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51: Episode "-2,0,1" (Bases; Exotic Bases)
424 Plays4 years ago
A numerical base is a system of representing numbers using a sequence of symbols. However, like any mathematical concept, it can be extended and re-imagined in many different forms. A term used occasionally in mathematics is the term 'exotic', which just means 'different than usual in an odd or quirky way'. In this episode we are covering exotic bases. We will start with something very familiar (viz., decimal points) as a continuation of our previous episode, and then progress to the more odd, such as non-integer and complex bases. So how can the base systems we covered last time be extended to represent fractional numbers? How can fractional numbers be used as a base for integers? And what is pi plus e times i in base i + 1?
This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.
[Featuring: Sofía Baca; Merryl Flaherty]
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Transcript
Introduction to Numerical Bases
00:00:00
Speaker
A numerical base is a system of representing numbers using a sequence of symbols. However, like any mathematical concept, it can be extended and reimagined in many different forms. A term used occasionally in mathematics is a term exotic, which just means different than usual in an odd or quirky way.
Exploration of Exotic Numerical Bases
00:00:16
Speaker
In this episode, we are covering exotic bases. We will start with something very familiar, that is to say, decimal points, as a continuation of a previous episode, and then progress to the more odd, such as non-integer and complex bases.
00:00:28
Speaker
So how can the base systems we covered last time be extended to represent fractional numbers? How can fractional numbers be used as a base for integers? And what is pi plus e times i in base i plus 1? All of this and more on this episode of Breaking Math. Episode 51, episode negative 2, 0, 1.
Guest Introduction: Meryl from UNM
00:00:53
Speaker
This is Sophia, and with me I have Meryl, who's a graduate student at UNM. Meryl, thank you for being on the program. Thank you. All right, so before we get into the subject, I got a few plugs. Patreon, we're on patreon.com slash Breaking Math if you want to support this show monthly. One dollar and more for outlines and ad-free episodes. We also have a poster store on Facebook at facebook.com slash Breaking Math podcast. The poster tier on Patreon will give you a podcast, we'll give you a poster for like
00:01:22
Speaker
I think it's like $35, but the poster store on Facebook, it's considerably less. It's $15.15 for a tensor poster, plus 450 shipping and handling, which is $19.65 anywhere within the United States. This tensor poster has the mathematics behind tensors and a little introduction to general relativity on it.
00:01:42
Speaker
What else? Oh yeah, on Facebook you can find updates about the show. We're also on Twitter at BreakingMathPod. And our website is breakingmathpodcast.app. That is breakingmathpodcast.app. Meryl, welcome to the show. It's good to be here, Sophia. We sound like a news people. I know, right? So, Bazes, any thoughts about Bazes?
Base Systems Overview
00:02:03
Speaker
Well, as a computer engineering student in, now I'm not a computer engineering student anymore because I got my bachelor's in that and went on to go for a PhD in pure math, I feel like I at least know something about bases having to deal with them all throughout a computer engineering degree.
00:02:22
Speaker
And of course the base that you deal with mostly is like, I mean, two or eight sometimes. Maybe eight, usually two or 16. If you're lucky 10. So last time we talked about what a base is. We talked about base five and how like, for example, two, four in base five is equal to 14. If you want to more about that, just go to episode 50, which is our previous episode.
00:02:48
Speaker
But right now we're going to be talking about how to represent fractions and bases.
Unusual Number Systems and Radix Points
00:02:54
Speaker
And then it's actually only a short lead there to using bases that aren't even integers. They're not whole numbers. And they're not even bases that are complex. All right, so without further ado, here's the episode. All right, we're going to do a review of bases. Base 10. What do we use for base 10? The Hindu Arabic number system, which is what?
00:03:18
Speaker
It's, you know, one through nine, or I guess zero through nine. And just, you know, you have your tens, your ones place, your tens place, your hundreds place, and so on. Yeah. And so in a example of a system that isn't base 10 is telephone, which is a language of the telephone people from New Guinea, which is base 27. They have different numbers from the left pinky to the left thumb, the wrist, lower arm, and so forth on the left and right side of their body.
00:03:47
Speaker
meaning that they just have to remember which part of their body is to remember the number less than 27. And a large number in telephone is 14 times 27, which would be 378.
