknown:
Well, in the way you claim that, I think, is is the internalization of these patterns in relationships. Right? Because our minds like, right now, you just, like, scan across the room, and your brain instantly conceptualizes what it sees into, like, okay, computer monitor, water bottle. Right? And, like, you don't even have to think about it. It's just Wow. You know the pattern
[00:03:52] Unknown:
of What? The water bottle when you see it. And so Yeah. Go ahead. This is gonna get this is gonna get good because you don't see a water bottle. You don't you what you see, right, is you see a bunch of pixels and then you probably see you see a a sparse Well, hold on. Meaning. Hold on. You see something very hold on. You see you see very, limited amount of images that a very small percentage of the thing you think you're seeing. And then your brain does the rest. Your your brain actually draws, interpolates. It tries to figure out what's in between those spaces.
Right? Cool. Yeah. Yeah. And so and, Gary, you know what better than I do. Right? Because of your experience and your experience, you know, military. We don't see for, truth. We see for survival. Yeah. So in other words, the way that our brains work in terms in doing that interpolation is it tells us sort of the most scary thing it could possibly be. It doesn't actually tell us what the thing is. It's telling us what it could be. Right? What threats could it be? And so, like, yeah, like, it's like if you, like how people know that a sniper is watching them from five miles away somehow. Right? And they've they've done these studies and they know, like, that they're in a site. Yeah. The animals not somehow know that they're in the sites of a sniper.
[00:05:21] Unknown:
Right? And and that that part of your like, it's, it's interesting. I was at a a a school informational thing. Right? Like, a new private school, that we're looking at, and and the woman presenting was talking about neuroplasticity. Right? She's like, there's, like, a definitive, you know, quote, unquote science now that is, like, sort of, you can rewire your brain. Right? And this is even, like, referenced in the bible and in some reference of, like, rewiring your brain. But it's, like, also, people have proven, sort of as much as you can prove anything that, like, the channels in your brain can be sort of adapted and and perform. And this is I think this is applicable just like anything. Right? Like, it's like going through the motions and getting the repetitions of anything will make you better at it.
[00:06:07] Unknown:
You can't rewire your brain, though. You can continue to wire. You can build on basically the chain that you have. Right? That's true. So like most people think I have my DNA and I this is it. I I stuck with what I got. Right? Maybe I can learn to ride a bike and I can pick up a couple new skills along the way but no you actually can you know so rewire to me sounds like you can redo it and maybe that's you'd have to be able to erase it probably to be able to redo it. But, you can do a lot. You can build quite a bit. Right? You can like, what it takes is things that are hard.
You know? And, it almost doesn't matter what it is in a certain way. Like, if you just committed to learning how to write your name with your offhand and you did that. You could do it over and over and over again until you get it right. If you spent like like ten minutes a day for thirty days with your offhand trying to write your name over and over again. It's not that it's not about getting it right. It's about that's how you build that's how you build new like neurons in your brain to build new skills. Right? You don't like actually give a shit if you write if you figure out how to even write your own name legibly.
You're going through it. It's kinda like, why football players run through tires. Right? They don't you don't run through tire in a game. Right. But somehow it's on every you know, every practice field's got the tires. Right? Right. Right. For the foot just for the footwork aspect of it to move around that thing. It's a yeah. It's a it's a training. It's a way to train. Right? It'd be funny if they did it in the game or they even did anything remotely like it in the game. But, like So, I mean, is this why you are
[00:07:52] Unknown:
constantly, like, going through equations? It's yeah. It's true. Me your notes. Like, you show me your notes of you're just doing math over and over and over again in different numbers, but the same calculations.
[00:08:05] Unknown:
It's one of the reasons. Yes. It's building calluses, it's building skill. It's not different from a musician learning scales or Right. Rip to just continuing. You know, there's a saying called practice makes perfect, but the true saying is practice makes permanent. So it's not that you're because you could practice something the wrong way and then you'll make that permanent. It's fine. Right? Right. Right. Yeah. Which is why like in what we're trying to do, if you're trying to create a fitness like, we're I'm trying to say that mathematics is a road to fitness.
And just like physical training is a road to fitness, but there's a billion podcasts and a billion books about how that is a road to fitness. That doesn't require you and I. The there are very few, there are very few people telling, talking to the people we care about Yep. Loudly that this matter. Very few people allocating time and energy to this conversation. Yep. And so that's where you and I have basically said, you know, you know, we want we want to make this make sure this conversation exists and we're gonna do our best. And it's hard. Like, for us, it's hard. And we're building new neurons trying to do it. It's very funny. Right? Yeah. You guys can kinda tell after a few episodes, it's not the easiest thing in the world. It's almost fun to watch us try to fail.
[00:09:33] Unknown:
What I think the so two points on just a recent topic was, like, one, the practice makes permanent thing. One of the things in the military is, like, because I I didn't grow up shooting, so I was actually a better shooter when they trained me because I had no bad habits to undo. It was like it was like, you know That's that's it's yeah. Tabla Buanca. You were an a ADIQ Bitcoiner, basically, ready to learn. Right? Exactly. And then the other thing with that was, one of this guy, he was, like, known in, like, he I I took, like, a strong man course from him. Right? Like, picking up stones and stuff. But one of the things that really stuck with me from this course was he said fitness is just a lifetime accumulation of reps.
He's like, it doesn't matter, like, heavy, speed, or or whatever, but, like, it depends on what you're going for. But it's like, do you you just have to do the reps over and over again. So I think that's probably
[00:10:23] Unknown:
I probably need to do more reps, personally. But that takes a lot of and it but that takes stamina. That takes skill. It takes a lot of skill to do one rep of an exercise correctly. In fact, I would almost bet that there's not a single lift that, like, my trainer wouldn't be able to show you how hard it could be with half the weight.
[00:10:45] Unknown:
Right. Right. Right. You know what I mean? And I'm not applying that in the math context, though. It's like, what what is what is, like okay. So there's, like, the couch to five k concept. Right? Where it's like, okay. You get up off your ass and you go walk first, and then you a mile maybe or even half mile or something like that. Like, what Yep. I wonder now, like, what is the Depends where you are. I so here's it's funny, man. Because I'm I have two daughters. One is a math major,
[00:11:11] Unknown:
and, you know, the other one's in tenth grade, and I just watched the progression.
[00:11:15] Unknown:
Right.
[00:11:16] Unknown:
My younger one is, like, rebellious. She's like, I just want you to know I don't care about math. You know? Yeah. Yeah. And, you know, and, like, last year when she's in ninth grade, she was getting, like, 15 on tests and stuff like that. And little by little That's good. Little by little, she starts caring and she starts working with me. And she still says she hates it, but and she's very good at it. Mhmm. And it's it's so it's just a matter of like wherever you are, right, you want to think about the next level. Right?
So if you are a complete, you've never seen you've never seen a number or even a logical concept. Right? You've never gone through a true false question or anything. You've never seen a question that challenged reason or anything. Right? Okay. Well, there's a there's a way there's a pathway for that. Right? You probably start looking at basic arithmetic and things like that. Right? Boolean logic. Yeah. I would say most people on average, if I ask what was the last thing you were really you remember being good at, right, they would all probably say seventh grade algebra or geometry or something like that. There was a moment where they actually thought they had it and then a week later it was over forever. You know, they got their their their public school curriculum just, you know, made sure they would never feel that way ever again.
You know? So, like, but that's so on average, I'm guessing that's where a lot of people are. Right? A lot of people in the Bitcoin space are, people with engineering degrees but have never looked at number theory or abstract algebra or anything like that. Now, these guys think they're math people. But they could benefit so much. Right? Yeah. In a lot of ways, I enjoy talking to total noobs more than people who think they know everything, you know, but yet are completely blunt you know, don't don't, you know, don't know this.
[00:13:26] Unknown:
That that's interesting. That makes me think of, like, like, you know, the the entry point of yeah. You're just a Bitcoiner. Right? You're just, like, whatever ADIQ Club out there that wants to, like, learn math and everything. It's like, go try to figure out why 21,000,000. Like, not why, but, like, how. Like, how does that actually work? And there's a great, post. I'll put it in the show notes, but it was by, Jameson Lop on, like, the supply limit and stuff. And he goes into the actual code and everything like that. And even that, like, simple concept of and I say simple because of the concept of 21,000,000, which first of all, it's not 21,000,000. It's like just a little bit under 21,000,000.
Yeah. There's like a reason behind that.
[00:14:09] Unknown:
And e even that was like I know some ears percolating already about there's some ears percolating getting excited to, like, one day we made to have this discussion of what's in between the supply limit and 21,000,000.
[00:14:23] Unknown:
What's in oh, what's in between? Go ahead. What's the remainder? Anyway, that's another question. But yeah. So go on. Sorry. I didn't get it. No. No. It it's just the even when I looked at it. Right? Like, as as in, you know, being in software and everything like that, I was like, oh, wait. I need to, like, give me let me, like, pause and, like, think about this for a second because it was not it's not just, like, 21,000,000. It's, like, written into the code somewhere. Right? It's, like, this, like, bit shift operation thing that happens, and it's based on, like, number halvings and all these other, metrics. It's based on blocks, essentially.
Like, the number of blocks. And, again, you have to, like, do the math to, like, calculate it out. It's like, okay. What actually happens? Right? So after video. For 210,000 blocks. Like, what actually happens here?
[00:15:11] Unknown:
Yeah. I mean, I I would sum it up as a it's a geometric series and it is kind of a geometric series. Okay. It's as simple as that, but it's it's still really a good it's a great exercise to go through, you know. If you don't have those words in your head, which you wouldn't unless you, you know, were trained to do math as a sort of a, you know, public, you know, government employee or whatever. Right? I mean, that's like the slave math they teach you, you know, learn what a gov on a geometric series is so you can do probability and, read worthless PhD papers and stuff like that. But, you know, it is still a powerful concept. I worked in annuities and every annuity is nothing more than a geometric series and it's a, it's basically just a sum. A series is a sum of things that, you know, sum of certain special things. Right? So in a series Okay. That makes sense.
Right, it's, you know, you're just adding, you're adding a block that is that has a factor of a of a number to a power. Right? So the first you know, 50 to 25 is one half. Yeah. Then you go then you then to 12 and a half. It's one fourth. Right? Yep. That's one half squared. So you have 50. You're almost gonna, like, factor out the 50 and then what you have is this one plus one half plus one half squared plus one half cubed. This is how you add up all. This is how you and then, you know, it's not 50. It's 210,000, actually. Right? It's 50 times Yeah. Times ten minutes, times twenty four hours, times six of them per hour. You know, you do that. It all comes looks like, oh my god. It comes out to 210,000 times this geometric series. Right?
And the end of this geometric series, right, at the limit and continuous, you know, like, that the limit is that turns out to be 20. Sorry. It turns out to be 10. Right? Or 10,000,000 in this case. So if if 210,000 times this number turns to be 21,000,000, which is I'm just trying to do the math in my head. I think it's a hundred.
[00:17:30] Unknown:
Yeah. I don't 210,000
[00:17:33] Unknown:
that much. Yeah. Anyway, but it's a two inch 10,000 blocks. So But there's some block time. Yeah. That's, like, a block time we're talking about. So it's, like, yeah, it actually just rolls up every 210,000 blocks. We do a we we we have we have the sorry. And I forgot about the actual reward, so times 50. So, you know, 200 times 50 then times one. So one plus one half plus one fourth plus one eighth plus one sixteenth blah blah blah blah blah all the way out to however many havings there are. What are there? 32 havings or something like that?
[00:18:07] Unknown:
Something like that. Let me see. I pulled up the Lop article here. By the way, if you're looking if you're if we don't have it in the show notes, which we should, it's how is the 21,000,000 Bitcoin cap defined and enforced. It was right on 01/12/2022 by Jameson Lop.
[00:18:21] Unknown:
Gary, is there a way to pause? I I actually have a water guy at my at my house.
[00:18:28] Unknown:
Alright. We're back rolling again. We're back. Okay. So You, listener, dear listener, would not know the, lengthy duration of which I sat here reading I didn't throw up. About the block. I didn't I didn't throw up. We're all good.
[00:18:43] Unknown:
That's inside. That's an inside reference.
[00:18:46] Unknown:
Back back to the so the block, though. Right? And I actually have the the the equation in in LOPS. We stopped so Gary could read the entire article. Yes. That is why. But there's this entire, like, this entire equation that that pops up. Right? And it's basically it's
[00:19:06] Unknown:
go ahead. What I was gonna say is I know Gary and I are looking at it, but do you guys know that it's the sum it's the big sum that looks like an e, you know, Greek letter sigma. Yep. From I goes from zero to 32 and it's got the 210,000 and then it has 50 divided by two to the the I power. Right? Yep. And what I'm saying is you can pull the 210,000 out. Right? You can even pull the 50 out and then what you're left with is this sum of one over two I which is a geometric series. And that's that that has a standard solution of one minus one divided by one half which is it's fucking so stupid. I I think the answer is two. I just have to do I I I'm so this is like the dumbest thing to be able to do in my head.
One over I think I'm pretty sure the answer is two. So, yeah, you'd multiply and and it is. So if that if that answer is two, and two times 50 is a hundred, and a hundred times 210,000 is 21,000,000. But that's two at the at, like, the limit. But how it actually happens when you hit the thirty second block is that you're not gonna complete all 21 sats.
[00:20:18] Unknown:
Right. And and and it's it's due to this thing called a right shift, which is, like, if you conceptualize we'll do a simple example of what, like, if you think about the number eight in binary is one zero zero zero. Right? Yeah. Because you have the ones place is zero, twos place is zero, the fours place is zero, and then the eight place is one. And and it the bit shift right is taking all those bits and just like if you're to, like, just shift them to the right. Right? So the one would slide into the four place, and then all the zeros slide to the right as well. So you end up with having four. And so what happens, like, in the actual code, this this subsidy number, right, because the the number that of new issued Bitcoin is actually called the subsidy, technically speaking.
The shift happens and how that's calculated is you're shifting by the number of halvings that occurred. And the number of halvings that occur is the block height divided by this halving interval, which is just so happens to be 210,000. Right? So it's like you're shifting it. So if you if you shift right one time if you start with eight, you shift right one time, it's dividing by two. And every time you shift that right, it's another division by two.
[00:21:31] Unknown:
Yeah. I actually think I can I actually might this is at the risk of another Icarus moment here? I think I can actually explain why it doesn't. So when you have a sum of a series. Right? Yeah. You have let's say you're, you know, you're you have the the sum of nine terms. And this is a, like, it's a matter of is your starting point time is your starting point zero or is your starting point one? And if your starting point is zero, then what happens is you finish before really you get to the end. Whereas, this is gonna confuse people, but like if your starting point is actually at one, then you finish at the actual end of this thing and, you end up with one more you would basically end up with one more term in your sum.
So if you went from zero to the absolute end, I think you end up with the you end up with this extra term in your sum that completes the you know, that that you end up with that last term. So you have 32 halvings. I'm guessing I mean, I would like to I guess somebody I'll challenge somebody. Does does adding a thirty third term to the sum complete? You know, because what basically, when I what I came up with, that factor of two that multiplied it by 210,000 times 50, that's based on infinite sum. So that's that was like if you carried it out to infinity, you know it's just one of those properties.
So knowing something's a geometric series remember how we said knowing something's a group is important because it allows you to know a lot of things? So knowing, being able to recognize something as a geometric series allows me to look at that sum over one of two I. And if I took it to infinity, that would evaluate to two, meaning one plus one half plus one fourth, plus one eighth, plus one sixteenth, all the way out to infinity, I know that I know the answer is two. Okay. So is Because I know it's a geometric series and I know how that, you know, then I've internalized all that. Is there an implication to putting a limitation on a geometric series? Right? Like the at least We don't live in infinity. That's the thing. So there are no there's never going to be infinity blocks. Right? Right. I mean, there are gonna be infinity blocks. There's just not gonna be infinity reward ish reward blocks. Blocks reward. Right? So that's just what Satoshi decided is, you know, what there's no point in going more than one set.
Right? Yeah. Not I'm sure he didn't call it that. I'm sure there's no point in going more than one over one hundred millionth of a reward.
[00:24:17] Unknown:
I'm guessing he just didn't wanna I'm get or he could've it could've literally just been the precision on his computer that had him decide to stop it there. That's exactly where that that's I was getting into. Right? It's because there's even, like, in the in the the code block that you see, one of the checks,
[00:24:34] Unknown:
is It wasn't like the messiah or anything that could just, like, conjured some perfect thing. There's a lot of luck involved and that was yeah. But but the having piece of it is if you if it's if it's greater than 64.
[00:24:46] Unknown:
Right? Meaning, like, that that that bit is, like, way, to the bit is, like, you you can't shift it anymore without starting over from the get go. Right? It does it doesn't check for that. So it Yeah. Yeah. Is this limitation. And in in general, like, in in computer science and stuff, like, a lot of things are built around, like,
[00:25:07] Unknown:
four, eight, like, powers of two, essentially. Yeah. In the base 10 system, the bit shift there is the I'd same analog would just be multiplying or be divided by 10. So in a bit shift in a binary number, moving one to the left is just is basically multiplying or dividing by two. You know? And we talked last week about how yeah. How a number is a polynomial of its base. Right? And
[00:25:31] Unknown:
Well, and and to put a concrete example, 400 to 40 is the same concept. Right? If you were if you remained three digits, 400 to forty forty would be divided by 10 and you're just shifting that four to the right one. Exactly. Right? And if it's a left shift, it's it's a multiplication of what the base that oh, man. So that would hold true for any sort of, like, base system. Right? So, like, because it's base two binary, a bit shift left or right is multiplying or dividing by two because the base is two. That's right. So in hexadecimal, if you if you shift if you do a shift left or right, you're multiply or dividing by 16. Exactly. Oh, interesting.
I I think that has a significant implication for, like, writing the actual code that does this math and, like, the shortcuts thereof. It probably does. It's probably the reason we have a hundred thousand,
[00:26:25] Unknown:
units to a Bitcoin. It's probably. I don't know if that's true. I'm just saying that. Million. What's that?
[00:26:32] Unknown:
About a million units?
[00:26:33] Unknown:
Two one yeah. A hundred hundred million. Right? Yeah. Yeah. It's like, yeah, dude, we get on mic and with our brain wipes out. Don't know basic facts anymore. Right? It happens to me all the time. Yeah. That's interesting. Okay. Like, I couldn't divide by one half just now. I forgot. I was like, oh, so because I, you know, the getting better at tuning out, caring if I'm super wrong about things. But, like, yeah, it's funny how oh, really? Is it a hundred million? Really? Yeah.
[00:27:05] Unknown:
It's, it's it's the journey. Right? And it's it's trying to understand the grokking
[00:27:09] Unknown:
of it all. People don't real so that's back to doing hard things and building, like, building your skill and building sort of your neurons. You don't understand what like, you think you we think we know things and then a little bit extra pressure gets put on us and we, like, all of a sudden become low IQ versions of ourselves. Right? Like, and and that's and again, it's just like how do you build resilience to that. Right? You might think you know the math. You might think you understand the math. Try getting on a podcast and explain it when you have peep you know your people are gonna tell you and set up with color you fucking moron when you get it wrong. Right? Yeah. It's like that little bit of pressure really then puts it it puts to the test. That's why I it's one of the reasons I wanted to do this. It's one of the reasons it's important to teach. I actually think it's really I think being an expert in something or even being competent in something requires teaching
[00:28:08] Unknown:
because you wanna do That's like the rhetoric level of understanding. Right? It's like knowledge, understanding, and then rhetoric.
[00:28:16] Unknown:
Yeah. I mean, playing music is no different. Right? So you can play as much as you want. You're you know, I can play as much as I want in my basement. But, like, until you go out and play with other people and you'd be judged, you're under the pressure of being judged by others. Right? Yeah. It is, you know, it's a different story. And so it's another reason we motivate we say we motivate the math. Right? We're hopefully we're motivating people to actually pick up a pencil and do a little work on it too.
[00:28:44] Unknown:
Like, I would check What and what what work should we motivate them to do today? Right? Because we've I think we've talked a lot of, like, conceptual stuff at the beginning, but, you know, and I I even confess that, like, I I wasn't able to get to a lot of reading, in between now and the last episode. I think the geometric series
[00:29:00] Unknown:
is like, go on Wikipedia. Look it up.
[00:29:06] Unknown:
Geometric series.
[00:29:07] Unknown:
Yeah. Check it out. Look at, like, an arithmetic series and a geometric series. And check out the formula for, you know, check out the formula for the block having, right, for the block issuance. I do see it. Yeah. And how it gets to, yeah. There's a nice little picture there. I mean, there's really cool visuals. So that's not a bad definitely not a bad way to go here. Right? Yep.
[00:29:40] Unknown:
Interesting. And so so, yeah, the block subsidy is a is a geometric series. It's it's just
[00:29:49] Unknown:
Yeah. You wanna actually go you wanna go even you wanna take even more risk here. So check this out. Sure. All so remember we said it's a, it's the number. Any number in it is a is like a polynomial because you have Right. You you have the base. So you said the number eight is like really one zero zero zero. Yeah. Why is that? Because it's, it's one times two to the third plus one plus zero times two squared plus zero times two to the first. Right? Plus zero. So the remember last time we'd said that the ones and the zeros are coefficients? Yes. Right? So the ones and the zeros are coefficients, and then the two is the base that's getting raised to an exponent. Right? Yep. Now the coefficients are in a field.
And in a they're actually in the field of finding field mod modulo two in binary. All the coefficients are either zero or one.
[00:30:50] Unknown:
Right. Right. And and the same applies to a base 10 because zero is not Zero is or nine. Yeah. That's right. Is is whatever number modulo
[00:30:59] Unknown:
10. So that's one of the reasons we we we talk about fields. Oh, wow. Okay. Okay. It's because, you know, when you do algebra and you find like terms and then add and multiply the coefficients, you want to make, you know, you knowing they're in a field, you know that you know that multiplying two numbers is going to give you a number in your field. Right? You know that. And, you you know, you don't have to think about it.