00:04:00
Speaker
I can just imagine with like that particular base 27 system just having to do an entire dance to count and I kind of want to see that now. I mean you try it right now listeners and you I mean just like use your right index finger as a cursor and touch your left pinky left ring finger left middle finger left index
00:04:23
Speaker
thumb, wrist, lower arm, elbow, upper arm, shoulder, side of the neck, ear, left eye, nose, and then and then just all the way on the other side you use your right index finger to point to that. So really for any number less than 378 times 2 which is what 506, 556 you can use no 756 you can use you just remember two parts of your upper body.
00:04:55
Speaker
But also you can remember 576 now. But non-10 bases. So if we're in base 5, our positions go 1, 5, 25, 125. If we're in base 60, how did they go? It's like a 160, whatever, 60 squared is, I guess, 3600. Et cetera. Yeah. Key multiplied by 60. Base 10 has how many digits? 10. Base 5 has? 5. And base 60 has? 60. And there we've reviewed bases.
00:05:26
Speaker
Alright, so the first thing we're going to talk about is the radix point, known in decimal notation as the decimal point. And it's used, of course, to represent non-integer numbers. Okay, like I was just, you know, interjecting a minute ago, how we have our ones, tens, hundreds, places, or so on for base 10, or our
00:05:48
Speaker
going left, of course, right? Right. So, but what if we're going to the right of the decimal point, then we have, you know, our tenths places, our hundredths places. Notice the emphasis on the th because we're going with, you know, fractions this time. So it's like, say I were in binary, base two, then we would have our one half place, our one quarter place, one eighth, sixteenth,
00:06:19
Speaker
634 64th and so on. So what would a point one one be in base two? I mean, would you like to go through that? So let's say that we had the number point one one in base two. So we have past the radix point, we have a one. So that first one is in the one halves place. And why is it the one halves place in binary?
00:06:40
Speaker
So we have that pretty much any base has, especially integer bases has a ones place. So that is because it's whatever your radix number is. Yeah. It's like the, it's like home base kind of for the number. So it's that to the power of zero and that's just one. And then past the radix point, we start going to the negative ones play like the negative first power, negative second power, and so on.
00:07:10
Speaker
So yeah, because just a quick review, 3 to the negative 2, for example, is 1 divided by 3 to the power of 2, or 3 squared, which is 1 9th.
00:07:18
Speaker
Exactly. And so in this case, if we're talking about the number point one, one and binary, then we have so we have one, the first place past the radix point is one half, and then we have the next part, which is the one quarters place. So what we can say is that this is the same thing as three quarters or point seven, five, as we would write it in decimal.
00:07:44
Speaker
because it's one half plus one quarter, right? And there's one half times one plus one quarter times one, right? Exactly. And so point one zero one in the same venue would be one half plus zero times one quarter, which is just zero, plus one eighth, which is five eighths. So the development of these numerals is actually surprisingly late. We talked about how Hindu Arabic numerals had been around in the early
Historical Development of Decimals
00:08:10
Speaker
verse millennium.
00:08:12
Speaker
but now, but it wasn't until the first, until the 10th century A.D. by Abu'l-Hazan al-Uqli-Disi, a mathematician in kind of polymath who wrote a book called the Arithmetics of Al-Uqli-Disi, original title, Kitab al-Fusul fee al-Hizab al-Hindi. In every translation I've found, it says it's called the Arithmetics of Al-Uqli-Disi, but I noticed that there's this al-Hindi in there and I haven't had time to research the original Arabic but
00:08:40
Speaker
If something tells me that there's something, there's a reference to the Hindu Arabic numerals inherent to the title that hasn't been translated. And I think it's just weird how long it took. And I think part of the reason why is commerce isn't always, you know, okay, like you go to the store, you don't buy 10 eggs, you buy a dozen eggs, right? Right. So, I mean, I don't know, why do you think it took so long from the time that like Arabic numerals were introduced to talk about decimal points?
00:09:08
Speaker
Hmm, instead of, you know, breaking numbers into small numbers, you just use smaller units of measurement. And it's like, you know, you know, that 12 inches make a feet, you're not as likely to think about how three inches is a quarter of a foot.
00:09:25
Speaker
Oh yeah, and which is 0.25, which is two whole decimal points. So it gets very abstract, I suppose. Even the Sumerians, in my understanding, when they used fractions, they would just use fractions instead of using decimal points, even in base 60. Or if they did use it, it was not with a radix point. It was just understood to be divided by something else. So it is hard to kind of pin down the beginning of this, I suppose.