[00:31:27] Unknown:
Right.
[00:31:30] Unknown:
If you multiply one and one in a base two field, you are going to get a number and you're gonna get one. If If you multiply one by zero, you're gonna get zero. Right? If you multiply zero by zero, you're gonna get zero. There's no outcome where you multiply one of those coefficients or even if you add them. Right? Because if you add one and one, you get zero.
[00:31:51] Unknown:
So, like So just but so I guess, is it does that like, base two is a field,
[00:31:57] Unknown:
but base Well, then that's not necessarily field. It's not that base two is a field. That there is a field there is a field that's, like, defined as basically, the field of numbers, maybe integers modulo two. And consist of the numbers that would consist in the numbers zero and one, and it would contain all of those properties of the operations, addition with commutivity around addition. Right? Mhmm. Identity inverse. And multiplication, definitely with commutivity, with an inverse. Right? With an identity and an inverse. So it it so happens.
Right? Knowing it's a knowing that it's a field basically it essentially means you could turn your brain off and do all the math you want without worrying whether it's legal. I mean, that's essentially what it allows you to do. Right? Which most peep which is the starting point for most people, and that's how they teach it to people for when they you start in third grade. You don't say, hey. By the way, there's a whole bunch of rules for whether or not this stuff is legal. They're like, we can you know, dividing by zero, we can't get around. So we're gonna make sure well, that's and that exception is enough. That's an easy rule to It's an easy rule, but it's also enough to make people wanna a lot of people just wanna puke when they do math. They're like, ugh, really? I gotta worry about this fucking zero thing? It's bullshit. You know? And, like but, like, we're not we're not you know, we try not to complicate it too much for I feel like at that level trying to
[00:33:36] Unknown:
I was trying to think of this field concept in, like, base 10. Right? Because I think it's very easy to, like, conceptualize
[00:33:43] Unknown:
base two. So it's the same deal. Right? So when you have, and if you if you think about how you were taught to do multiplication
[00:33:51] Unknown:
Yeah.
[00:33:53] Unknown:
Or subtraction where you borrowed from a 10 and you took, you know, you you might borrow from a tenths place. Right? It's this is this is all because of, you can do this. It's all modulo 10. Okay? You're dealing with this digit zero through nine, and it's all in a modulo 10. So nine plus nine, when you do when you actually do addition, you write down the number eight.
[00:34:19] Unknown:
Right. And then you carry it to the That's right.
[00:34:23] Unknown:
Oh. You carry that 10 into the 10 into the next tens place. It's a bit shift. You carry it into the shift you like into the shifted it's not bit because bit means base two. But whatever the base 10 equivalent of a bit is.
[00:34:37] Unknown:
Because it it it's Maybe yeah. Maybe they do call it. Yeah. Because It's not a computer science guy. Is not within that field, so you have to represent that a different way. And so you move it to a whole new place.
[00:34:47] Unknown:
It's all modular 10. Place. Everything's modular 10. 10 place is now a new
[00:34:52] Unknown:
representation of that field, but it is with a different, it is a different I don't think I'm gonna use the right word, but it's like a different piece of the polynomial that is defining it. Right?
[00:35:05] Unknown:
It's yeah. You move it where it belongs. Right. You're like, no. We're this is not part of addition. This is now part of, I guess, you know, you'd say maybe multiplication. Right? It's part of it's this is now part of multiplication where we have to go one to the left. Right? We're gonna hold this eight because there's no there's no 18 to write down here. Right? It doesn't make sense. See, the polynomial that represents the field there. Yeah. Well, the so in other words, because you have this polynomial representing the number, and the polynomial is coefficients times your base. Okay?
[00:35:43] Unknown:
So Yeah. Ten ten times 10 plus eight times 10 to the zero, which is one.
[00:35:50] Unknown:
Right? And so you yeah. And so when you say carry the one, you're just basically saying, yeah. Because I have, you know, I have an extra 10 that needs to go somewhere. Add a new non zero coefficient. But that's part of my power. No. So what I'm doing is I'm recognizing that it's in the power, not in the coefficient. The coefficient is the field is the member of the field. And this is, you know, very dangerous. We're in dangerous ground right now in this conversation. But I think it's it's okay to look at a polynomial. It's it's it's a bunch of coefficients times a bunch of one number to a power.
[00:36:27] Unknown:
Right? So yeah. Okay. So so let's let's let's see. So, like, the the the number, the quote, unquote number 418 Yes. Is literally saying Sum of three numbers. It's the it is the sum of three numbers, and those three numbers are each, if you represent it as a polynomial Yep. You are representing it as, the coefficient being the numbers that that we conceptualize generally as numbers, like the four What we call the digits. What we would call the Great. Yep. The the the four, the one, and the eight, but then it the four is actually the value of four times 10 to the second Yes.
Which is a hundred conceptually. Conceptually. Yes. And then one times 10 to the first, which is just 10. Yep. And then eight times 10 to the zero, which is just one Yeah. Which just gives you the eight. Yeah.
[00:37:23] Unknown:
Correct. So Oh. 4418 are is a series of coefficients. And I shouldn't say series when I just said it was a bunch of sum when I said that was a sum. Yeah. So it's a bad word. Four eighteen is a nothing more than a like, a string of coefficients.
[00:37:41] Unknown:
The okay. So this is helping me this is helping me internalize this concept of representing other, like, you can basically break everything down into a polynomial. Yeah. Like, that's and and I'm wondering now and and and I I don't wanna think too far ahead because, we're not there yet. I don't think. But, like, when in in my previous research into, like, Schnorr signatures and frost and stuff like that, like, a lot of the stuff has to do with, like, these polynomials and everything like that. Like, you're crafting, like, these polynomial structures.
And and I think that I think there's just, like, an unlock there for, like, understanding how a number can break down into a polynomial. Yeah. It is a polynomial. Yeah. A number is a polynomial of of a specific base, and each representation of the digits are
[00:38:30] Unknown:
fields. Like, there's a field Yeah. Aspect to that. So what you can do with a polynomial is essentially generalize a number. Say, I don't care what number it is, and I don't even care what base it's in. Imagine I can do all of these same operations. If I had some way of knowing, I could do all this math legally on any number, right, regardless of what its coefficients are, what its bases are, or Just because you know it's a polynomial? Out to. Or what do you mean? Like you know yeah. You know it's a polynomial. Right?
You know it's a polynomial. Represent it as an equation or polynomial. Represent it as a x to the n plus b x to the n minus one plus c x to the n minus two dot dot dot dot plus something times x to the first power, you know, plus a constant. Right? Right. You know you can always represent it that way.
[00:39:29] Unknown:
Right? And that that representation has certain properties. Is that am I correct in understanding that?
[00:39:36] Unknown:
Yeah. It it that representation is, like, what we know is what a number is. Right? And when we add numbers together, we've we don't ask why it's legal.
[00:39:47] Unknown:
Mhmm.
[00:39:48] Unknown:
Right? When you take 418 and add 512, you know, you you just add those you add all of you know, you sort the numbers in the right places, meaning you Mhmm. You combine like terms of the base to the power. Right? You combine the b squared term with the b term with the constant term, and then you add the and you're just adding coefficients. You don't say, well, I'm just adding coefficients. So, how am I allowed to do this? Right? You'd never ask that question. It's What I'm trying to think of the relationship of, like, the, So so but so hold on. So, like, being able to define it as a polynomial and being able to actually say it's so I can generalize a number to a polynomial.
Yeah. I can also I can also generalize the coefficients to a ring. And so now I really know that these co this math around these coefficients is gonna be okay and legit. Right?
[00:40:46] Unknown:
Hold on. Say that again. So you can generalize the coefficients
[00:40:49] Unknown:
to a ring. Exactly. So like we just said, the ring of, base two, right, was it was gonna contain zeros and ones only. And the sums and the sums will be in the ring. The multiple the products will be in the ring. They'll all have invert they'll all be invertible. Right?
[00:41:07] Unknown:
Yep.
[00:41:09] Unknown:
We'll we specify it that way. Right? So, like, you don't know that all of this is behind it. You don't know you don't you don't know underneath the iceberg that there is all of this making it okay. Right?
[00:41:22] Unknown:
And and the making it okay is because you can go through and, sort of prove these properties by by listing out all the equations.
[00:41:34] Unknown:
Well, you can basically go back to the axioms. And if you accept the axioms Yeah. Then
[00:41:43] Unknown:
The axiom meaning the definition
[00:41:45] Unknown:
of whatever it is you're you're specifying. Yeah. That's kind of like the the the the endpoint of Right. What you need to prove. Like, you know, you could say, hey, prove that there's such a thing called one. I mean, you you know, I mean, the axioms basically set this floor. Let's say, if you get here, you can you you know, you're good. You have, you know, you have proven this ring exists and that multiplication and addition are present. Right? And it's closed. As I know, you have other properties, you know, that's Like, is base two then, like because,
[00:42:19] Unknown:
you know, one and zero, like, we keep we keep hitting on that as, like, very critical for some of these axioms. Right? Because it's like the identity property of, like, a a multiplicative
[00:42:29] Unknown:
Yeah. And I would call I would look at them differently. So zero and one no, because zero and one, maybe they're just special. Maybe this is why base two is so special. Yes. That's what I was getting at. Like, base two, like, binary is, like, very spec because it Because it literally only contains the identity elements of the of the ring. That's literally what that what base two is. Yes. Exactly
[00:42:51] Unknown:
where I was going. Yeah. I was exactly going there because it's like, okay. There's these two special numbers, zero and one that come up in all these other definitions of all these axioms of these different things within mathematics. And maybe things is an imprecise word you hear, but it's like No. If you boil it all down now to the only group field ring that has just these two numbers, zero and one, like, that seems like it's it's like is there something special without being, like, the underpinning basis
[00:43:26] Unknown:
of everything else? It's probably why computers use that. It's I mean, it is the most powerful and simple
[00:43:35] Unknown:
think about what so Well, I don't know if that's why computers use that. Right? Like, the computers like, when you get down to, like, the circuitry level, it's it's you're you're essentially measuring electricity flow. Right? Like, it's it's a and and and I'm probably gonna get this a little wrong. Right? Areas, like, gone off. It's Exactly. Yes or no, true or false. But even that is more of a, like, you can have some measurements at least in electronics components. Right? Like, some measurements are just true. Like, is there electricity or is there not? Right? Like, just a a a straight up gate of, like Mhmm. Is it charged? Is it not charged? But then there are whole whole methodologies of measuring, like, what comes to mind immediately is something called pulse pulse with PWM, pulse with modulation, I think.
And that is, like, sort of measuring and converting from an analog signal. Right? Because, like, electricity is, like, flowing. Like, it there's there's, like, a a sine wave to it. But converting from that signal to a one or a zero based on stuff. Pulse with modulation. That's what it is. And modulation is just the way of taking an analog signal and converting it into a digital representation of that. Right? That's the modem. Like, the term modem comes from, like, modulation, demodulation. Trying to figure out if we've wiped out at this point. We might have. Yeah. Sure.