00:09:50
Speaker
So then you have Manuel Ben Jacob von Fiel, who lived from 1300 to 1377. He was a pioneer of exponential calculus. He used decimal fractions. And he was a French Jewish mathematician and astronomer. And he wrote this book called Six Wings. It was a book of astronomical tables, eclipses, and stuff that was used for 300 years by sailors and explorers until the 1600s. And he was the one who used the fractions for the first time.
00:10:18
Speaker
In 1585, Sivan Steven, who lived from 1548 to 1620, wrote a book called De Tienda. I think it's De Tienda. And I apologize if I've been butchering things. All these names are things I'm fairly unfamiliar with.
00:10:35
Speaker
But he lived in, this book was written in 1585, and it means of tenths. And in this system, to write 120.21, you'd write 120, then you write a circle around a zero, and then a two, and the circle around a one, and then a one.
00:10:56
Speaker
Meryl, isn't this great? Isn't this a great system? It's hideous. And I just want to say so you have 120 and then you have. So what is like the circle zero? Is that saying like the power zero is to the left? Is the circle one saying that the power to the negative one is to the left of that? It's just, you know, considering that you read from left to right, it seems pretty counterintuitive.
00:11:24
Speaker
Oh yeah, and it just seems counterintuitive too because like 3.1415 would be three circled zero, one circled two, four circled three, one circled four, five, and it just, it gets, why?
00:11:40
Speaker
white but so it didn't take long for people to write a bar over the units digit so 120.21 would be one two zero with a line over it two one and then superscript which is actually still using prices like with the 599 with a 99 up into the right of the five and then of course decimals came into play at some point it was in the next century except for in Europe where they use a comma yeah so certain places in Europe I believe
00:12:07
Speaker
So we talked a little bit already about non-ten bases for decimal points, or I suppose radix points, but just something kind of interesting.
Patterns in Dividing by Base Numerals
00:12:17
Speaker
We're just going to give you a couple of facts about that. If you have a calculator right now, if you type in 12345 divided by 99999,
00:12:29
Speaker
Of course you don't have a calculator, you have a phone, but type it into the calculator app and you'll get 0.1234512345 and it'll have 12345 over and over ad infinitum. If you notice, that's because I had 12345, five numerals over 99999, which is five numerals.
00:12:46
Speaker
And surprisingly, this works in other bases. So like, for example, 3, 4 divided by 4, 4 in base 5 would be 0.343434. And one interesting thing about binary is that pi in base binary, actually base 16, which is closely related to binary,
00:13:05
Speaker
You can write out the decimals of pi, and there's actually a definite expression for each decimal. It's like a summation or something, but there's a formula for each decimal. And as far as I know, there's no known ways of doing that in other bases. I don't know if it's been proved that it can't be done, though.
Non-Integer and Non-Standard Bases
00:13:24
Speaker
Alright, so how would we do non-integer bases? As I was doing this podcast, I was just thinking about it. In the last episode, let's say we wanted to convert to base 2. We did it from right to left, from the units digit to the 1, 2, 4, 8, 16, etc.
00:13:45
Speaker
and what you do is you divide by two so like if you 17 in base two 17 divided by two is uh eight remainder one so the last digit is one eight divided by two is uh four remainder zero so you take the remainder which is zero which is the second digit from the right and so on so um then you get a one one one zero one i believe or something like that oh yeah no it's just one zero zero zero one yeah just take a 16 tack and one at the end and you got it
00:14:14
Speaker
Yeah, but so there's another way that we could have done that. Let's say we want to convert 17 to base 3 using the largest power 3 that fits into 17 is what? That should be 9.
00:14:25
Speaker
Yeah, 9. And that goes into 17 one time, right? Right. And 17 minus 9 is 8. And then, OK, so the power over from 9 is 3. And 3 goes into 8 twice. So the number so far is 1, 2. And then the last digit is just 2, because 1 goes into 2 twice.
00:14:48
Speaker
Right, so the last digit kind of acts as a remainder in this case. Yeah, which is kind of weird. I suppose that makes sense, yeah, every single time it's going to be a remainder. But basically, there's nothing in this algorithm that stops us from doing it with non integers. And so basically, if we're doing base pi, we just have to keep multiplying pi by itself until we're less than a number and then divide and so on.