[00:44:58] Unknown:
I think we really got something though. Like, I think, like, holy shit. Like, figuring like, actually, this whole idea of a ring with, basically, where you can find a field modulo two containing just the identity elements of, you know, the two operations Yeah. That's gonna be a like, I'm probably not gonna sleep for the next three nights. Thinking about that? Thinking about that. Yeah. Like, that is something that is like a wild explosion.
[00:45:26] Unknown:
And maybe Well, because what are the implications? Right? Because if you if you're if you're using those two numbers within this field ring group, whatever you would I mean, it's all of this. Right? You're limiting your coefficients, and the thing is there's nothing nicer.
[00:45:38] Unknown:
There if you guys have ever done, like, ugly algebra. Right? Mhmm. There's nothing nicer than adding two numbers together and getting zero and then just getting getting to getting to remove it. Just it's gone. Yeah. Yeah. So, like, you know, taking a taking, like, a hundred term what would be, like, a hundred term polynomial and realizing it's only two terms? Because you and which is Zero. Yeah. Which is what you can do, you know, with binomial expansion when you have your coefficients are in finite fields and have character have a non zero characteristic. Again Yeah. Getting on the wave here, but, like, it enables you then to wipe out a lot of those intermediate terms because they just add to zero. So a coefficient of zero means bye bye.
[00:46:26] Unknown:
And this takes me back to one of the the going through the AES cipher and this understanding cryptography PDF that that I've been going through is, like, one of the ways that they're able to take it and represent it in a binary fashion is by using, you know, base two as, like, the polynomial representation of whatever numbers that they're using. Right? Like Yes. Absolutely. You can take it anything can be reduced to a base two polynomial.
[00:46:52] Unknown:
Yes. And And it's so intuitive sometimes to look at things that way.
[00:46:58] Unknown:
It's so intuitive to just Let me let me try to phrase this in like a a so you can take any number and and and factor it is I think is the correct term. Factor it into a polynomial using
[00:47:11] Unknown:
base two, which means just like a one or a zero. True, but it makes that I I don't know. I actually don't know if that statement is true, but it might it very well might be.
[00:47:22] Unknown:
Well, I mean, it's just like, that's how we represent numbers in computers. Right? It's like It's yeah. It seems to make sense, though. Right? Yeah. There's I wonder if there's a limitation there. Like, I I I see what you're saying with, like, I don't know if that's true, like, ad infinitum. Yeah. But I'm I'm confident that's true at least within a certain limitation.
[00:47:45] Unknown:
Like, you have I mean, in a again, not to get maybe to shift gears slightly, but to stay on this topic Yeah. Yeah. Is that, there's an idea of a irreducible polynomial, polynomial that can't be reduced. So for example, I think what most people could relate to is x squared plus one. It it in if with only real numbers, right, say, you can't reduce that. You need complex numbers to reduce x squared plus one. Right? Because
[00:48:17] Unknown:
you'd need x inverse of a square?
[00:48:19] Unknown:
You need a negative one you need x squared to equal negative one. And you yeah. You can't. So what the power of an irreducible polynomial is that it acts like a prime it's like the polynomial generalization of a prime number.
[00:48:36] Unknown:
And in Now Yeah. Hold on a sec. Like, is that if you were if you factor a prime number.
[00:48:47] Unknown:
Can't.
[00:48:49] Unknown:
Because it is just an irreducible polynomial at that point.
[00:48:52] Unknown:
Yeah. Like, prime you could there's no, like, two you know, three is just three. Right? Remember, every number is the product of primes to a power.
[00:49:02] Unknown:
Right? Right. Right. But prime numbers are always odd. Right? And that's because Separate it unless it's two. Well yeah. Correct. Good. Good. Thank you. But just in general, like, you just said x x squared plus one. Right? Like, that plus one, I think, like, there's something to that because everything like, it's an odd number all the time. Right? So it's like squared plus two has no solution either in real numbers either because you would need x squared equal negative two, which,
[00:49:28] Unknown:
you know.
[00:49:32] Unknown:
Now you need yeah. You know. Yeah. I feel like we're wiping out on the wave here.
[00:49:36] Unknown:
Right. It's just the bottom line is I think it's it's I think you can since we talked about polynomials, it's a way as a way to generalize a number. There are polynomials that can't be reduced, which you can generalize to prime you can generalize to primes. And we use them in in cryptography. And I you know, this is like the first thing I noticed in this book of understanding cryptography, right, was that it's not only we don't we don't just pick big numbers as generators, we pick polynomials as, you know, as something. You know?
So there's So the Part of the cryptography and the part of it involves an irreducible involves a polynomial that cannot be reduced as sort of the basis for generating, like, maybe it's for generating, the points in the elliptic curve. I can't keep it fully straight. It's used some there it is it is used in a way I can't full I can't articulate here. I shouldn't have gone here.
[00:50:39] Unknown:
There's is it like, it's like, it's not an equivalence. Right?
[00:50:45] Unknown:
But, like, prime numbers It's used as a modulo. Like, you know how we say, oh, we've find a field modulo 11 and we divide everything by 11. So it's used as actually the modulus, but you don't define the number. You just you just say it's you you say it's a polynomial. Just to find the polynomial. X squared plus one is the modulus and then you can say for any x, right, you can Yeah. It doesn't you know, you don't have to, you know, you don't have to restrict it. You can generalize it to any x. Anyway, you know, let's not get let's not get too dragged into this. Well, in this we're we're we're
[00:51:26] Unknown:
about at the time that we usually, like Yeah. Rip. And so I wonder if this is a good pausing point. Man, wow. That was all that was, like, this was so for for not reading through the cryptography, Brooke, I think I like, there's a lot to take away from here and just understanding, like, one, more clarification to me on, like, what a polynomial is and the pieces of that and why that's important. And then I look forward to your, sleepless nights reporting back on, like, this this significance of, like, the zero and one.
[00:52:04] Unknown:
And I wonder I know you're trying to close it out. Let I I there's let me let me say real quick. Yeah. That, like, I really enjoyed doing this. I don't know if people are I mean, this is now to the people who've hung in for fifty two minutes. Okay. I'm assuming you guys either enjoy it or getting something out of it. Right? I certainly am. To me, this struggle is part of, like, my forty hours a week of math homework. Right? It's pushing me it's pushing me to be to my limits. Right? And it's pushing me to grow, and I'm personally enjoying it. But I'm not just gonna do this because I like to do it. I'm hoping I hope people are getting stuff out. I'm getting comments from people saying they're enjoying it.
I would just say keep that keep communicating with us. We have a page on Nostr, Motivate the Math. Most people know how to contact Gary and I and multiple plat multiple platforms, Nostr. So that's that, guys. K? I enjoy it. I hope I can see Gary has stamina for it. I can't tell if he's enjoying it or not.
[00:53:17] Unknown:
Oh, I'm a masochist. I enjoy every second of it. It it's it's a new the the my brain is doing new things. Right? Like, that's a that's an itch that's hard to scratch.
[00:53:29] Unknown:
I think it's important that we do this. Yeah. You know? Something like we're gonna like, just today, I'm gonna walk away with this discovery of the zero one ring and say, man, if I hadn't committed to this podcast, we I wouldn't have those words wouldn't have uttered. We wouldn't have connected on that.
[00:53:47] Unknown:
Right.
[00:53:48] Unknown:
Alright. Wanna read the boosts?
[00:53:50] Unknown:
Yeah. Let's read the boost. Topboost, Dan. If you haven't checked out Dan's cartoons, I I highly recommend checking out bread and toast. Like, they're just they're just, like, wonderful, wholesome little cartoons. But when Dan gave me Zapboost. Dan gave me my logo for my fish podcast,
[00:54:09] Unknown:
and he'd you know, this is was, like, two years in the making.
[00:54:12] Unknown:
He's he's just the man. It means a lot to me that Dan's in our in our, like, in this community here too. Thank you, Fundamentals and Average Gary, for doing this work. If you keep doing it, then I'll keep listening even if most of the stuff goes right over my head the first couple times hearing it. Love it. Stamina, just like you said. Next, God's death, 537 sats boosted. Thank you, gentlemen. Can you have slides in the thumbnail on fountain showing equations and such as you go through the episode? Visuals would help me understand a great deal. More great work.
Like, be on the lookout for for more, like, visual things. I'm working on it. Yeah. I I really am working on a way
[00:54:49] Unknown:
to supplement this with visual aids. So I would say be patient. You know, the it's after I, like, when when it's when Gary hit stop, I have this, like, I go into my head and be and start thinking to myself, man, I need to figure out how to do video. And, I actually did an I did a full video last week. Okay. I didn't post it. I just just to convince myself that I could do it, I haven't shared it with you yet either because for some reason I can't we can't I can't share it with you on the platforms you're on for some reason. We'll get there. We'll get there. We'll get there. But anyway, so it is in the works. Okay? It's in the works. I'm gonna figure out how to do this. We might have it might be a YouTube site where Yeah.
I start posting, like, exercise videos. You know? Not not not I just said not with I want I want you in, like, a Richard Simmons sweatband, right, for the exercise. Yeah. With a thigh master. Yeah. And a pencil.
[00:55:49] Unknown:
Last boots anonymous, a hundred sats. Insert boost comment here. Thank you, whoever you are. And, fundamentals, thank you again for for blowing my mind on this this mask thing. Good shit. Good stuff. Thanks.
Because it Yeah. This is, like, not an easy topic to talk about. And so I think it's prudent.
[00:00:08] Unknown:
If it if it was, right, there'd be a lot better people than you and I doing it professionally.
[00:00:14] Unknown:
Right? Well, hold on. Wait. You're saying that if it was easy, there would be better people doing what we're doing. There would be, like, if it was easy, right Yeah. It would be scaled.
[00:00:24] Unknown:
Like, the influencer class would already figure out how to productionalize these conversations. Right? I mean, you see a lot of math channels. You see, you know, you'd see it. There's a few cryptography channels that I found now. You do see them. They try. Right? But it's hard. And the hardest thing is it's it may be easy to teach it may be easy to make videos of a math subject. Right? Right. But what's difficult to get across I I don't I'm what I'm more interested in doing here, right, is in keeping alive this notion that we should not be just trusting math. Right? And how do you do that is by giving examples as to why we should be, why should we worry that we've trusted, like, we have many of us here have trusted our life savings on our protocol, right, that has a lot of reliances on things that we might want to make sure in our own minds we're very comfortable with.
[00:01:29] Unknown:
Well, and it's not even it's not even just, you know, Bitcoiners or or or or anything like that. Right? Like, the entire modern infrastructure of the web is, like, fund like, the the fun one of the fundamental pillars is this mathematics called cryptography. And then even further than that, though, I think it's math is is is an explanation or is is the closest approximation of using human language. Right? Because math is essentially just a language of explaining things
[00:02:03] Unknown:
to, like, how the world works. Right? Because it's like you're measuring all the any science has, like, numbers in it. Like, any science has numbers. And then then there's other words to science, but there's always numbers. And as we know we now know after three or four episodes, math isn't necessarily just numbers. Right? Right. Like, we had we did two episodes about, like, what a group is and without really mentioning any numbers except for zero and one maybe. Right? And you can go through the entire textbook, a lot of math textbooks, not see any numbers. Right?
So, like, it's what's going on. Right? It's not just number crunching. Right?
[00:02:38] Unknown:
Yeah. It's like pattern pattern recognition. Right? Like, humans were great at, like, pattern recognition and and deducing patterns in relationships.
[00:02:46] Unknown:
Some of us. Some of us. Quite good at it. Quite good at it. Right. Right? So, you know, so what you're getting at here is that is the second thing I think is really important. It's probably more important, but you don't get there without, I think, the fear. It's like the fear of yielding your power is, I think, the motive the real motivator to then this notion that math is access to power. You know? Like, so once you get over, okay, I don't want to yield my power to somebody else, that's really important. Mhmm. Then you say, well, okay, what about this source of power? Can I claim it? Can I claim this power for myself, and what might that make possible for myself?
Right?
[00:03:30] Unknown:
Well, in the way you claim that, I think, is is the internalization of these patterns in relationships. Right? Because our minds like, right now, you just, like, scan across the room, and your brain instantly conceptualizes what it sees into, like, okay, computer monitor, water bottle. Right? And, like, you don't even have to think about it. It's just Wow. You know the pattern
[00:03:52] Unknown:
of What? The water bottle when you see it. And so Yeah. Go ahead. This is gonna get this is gonna get good because you don't see a water bottle. You don't you what you see, right, is you see a bunch of pixels and then you probably see you see a a sparse Well, hold on. Meaning. Hold on. You see something very hold on. You see you see very, limited amount of images that a very small percentage of the thing you think you're seeing. And then your brain does the rest. Your your brain actually draws, interpolates. It tries to figure out what's in between those spaces.
Right? Cool. Yeah. Yeah. And so and, Gary, you know what better than I do. Right? Because of your experience and your experience, you know, military. We don't see for, truth. We see for survival. Yeah. So in other words, the way that our brains work in terms in doing that interpolation is it tells us sort of the most scary thing it could possibly be. It doesn't actually tell us what the thing is. It's telling us what it could be. Right? What threats could it be? And so, like, yeah, like, it's like if you, like how people know that a sniper is watching them from five miles away somehow. Right? And they've they've done these studies and they know, like, that they're in a site. Yeah. The animals not somehow know that they're in the sites of a sniper.
[00:05:21] Unknown:
Right? And and that that part of your like, it's, it's interesting. I was at a a a school informational thing. Right? Like, a new private school, that we're looking at, and and the woman presenting was talking about neuroplasticity. Right? She's like, there's, like, a definitive, you know, quote, unquote science now that is, like, sort of, you can rewire your brain. Right? And this is even, like, referenced in the bible and in some reference of, like, rewiring your brain. But it's, like, also, people have proven, sort of as much as you can prove anything that, like, the channels in your brain can be sort of adapted and and perform. And this is I think this is applicable just like anything. Right? Like, it's like going through the motions and getting the repetitions of anything will make you better at it.
[00:06:07] Unknown:
You can't rewire your brain, though. You can continue to wire. You can build on basically the chain that you have. Right? That's true. So like most people think I have my DNA and I this is it. I I stuck with what I got. Right? Maybe I can learn to ride a bike and I can pick up a couple new skills along the way but no you actually can you know so rewire to me sounds like you can redo it and maybe that's you'd have to be able to erase it probably to be able to redo it. But, you can do a lot. You can build quite a bit. Right? You can like, what it takes is things that are hard.
You know? And, it almost doesn't matter what it is in a certain way. Like, if you just committed to learning how to write your name with your offhand and you did that. You could do it over and over and over again until you get it right. If you spent like like ten minutes a day for thirty days with your offhand trying to write your name over and over again. It's not that it's not about getting it right. It's about that's how you build that's how you build new like neurons in your brain to build new skills. Right? You don't like actually give a shit if you write if you figure out how to even write your own name legibly.
You're going through it. It's kinda like, why football players run through tires. Right? They don't you don't run through tire in a game. Right. But somehow it's on every you know, every practice field's got the tires. Right? Right. Right. For the foot just for the footwork aspect of it to move around that thing. It's a yeah. It's a it's a training. It's a way to train. Right? It'd be funny if they did it in the game or they even did anything remotely like it in the game. But, like So, I mean, is this why you are
[00:07:52] Unknown:
constantly, like, going through equations? It's yeah. It's true. Me your notes. Like, you show me your notes of you're just doing math over and over and over again in different numbers, but the same calculations.
[00:08:05] Unknown:
It's one of the reasons. Yes. It's building calluses, it's building skill. It's not different from a musician learning scales or Right. Rip to just continuing. You know, there's a saying called practice makes perfect, but the true saying is practice makes permanent. So it's not that you're because you could practice something the wrong way and then you'll make that permanent. It's fine. Right? Right. Right. Yeah. Which is why like in what we're trying to do, if you're trying to create a fitness like, we're I'm trying to say that mathematics is a road to fitness.
And just like physical training is a road to fitness, but there's a billion podcasts and a billion books about how that is a road to fitness. That doesn't require you and I. The there are very few, there are very few people telling, talking to the people we care about Yep. Loudly that this matter. Very few people allocating time and energy to this conversation. Yep. And so that's where you and I have basically said, you know, you know, we want we want to make this make sure this conversation exists and we're gonna do our best. And it's hard. Like, for us, it's hard. And we're building new neurons trying to do it. It's very funny. Right? Yeah. You guys can kinda tell after a few episodes, it's not the easiest thing in the world. It's almost fun to watch us try to fail.
[00:09:33] Unknown:
What I think the so two points on just a recent topic was, like, one, the practice makes permanent thing. One of the things in the military is, like, because I I didn't grow up shooting, so I was actually a better shooter when they trained me because I had no bad habits to undo. It was like it was like, you know That's that's it's yeah. Tabla Buanca. You were an a ADIQ Bitcoiner, basically, ready to learn. Right? Exactly. And then the other thing with that was, one of this guy, he was, like, known in, like, he I I took, like, a strong man course from him. Right? Like, picking up stones and stuff. But one of the things that really stuck with me from this course was he said fitness is just a lifetime accumulation of reps.
He's like, it doesn't matter, like, heavy, speed, or or whatever, but, like, it depends on what you're going for. But it's like, do you you just have to do the reps over and over again. So I think that's probably
[00:10:23] Unknown:
I probably need to do more reps, personally. But that takes a lot of and it but that takes stamina. That takes skill. It takes a lot of skill to do one rep of an exercise correctly. In fact, I would almost bet that there's not a single lift that, like, my trainer wouldn't be able to show you how hard it could be with half the weight.
[00:10:45] Unknown:
Right. Right. Right. You know what I mean? And I'm not applying that in the math context, though. It's like, what what is what is, like okay. So there's, like, the couch to five k concept. Right? Where it's like, okay. You get up off your ass and you go walk first, and then you a mile maybe or even half mile or something like that. Like, what Yep. I wonder now, like, what is the Depends where you are. I so here's it's funny, man. Because I'm I have two daughters. One is a math major,
[00:11:11] Unknown:
and, you know, the other one's in tenth grade, and I just watched the progression.
[00:11:15] Unknown:
Right.
[00:11:16] Unknown:
My younger one is, like, rebellious. She's like, I just want you to know I don't care about math. You know? Yeah. Yeah. And, you know, and, like, last year when she's in ninth grade, she was getting, like, 15 on tests and stuff like that. And little by little That's good. Little by little, she starts caring and she starts working with me. And she still says she hates it, but and she's very good at it. Mhmm. And it's it's so it's just a matter of like wherever you are, right, you want to think about the next level. Right?
So if you are a complete, you've never seen you've never seen a number or even a logical concept. Right? You've never gone through a true false question or anything. You've never seen a question that challenged reason or anything. Right? Okay. Well, there's a there's a way there's a pathway for that. Right? You probably start looking at basic arithmetic and things like that. Right? Boolean logic. Yeah. I would say most people on average, if I ask what was the last thing you were really you remember being good at, right, they would all probably say seventh grade algebra or geometry or something like that. There was a moment where they actually thought they had it and then a week later it was over forever. You know, they got their their their public school curriculum just, you know, made sure they would never feel that way ever again.
You know? So, like, but that's so on average, I'm guessing that's where a lot of people are. Right? A lot of people in the Bitcoin space are, people with engineering degrees but have never looked at number theory or abstract algebra or anything like that. Now, these guys think they're math people. But they could benefit so much. Right? Yeah. In a lot of ways, I enjoy talking to total noobs more than people who think they know everything, you know, but yet are completely blunt you know, don't don't, you know, don't know this.
[00:13:26] Unknown:
That that's interesting. That makes me think of, like, like, you know, the the entry point of yeah. You're just a Bitcoiner. Right? You're just, like, whatever ADIQ Club out there that wants to, like, learn math and everything. It's like, go try to figure out why 21,000,000. Like, not why, but, like, how. Like, how does that actually work? And there's a great, post. I'll put it in the show notes, but it was by, Jameson Lop on, like, the supply limit and stuff. And he goes into the actual code and everything like that. And even that, like, simple concept of and I say simple because of the concept of 21,000,000, which first of all, it's not 21,000,000. It's like just a little bit under 21,000,000.
Yeah. There's like a reason behind that.
[00:14:09] Unknown:
And e even that was like I know some ears percolating already about there's some ears percolating getting excited to, like, one day we made to have this discussion of what's in between the supply limit and 21,000,000.
[00:14:23] Unknown:
What's in oh, what's in between? Go ahead. What's the remainder? Anyway, that's another question. But yeah. So go on. Sorry. I didn't get it. No. No. It it's just the even when I looked at it. Right? Like, as as in, you know, being in software and everything like that, I was like, oh, wait. I need to, like, give me let me, like, pause and, like, think about this for a second because it was not it's not just, like, 21,000,000. It's, like, written into the code somewhere. Right? It's, like, this, like, bit shift operation thing that happens, and it's based on, like, number halvings and all these other, metrics. It's based on blocks, essentially.
Like, the number of blocks. And, again, you have to, like, do the math to, like, calculate it out. It's like, okay. What actually happens? Right? So after video. For 210,000 blocks. Like, what actually happens here?
[00:15:11] Unknown:
Yeah. I mean, I I would sum it up as a it's a geometric series and it is kind of a geometric series. Okay. It's as simple as that, but it's it's still really a good it's a great exercise to go through, you know. If you don't have those words in your head, which you wouldn't unless you, you know, were trained to do math as a sort of a, you know, public, you know, government employee or whatever. Right? I mean, that's like the slave math they teach you, you know, learn what a gov on a geometric series is so you can do probability and, read worthless PhD papers and stuff like that. But, you know, it is still a powerful concept. I worked in annuities and every annuity is nothing more than a geometric series and it's a, it's basically just a sum. A series is a sum of things that, you know, sum of certain special things. Right? So in a series Okay. That makes sense.
Right, it's, you know, you're just adding, you're adding a block that is that has a factor of a of a number to a power. Right? So the first you know, 50 to 25 is one half. Yeah. Then you go then you then to 12 and a half. It's one fourth. Right? Yep. That's one half squared. So you have 50. You're almost gonna, like, factor out the 50 and then what you have is this one plus one half plus one half squared plus one half cubed. This is how you add up all. This is how you and then, you know, it's not 50. It's 210,000, actually. Right? It's 50 times Yeah. Times ten minutes, times twenty four hours, times six of them per hour. You know, you do that. It all comes looks like, oh my god. It comes out to 210,000 times this geometric series. Right?
And the end of this geometric series, right, at the limit and continuous, you know, like, that the limit is that turns out to be 20. Sorry. It turns out to be 10. Right? Or 10,000,000 in this case. So if if 210,000 times this number turns to be 21,000,000, which is I'm just trying to do the math in my head. I think it's a hundred.
[00:17:30] Unknown:
Yeah. I don't 210,000
[00:17:33] Unknown:
that much. Yeah. Anyway, but it's a two inch 10,000 blocks. So But there's some block time. Yeah. That's, like, a block time we're talking about. So it's, like, yeah, it actually just rolls up every 210,000 blocks. We do a we we we have we have the sorry. And I forgot about the actual reward, so times 50. So, you know, 200 times 50 then times one. So one plus one half plus one fourth plus one eighth plus one sixteenth blah blah blah blah blah all the way out to however many havings there are. What are there? 32 havings or something like that?
[00:18:07] Unknown:
Something like that. Let me see. I pulled up the Lop article here. By the way, if you're looking if you're if we don't have it in the show notes, which we should, it's how is the 21,000,000 Bitcoin cap defined and enforced. It was right on 01/12/2022 by Jameson Lop.
[00:18:21] Unknown:
Gary, is there a way to pause? I I actually have a water guy at my at my house.
[00:18:28] Unknown:
Alright. We're back rolling again. We're back. Okay. So You, listener, dear listener, would not know the, lengthy duration of which I sat here reading I didn't throw up. About the block. I didn't I didn't throw up. We're all good.
[00:18:43] Unknown:
That's inside. That's an inside reference.
[00:18:46] Unknown:
Back back to the so the block, though. Right? And I actually have the the the equation in in LOPS. We stopped so Gary could read the entire article. Yes. That is why. But there's this entire, like, this entire equation that that pops up. Right? And it's basically it's
[00:19:06] Unknown:
go ahead. What I was gonna say is I know Gary and I are looking at it, but do you guys know that it's the sum it's the big sum that looks like an e, you know, Greek letter sigma. Yep. From I goes from zero to 32 and it's got the 210,000 and then it has 50 divided by two to the the I power. Right? Yep. And what I'm saying is you can pull the 210,000 out. Right? You can even pull the 50 out and then what you're left with is this sum of one over two I which is a geometric series. And that's that that has a standard solution of one minus one divided by one half which is it's fucking so stupid. I I think the answer is two. I just have to do I I I'm so this is like the dumbest thing to be able to do in my head.
One over I think I'm pretty sure the answer is two. So, yeah, you'd multiply and and it is. So if that if that answer is two, and two times 50 is a hundred, and a hundred times 210,000 is 21,000,000. But that's two at the at, like, the limit. But how it actually happens when you hit the thirty second block is that you're not gonna complete all 21 sats.
[00:20:18] Unknown:
Right. And and and it's it's due to this thing called a right shift, which is, like, if you conceptualize we'll do a simple example of what, like, if you think about the number eight in binary is one zero zero zero. Right? Yeah. Because you have the ones place is zero, twos place is zero, the fours place is zero, and then the eight place is one. And and it the bit shift right is taking all those bits and just like if you're to, like, just shift them to the right. Right? So the one would slide into the four place, and then all the zeros slide to the right as well. So you end up with having four. And so what happens, like, in the actual code, this this subsidy number, right, because the the number that of new issued Bitcoin is actually called the subsidy, technically speaking.
The shift happens and how that's calculated is you're shifting by the number of halvings that occurred. And the number of halvings that occur is the block height divided by this halving interval, which is just so happens to be 210,000. Right? So it's like you're shifting it. So if you if you shift right one time if you start with eight, you shift right one time, it's dividing by two. And every time you shift that right, it's another division by two.
[00:21:31] Unknown:
Yeah. I actually think I can I actually might this is at the risk of another Icarus moment here? I think I can actually explain why it doesn't. So when you have a sum of a series. Right? Yeah. You have let's say you're, you know, you're you have the the sum of nine terms. And this is a, like, it's a matter of is your starting point time is your starting point zero or is your starting point one? And if your starting point is zero, then what happens is you finish before really you get to the end. Whereas, this is gonna confuse people, but like if your starting point is actually at one, then you finish at the actual end of this thing and, you end up with one more you would basically end up with one more term in your sum.
So if you went from zero to the absolute end, I think you end up with the you end up with this extra term in your sum that completes the you know, that that you end up with that last term. So you have 32 halvings. I'm guessing I mean, I would like to I guess somebody I'll challenge somebody. Does does adding a thirty third term to the sum complete? You know, because what basically, when I what I came up with, that factor of two that multiplied it by 210,000 times 50, that's based on infinite sum. So that's that was like if you carried it out to infinity, you know it's just one of those properties.
So knowing something's a geometric series remember how we said knowing something's a group is important because it allows you to know a lot of things? So knowing, being able to recognize something as a geometric series allows me to look at that sum over one of two I. And if I took it to infinity, that would evaluate to two, meaning one plus one half plus one fourth, plus one eighth, plus one sixteenth, all the way out to infinity, I know that I know the answer is two. Okay. So is Because I know it's a geometric series and I know how that, you know, then I've internalized all that. Is there an implication to putting a limitation on a geometric series? Right? Like the at least We don't live in infinity. That's the thing. So there are no there's never going to be infinity blocks. Right? Right. I mean, there are gonna be infinity blocks. There's just not gonna be infinity reward ish reward blocks. Blocks reward. Right? So that's just what Satoshi decided is, you know, what there's no point in going more than one set.
Right? Yeah. Not I'm sure he didn't call it that. I'm sure there's no point in going more than one over one hundred millionth of a reward.
[00:24:17] Unknown:
I'm guessing he just didn't wanna I'm get or he could've it could've literally just been the precision on his computer that had him decide to stop it there. That's exactly where that that's I was getting into. Right? It's because there's even, like, in the in the the code block that you see, one of the checks,
[00:24:34] Unknown:
is It wasn't like the messiah or anything that could just, like, conjured some perfect thing. There's a lot of luck involved and that was yeah. But but the having piece of it is if you if it's if it's greater than 64.
[00:24:46] Unknown:
Right? Meaning, like, that that that bit is, like, way, to the bit is, like, you you can't shift it anymore without starting over from the get go. Right? It does it doesn't check for that. So it Yeah. Yeah. Is this limitation. And in in general, like, in in computer science and stuff, like, a lot of things are built around, like,
[00:25:07] Unknown:
four, eight, like, powers of two, essentially. Yeah. In the base 10 system, the bit shift there is the I'd same analog would just be multiplying or be divided by 10. So in a bit shift in a binary number, moving one to the left is just is basically multiplying or dividing by two. You know? And we talked last week about how yeah. How a number is a polynomial of its base. Right? And
[00:25:31] Unknown:
Well, and and to put a concrete example, 400 to 40 is the same concept. Right? If you were if you remained three digits, 400 to forty forty would be divided by 10 and you're just shifting that four to the right one. Exactly. Right? And if it's a left shift, it's it's a multiplication of what the base that oh, man. So that would hold true for any sort of, like, base system. Right? So, like, because it's base two binary, a bit shift left or right is multiplying or dividing by two because the base is two. That's right. So in hexadecimal, if you if you shift if you do a shift left or right, you're multiply or dividing by 16. Exactly. Oh, interesting.
I I think that has a significant implication for, like, writing the actual code that does this math and, like, the shortcuts thereof. It probably does. It's probably the reason we have a hundred thousand,
[00:26:25] Unknown:
units to a Bitcoin. It's probably. I don't know if that's true. I'm just saying that. Million. What's that?
[00:26:32] Unknown:
About a million units?
[00:26:33] Unknown:
Two one yeah. A hundred hundred million. Right? Yeah. Yeah. It's like, yeah, dude, we get on mic and with our brain wipes out. Don't know basic facts anymore. Right? It happens to me all the time. Yeah. That's interesting. Okay. Like, I couldn't divide by one half just now. I forgot. I was like, oh, so because I, you know, the getting better at tuning out, caring if I'm super wrong about things. But, like, yeah, it's funny how oh, really? Is it a hundred million? Really? Yeah.
[00:27:05] Unknown:
It's, it's it's the journey. Right? And it's it's trying to understand the grokking
[00:27:09] Unknown:
of it all. People don't real so that's back to doing hard things and building, like, building your skill and building sort of your neurons. You don't understand what like, you think you we think we know things and then a little bit extra pressure gets put on us and we, like, all of a sudden become low IQ versions of ourselves. Right? Like, and and that's and again, it's just like how do you build resilience to that. Right? You might think you know the math. You might think you understand the math. Try getting on a podcast and explain it when you have peep you know your people are gonna tell you and set up with color you fucking moron when you get it wrong. Right? Yeah. It's like that little bit of pressure really then puts it it puts to the test. That's why I it's one of the reasons I wanted to do this. It's one of the reasons it's important to teach. I actually think it's really I think being an expert in something or even being competent in something requires teaching
[00:28:08] Unknown:
because you wanna do That's like the rhetoric level of understanding. Right? It's like knowledge, understanding, and then rhetoric.
[00:28:16] Unknown:
Yeah. I mean, playing music is no different. Right? So you can play as much as you want. You're you know, I can play as much as I want in my basement. But, like, until you go out and play with other people and you'd be judged, you're under the pressure of being judged by others. Right? Yeah. It is, you know, it's a different story. And so it's another reason we motivate we say we motivate the math. Right? We're hopefully we're motivating people to actually pick up a pencil and do a little work on it too.
[00:28:44] Unknown:
Like, I would check What and what what work should we motivate them to do today? Right? Because we've I think we've talked a lot of, like, conceptual stuff at the beginning, but, you know, and I I even confess that, like, I I wasn't able to get to a lot of reading, in between now and the last episode. I think the geometric series
[00:29:00] Unknown:
is like, go on Wikipedia. Look it up.
[00:29:06] Unknown:
Geometric series.
[00:29:07] Unknown:
Yeah. Check it out. Look at, like, an arithmetic series and a geometric series. And check out the formula for, you know, check out the formula for the block having, right, for the block issuance. I do see it. Yeah. And how it gets to, yeah. There's a nice little picture there. I mean, there's really cool visuals. So that's not a bad definitely not a bad way to go here. Right? Yep.
[00:29:40] Unknown:
Interesting. And so so, yeah, the block subsidy is a is a geometric series. It's it's just
[00:29:49] Unknown:
Yeah. You wanna actually go you wanna go even you wanna take even more risk here. So check this out. Sure. All so remember we said it's a, it's the number. Any number in it is a is like a polynomial because you have Right. You you have the base. So you said the number eight is like really one zero zero zero. Yeah. Why is that? Because it's, it's one times two to the third plus one plus zero times two squared plus zero times two to the first. Right? Plus zero. So the remember last time we'd said that the ones and the zeros are coefficients? Yes. Right? So the ones and the zeros are coefficients, and then the two is the base that's getting raised to an exponent. Right? Yep. Now the coefficients are in a field.
And in a they're actually in the field of finding field mod modulo two in binary. All the coefficients are either zero or one.
[00:30:50] Unknown:
Right. Right. And and the same applies to a base 10 because zero is not Zero is or nine. Yeah. That's right. Is is whatever number modulo
[00:30:59] Unknown:
10. So that's one of the reasons we we we talk about fields. Oh, wow. Okay. Okay. It's because, you know, when you do algebra and you find like terms and then add and multiply the coefficients, you want to make, you know, you knowing they're in a field, you know that you know that multiplying two numbers is going to give you a number in your field. Right? You know that. And, you you know, you don't have to think about it.
[00:31:27] Unknown:
Right.
[00:31:30] Unknown:
If you multiply one and one in a base two field, you are going to get a number and you're gonna get one. If If you multiply one by zero, you're gonna get zero. Right? If you multiply zero by zero, you're gonna get zero. There's no outcome where you multiply one of those coefficients or even if you add them. Right? Because if you add one and one, you get zero.
[00:31:51] Unknown:
So, like So just but so I guess, is it does that like, base two is a field,
[00:31:57] Unknown:
but base Well, then that's not necessarily field. It's not that base two is a field. That there is a field there is a field that's, like, defined as basically, the field of numbers, maybe integers modulo two. And consist of the numbers that would consist in the numbers zero and one, and it would contain all of those properties of the operations, addition with commutivity around addition. Right? Mhmm. Identity inverse. And multiplication, definitely with commutivity, with an inverse. Right? With an identity and an inverse. So it it so happens.
Right? Knowing it's a knowing that it's a field basically it essentially means you could turn your brain off and do all the math you want without worrying whether it's legal. I mean, that's essentially what it allows you to do. Right? Which most peep which is the starting point for most people, and that's how they teach it to people for when they you start in third grade. You don't say, hey. By the way, there's a whole bunch of rules for whether or not this stuff is legal. They're like, we can you know, dividing by zero, we can't get around. So we're gonna make sure well, that's and that exception is enough. That's an easy rule to It's an easy rule, but it's also enough to make people wanna a lot of people just wanna puke when they do math. They're like, ugh, really? I gotta worry about this fucking zero thing? It's bullshit. You know? And, like but, like, we're not we're not you know, we try not to complicate it too much for I feel like at that level trying to
[00:33:36] Unknown:
I was trying to think of this field concept in, like, base 10. Right? Because I think it's very easy to, like, conceptualize
[00:33:43] Unknown:
base two. So it's the same deal. Right? So when you have, and if you if you think about how you were taught to do multiplication
[00:33:51] Unknown:
Yeah.
[00:33:53] Unknown:
Or subtraction where you borrowed from a 10 and you took, you know, you you might borrow from a tenths place. Right? It's this is this is all because of, you can do this. It's all modulo 10. Okay? You're dealing with this digit zero through nine, and it's all in a modulo 10. So nine plus nine, when you do when you actually do addition, you write down the number eight.
[00:34:19] Unknown:
Right. And then you carry it to the That's right.
[00:34:23] Unknown:
Oh. You carry that 10 into the 10 into the next tens place. It's a bit shift. You carry it into the shift you like into the shifted it's not bit because bit means base two. But whatever the base 10 equivalent of a bit is.
[00:34:37] Unknown:
Because it it it's Maybe yeah. Maybe they do call it. Yeah. Because It's not a computer science guy. Is not within that field, so you have to represent that a different way. And so you move it to a whole new place.
[00:34:47] Unknown:
It's all modular 10. Place. Everything's modular 10. 10 place is now a new
[00:34:52] Unknown:
representation of that field, but it is with a different, it is a different I don't think I'm gonna use the right word, but it's like a different piece of the polynomial that is defining it. Right?
[00:35:05] Unknown:
It's yeah. You move it where it belongs. Right. You're like, no. We're this is not part of addition. This is now part of, I guess, you know, you'd say maybe multiplication. Right? It's part of it's this is now part of multiplication where we have to go one to the left. Right? We're gonna hold this eight because there's no there's no 18 to write down here. Right? It doesn't make sense. See, the polynomial that represents the field there. Yeah. Well, the so in other words, because you have this polynomial representing the number, and the polynomial is coefficients times your base. Okay?
[00:35:43] Unknown:
So Yeah. Ten ten times 10 plus eight times 10 to the zero, which is one.
[00:35:50] Unknown:
Right? And so you yeah. And so when you say carry the one, you're just basically saying, yeah. Because I have, you know, I have an extra 10 that needs to go somewhere. Add a new non zero coefficient. But that's part of my power. No. So what I'm doing is I'm recognizing that it's in the power, not in the coefficient. The coefficient is the field is the member of the field. And this is, you know, very dangerous. We're in dangerous ground right now in this conversation. But I think it's it's okay to look at a polynomial. It's it's it's a bunch of coefficients times a bunch of one number to a power.
[00:36:27] Unknown:
Right? So yeah. Okay. So so let's let's let's see. So, like, the the the number, the quote, unquote number 418 Yes. Is literally saying Sum of three numbers. It's the it is the sum of three numbers, and those three numbers are each, if you represent it as a polynomial Yep. You are representing it as, the coefficient being the numbers that that we conceptualize generally as numbers, like the four What we call the digits. What we would call the Great. Yep. The the the four, the one, and the eight, but then it the four is actually the value of four times 10 to the second Yes.
Which is a hundred conceptually. Conceptually. Yes. And then one times 10 to the first, which is just 10. Yep. And then eight times 10 to the zero, which is just one Yeah. Which just gives you the eight. Yeah.
[00:37:23] Unknown:
Correct. So Oh. 4418 are is a series of coefficients. And I shouldn't say series when I just said it was a bunch of sum when I said that was a sum. Yeah. So it's a bad word. Four eighteen is a nothing more than a like, a string of coefficients.
[00:37:41] Unknown:
The okay. So this is helping me this is helping me internalize this concept of representing other, like, you can basically break everything down into a polynomial. Yeah. Like, that's and and I'm wondering now and and and I I don't wanna think too far ahead because, we're not there yet. I don't think. But, like, when in in my previous research into, like, Schnorr signatures and frost and stuff like that, like, a lot of the stuff has to do with, like, these polynomials and everything like that. Like, you're crafting, like, these polynomial structures.
And and I think that I think there's just, like, an unlock there for, like, understanding how a number can break down into a polynomial. Yeah. It is a polynomial. Yeah. A number is a polynomial of of a specific base, and each representation of the digits are
[00:38:30] Unknown:
fields. Like, there's a field Yeah. Aspect to that. So what you can do with a polynomial is essentially generalize a number. Say, I don't care what number it is, and I don't even care what base it's in. Imagine I can do all of these same operations. If I had some way of knowing, I could do all this math legally on any number, right, regardless of what its coefficients are, what its bases are, or Just because you know it's a polynomial? Out to. Or what do you mean? Like you know yeah. You know it's a polynomial. Right?
You know it's a polynomial. Represent it as an equation or polynomial. Represent it as a x to the n plus b x to the n minus one plus c x to the n minus two dot dot dot dot plus something times x to the first power, you know, plus a constant. Right? Right. You know you can always represent it that way.
[00:39:29] Unknown:
Right? And that that representation has certain properties. Is that am I correct in understanding that?
[00:39:36] Unknown:
Yeah. It it that representation is, like, what we know is what a number is. Right? And when we add numbers together, we've we don't ask why it's legal.
[00:39:47] Unknown:
Mhmm.
[00:39:48] Unknown:
Right? When you take 418 and add 512, you know, you you just add those you add all of you know, you sort the numbers in the right places, meaning you Mhmm. You combine like terms of the base to the power. Right? You combine the b squared term with the b term with the constant term, and then you add the and you're just adding coefficients. You don't say, well, I'm just adding coefficients. So, how am I allowed to do this? Right? You'd never ask that question. It's What I'm trying to think of the relationship of, like, the, So so but so hold on. So, like, being able to define it as a polynomial and being able to actually say it's so I can generalize a number to a polynomial.
Yeah. I can also I can also generalize the coefficients to a ring. And so now I really know that these co this math around these coefficients is gonna be okay and legit. Right?
[00:40:46] Unknown:
Hold on. Say that again. So you can generalize the coefficients
[00:40:49] Unknown:
to a ring. Exactly. So like we just said, the ring of, base two, right, was it was gonna contain zeros and ones only. And the sums and the sums will be in the ring. The multiple the products will be in the ring. They'll all have invert they'll all be invertible. Right?
[00:41:07] Unknown:
Yep.
[00:41:09] Unknown:
We'll we specify it that way. Right? So, like, you don't know that all of this is behind it. You don't know you don't you don't know underneath the iceberg that there is all of this making it okay. Right?
[00:41:22] Unknown:
And and the making it okay is because you can go through and, sort of prove these properties by by listing out all the equations.
[00:41:34] Unknown:
Well, you can basically go back to the axioms. And if you accept the axioms Yeah. Then
[00:41:43] Unknown:
The axiom meaning the definition
[00:41:45] Unknown:
of whatever it is you're you're specifying. Yeah. That's kind of like the the the the endpoint of Right. What you need to prove. Like, you know, you could say, hey, prove that there's such a thing called one. I mean, you you know, I mean, the axioms basically set this floor. Let's say, if you get here, you can you you know, you're good. You have, you know, you have proven this ring exists and that multiplication and addition are present. Right? And it's closed. As I know, you have other properties, you know, that's Like, is base two then, like because,
[00:42:19] Unknown:
you know, one and zero, like, we keep we keep hitting on that as, like, very critical for some of these axioms. Right? Because it's like the identity property of, like, a a multiplicative
[00:42:29] Unknown:
Yeah. And I would call I would look at them differently. So zero and one no, because zero and one, maybe they're just special. Maybe this is why base two is so special. Yes. That's what I was getting at. Like, base two, like, binary is, like, very spec because it Because it literally only contains the identity elements of the of the ring. That's literally what that what base two is. Yes. Exactly
[00:42:51] Unknown:
where I was going. Yeah. I was exactly going there because it's like, okay. There's these two special numbers, zero and one that come up in all these other definitions of all these axioms of these different things within mathematics. And maybe things is an imprecise word you hear, but it's like No. If you boil it all down now to the only group field ring that has just these two numbers, zero and one, like, that seems like it's it's like is there something special without being, like, the underpinning basis
[00:43:26] Unknown:
of everything else? It's probably why computers use that. It's I mean, it is the most powerful and simple
[00:43:35] Unknown:
think about what so Well, I don't know if that's why computers use that. Right? Like, the computers like, when you get down to, like, the circuitry level, it's it's you're you're essentially measuring electricity flow. Right? Like, it's it's a and and and I'm probably gonna get this a little wrong. Right? Areas, like, gone off. It's Exactly. Yes or no, true or false. But even that is more of a, like, you can have some measurements at least in electronics components. Right? Like, some measurements are just true. Like, is there electricity or is there not? Right? Like, just a a a straight up gate of, like Mhmm. Is it charged? Is it not charged? But then there are whole whole methodologies of measuring, like, what comes to mind immediately is something called pulse pulse with PWM, pulse with modulation, I think.
And that is, like, sort of measuring and converting from an analog signal. Right? Because, like, electricity is, like, flowing. Like, it there's there's, like, a a sine wave to it. But converting from that signal to a one or a zero based on stuff. Pulse with modulation. That's what it is. And modulation is just the way of taking an analog signal and converting it into a digital representation of that. Right? That's the modem. Like, the term modem comes from, like, modulation, demodulation. Trying to figure out if we've wiped out at this point. We might have. Yeah. Sure.
[00:44:58] Unknown:
I think we really got something though. Like, I think, like, holy shit. Like, figuring like, actually, this whole idea of a ring with, basically, where you can find a field modulo two containing just the identity elements of, you know, the two operations Yeah. That's gonna be a like, I'm probably not gonna sleep for the next three nights. Thinking about that? Thinking about that. Yeah. Like, that is something that is like a wild explosion.
[00:45:26] Unknown:
And maybe Well, because what are the implications? Right? Because if you if you're if you're using those two numbers within this field ring group, whatever you would I mean, it's all of this. Right? You're limiting your coefficients, and the thing is there's nothing nicer.
[00:45:38] Unknown:
There if you guys have ever done, like, ugly algebra. Right? Mhmm. There's nothing nicer than adding two numbers together and getting zero and then just getting getting to getting to remove it. Just it's gone. Yeah. Yeah. So, like, you know, taking a taking, like, a hundred term what would be, like, a hundred term polynomial and realizing it's only two terms? Because you and which is Zero. Yeah. Which is what you can do, you know, with binomial expansion when you have your coefficients are in finite fields and have character have a non zero characteristic. Again Yeah. Getting on the wave here, but, like, it enables you then to wipe out a lot of those intermediate terms because they just add to zero. So a coefficient of zero means bye bye.
[00:46:26] Unknown:
And this takes me back to one of the the going through the AES cipher and this understanding cryptography PDF that that I've been going through is, like, one of the ways that they're able to take it and represent it in a binary fashion is by using, you know, base two as, like, the polynomial representation of whatever numbers that they're using. Right? Like Yes. Absolutely. You can take it anything can be reduced to a base two polynomial.
[00:46:52] Unknown:
Yes. And And it's so intuitive sometimes to look at things that way.
[00:46:58] Unknown:
It's so intuitive to just Let me let me try to phrase this in like a a so you can take any number and and and factor it is I think is the correct term. Factor it into a polynomial using
[00:47:11] Unknown:
base two, which means just like a one or a zero. True, but it makes that I I don't know. I actually don't know if that statement is true, but it might it very well might be.
[00:47:22] Unknown:
Well, I mean, it's just like, that's how we represent numbers in computers. Right? It's like It's yeah. It seems to make sense, though. Right? Yeah. There's I wonder if there's a limitation there. Like, I I I see what you're saying with, like, I don't know if that's true, like, ad infinitum. Yeah. But I'm I'm confident that's true at least within a certain limitation.
[00:47:45] Unknown:
Like, you have I mean, in a again, not to get maybe to shift gears slightly, but to stay on this topic Yeah. Yeah. Is that, there's an idea of a irreducible polynomial, polynomial that can't be reduced. So for example, I think what most people could relate to is x squared plus one. It it in if with only real numbers, right, say, you can't reduce that. You need complex numbers to reduce x squared plus one. Right? Because
[00:48:17] Unknown:
you'd need x inverse of a square?
[00:48:19] Unknown:
You need a negative one you need x squared to equal negative one. And you yeah. You can't. So what the power of an irreducible polynomial is that it acts like a prime it's like the polynomial generalization of a prime number.
[00:48:36] Unknown:
And in Now Yeah. Hold on a sec. Like, is that if you were if you factor a prime number.
[00:48:47] Unknown:
Can't.
[00:48:49] Unknown:
Because it is just an irreducible polynomial at that point.
[00:48:52] Unknown:
Yeah. Like, prime you could there's no, like, two you know, three is just three. Right? Remember, every number is the product of primes to a power.
[00:49:02] Unknown:
Right? Right. Right. But prime numbers are always odd. Right? And that's because Separate it unless it's two. Well yeah. Correct. Good. Good. Thank you. But just in general, like, you just said x x squared plus one. Right? Like, that plus one, I think, like, there's something to that because everything like, it's an odd number all the time. Right? So it's like squared plus two has no solution either in real numbers either because you would need x squared equal negative two, which,
[00:49:28] Unknown:
you know.
[00:49:32] Unknown:
Now you need yeah. You know. Yeah. I feel like we're wiping out on the wave here.
[00:49:36] Unknown:
Right. It's just the bottom line is I think it's it's I think you can since we talked about polynomials, it's a way as a way to generalize a number. There are polynomials that can't be reduced, which you can generalize to prime you can generalize to primes. And we use them in in cryptography. And I you know, this is like the first thing I noticed in this book of understanding cryptography, right, was that it's not only we don't we don't just pick big numbers as generators, we pick polynomials as, you know, as something. You know?
So there's So the Part of the cryptography and the part of it involves an irreducible involves a polynomial that cannot be reduced as sort of the basis for generating, like, maybe it's for generating, the points in the elliptic curve. I can't keep it fully straight. It's used some there it is it is used in a way I can't full I can't articulate here. I shouldn't have gone here.
[00:50:39] Unknown:
There's is it like, it's like, it's not an equivalence. Right?
[00:50:45] Unknown:
But, like, prime numbers It's used as a modulo. Like, you know how we say, oh, we've find a field modulo 11 and we divide everything by 11. So it's used as actually the modulus, but you don't define the number. You just you just say it's you you say it's a polynomial. Just to find the polynomial. X squared plus one is the modulus and then you can say for any x, right, you can Yeah. It doesn't you know, you don't have to, you know, you don't have to restrict it. You can generalize it to any x. Anyway, you know, let's not get let's not get too dragged into this. Well, in this we're we're we're
[00:51:26] Unknown:
about at the time that we usually, like Yeah. Rip. And so I wonder if this is a good pausing point. Man, wow. That was all that was, like, this was so for for not reading through the cryptography, Brooke, I think I like, there's a lot to take away from here and just understanding, like, one, more clarification to me on, like, what a polynomial is and the pieces of that and why that's important. And then I look forward to your, sleepless nights reporting back on, like, this this significance of, like, the zero and one.
[00:52:04] Unknown:
And I wonder I know you're trying to close it out. Let I I there's let me let me say real quick. Yeah. That, like, I really enjoyed doing this. I don't know if people are I mean, this is now to the people who've hung in for fifty two minutes. Okay. I'm assuming you guys either enjoy it or getting something out of it. Right? I certainly am. To me, this struggle is part of, like, my forty hours a week of math homework. Right? It's pushing me it's pushing me to be to my limits. Right? And it's pushing me to grow, and I'm personally enjoying it. But I'm not just gonna do this because I like to do it. I'm hoping I hope people are getting stuff out. I'm getting comments from people saying they're enjoying it.
I would just say keep that keep communicating with us. We have a page on Nostr, Motivate the Math. Most people know how to contact Gary and I and multiple plat multiple platforms, Nostr. So that's that, guys. K? I enjoy it. I hope I can see Gary has stamina for it. I can't tell if he's enjoying it or not.
[00:53:17] Unknown:
Oh, I'm a masochist. I enjoy every second of it. It it's it's a new the the my brain is doing new things. Right? Like, that's a that's an itch that's hard to scratch.
[00:53:29] Unknown:
I think it's important that we do this. Yeah. You know? Something like we're gonna like, just today, I'm gonna walk away with this discovery of the zero one ring and say, man, if I hadn't committed to this podcast, we I wouldn't have those words wouldn't have uttered. We wouldn't have connected on that.
[00:53:47] Unknown:
Right.
[00:53:48] Unknown:
Alright. Wanna read the boosts?
[00:53:50] Unknown:
Yeah. Let's read the boost. Topboost, Dan. If you haven't checked out Dan's cartoons, I I highly recommend checking out bread and toast. Like, they're just they're just, like, wonderful, wholesome little cartoons. But when Dan gave me Zapboost. Dan gave me my logo for my fish podcast,
[00:54:09] Unknown:
and he'd you know, this is was, like, two years in the making.
[00:54:12] Unknown:
He's he's just the man. It means a lot to me that Dan's in our in our, like, in this community here too. Thank you, Fundamentals and Average Gary, for doing this work. If you keep doing it, then I'll keep listening even if most of the stuff goes right over my head the first couple times hearing it. Love it. Stamina, just like you said. Next, God's death, 537 sats boosted. Thank you, gentlemen. Can you have slides in the thumbnail on fountain showing equations and such as you go through the episode? Visuals would help me understand a great deal. More great work.
Like, be on the lookout for for more, like, visual things. I'm working on it. Yeah. I I really am working on a way
[00:54:49] Unknown:
to supplement this with visual aids. So I would say be patient. You know, the it's after I, like, when when it's when Gary hit stop, I have this, like, I go into my head and be and start thinking to myself, man, I need to figure out how to do video. And, I actually did an I did a full video last week. Okay. I didn't post it. I just just to convince myself that I could do it, I haven't shared it with you yet either because for some reason I can't we can't I can't share it with you on the platforms you're on for some reason. We'll get there. We'll get there. We'll get there. But anyway, so it is in the works. Okay? It's in the works. I'm gonna figure out how to do this. We might have it might be a YouTube site where Yeah.
I start posting, like, exercise videos. You know? Not not not I just said not with I want I want you in, like, a Richard Simmons sweatband, right, for the exercise. Yeah. With a thigh master. Yeah. And a pencil.
[00:55:49] Unknown:
Last boots anonymous, a hundred sats. Insert boost comment here. Thank you, whoever you are. And, fundamentals, thank you again for for blowing my mind on this this mask thing. Good shit. Good stuff. Thanks.
Introduction and the Challenge of Discussing Math
The Importance of Questioning Math and Cryptography
Math as a Source of Power and Pattern Recognition
Neuroplasticity and Building Skills
The Journey of Learning Math
Understanding Bitcoin's 21 Million Limit
Geometric Series and Bitcoin's Block Reward
Exploring Polynomials and Fields
Binary and the Special Nature of Base Two
Irreducible Polynomials and Cryptography