00:15:12
Speaker
And we'll leave this as an exercise to the reader. We might talk about this in a problem episode. But just kind of for fun, one half in base pi is 0.11211021020. And in fact, this number I'm pretty sure would be transcendental. Don't quote me on that because God knows that not every postulate is correct. I actually do have a question about this one.
00:15:36
Speaker
So I noticed that it's base pi, but your digits are all 0, 1, 2, 3. Yeah. And I was just using the convention in this case that all the digits would still be decimal. I mean, it would still be integers and all integers less than pi. Okay. I can see that then.
00:15:55
Speaker
Yeah, and so E in base pi would be 2.2021201002. But what's cool is that 1 divided by pi in base pi is just 0.1, and 1 divided by pi squared in base pi is 0.01. But 1 divided by pi squared times 3 is 0.1022012202, and it just goes on forever. So it's just weird to add two things that have finite representation together and get something that has infinite representation.
00:16:22
Speaker
And something I actually kind of noticed about that is because, remember, pi is transcendental over the rationals. What we're doing here is that we're just taking, say,
00:16:39
Speaker
taking pi, plugging it into a polynomial, and then you're getting something based on that. So it almost makes sense that what you get in base pi would be some sort of, say, repeating decimal. Yeah, I didn't even think about it that way. But yeah, that totally makes sense.
00:17:01
Speaker
And yeah, that kind of covers that aspect. However, there's another way we could write, we could do fractional bases. Okay, so this method of fractional bases only applies to rational numbers. And do you want to talk about what rational numbers are super quick?
00:17:17
Speaker
So if you study math, the definition of the rationals that we get is that we have a divided by b, a and b are just some integers and b is not zero.
00:17:33
Speaker
Yeah, so 3 quarters, 1 over 307, 307 divided by 308. But yeah, et cetera. So in base 10, you ever play Mancala? Yeah. So Mancala is this game where you have a bunch of different wells with little stones in them. And this is nothing like Mancala. The rules of Mancala don't apply here. But I just wanted to just imagine in front of you that you have a board filled with a few different wells that you could place stones in, small colored stones.
00:18:02
Speaker
Now let's say that we're counting in base 10. We put one stone in the furthest right well. We add another stone. Now we have two in the furthest right well. We continue until we have nine. But then when we have ten, we take out all the ten stones from the well that has ten stones. And in doing that, add one stone to the well right next to it. So now we have a well with one stone next to a well with no stones in it, which is kind of like the number ten. One zero. Are you with me so far?
00:18:27
Speaker
Uh, yeah, actually I've never even heard this explanation before so I need to remember to buy a Mancala set if I ever end up teaching digital logic.
00:18:36
Speaker
Yeah, and also then people can play Mancala as a trade, because Mancala is fun. And then for all you don't know, Mancala is a really old game from I think Africa, where you just take, but I'm pretty sure a ton of cultures have their own version of it, where you move stones around and you have to collect as many stones as you can, and it's the whole thing.
00:18:59
Speaker
All right, let's say that we're in base three, right? When we start with one stone on the right, two stones on the right, when we have three stones on the right, we replace those three stones with one stone next to it, right? So we have one zero, meaning three in base three. Then one one, one two, one three, but one three doesn't exist in base three. So you take the three stones and replace it with stones next to it. So you have two stones next to no stones, or two zero in base three, which is six in our system.
00:19:27
Speaker
So you have 2-0, 2-1, 2-2, and then when you get to 2-3, you take away the three stones and you add one stone to the well next to it, which now has three stones, so you have 3-0. However, three can't exist in base 3, so you take away those three stones and you add a one to the well to its left. So you have 1-0-0 in base 3. And that's just 9 as we know it.
00:19:52
Speaker
So, let's say, though, that instead of taking out three stones and adding one to the well next to it, or taking out ten stones and adding it to the well next to it, or taking out sixty-four stones and adding it to the well next to it, this is a Big Macaulay set, suppose we took out three stones and added two to the one next to it.
00:20:12
Speaker
In this episode of the podcast, we talk a little bit about infinite bases, and that's why we're recommending the brand new Infinity course from the folks over at Brilliant. It's of course all about the math of infinity. Explore exotic concepts such as infinite area and more on this course. So what is Brilliant? Brilliant is a website that teaches various concepts using text, videos, diagrams, and interactive demonstrations.
00:20:32
Speaker
With an approach emphasizing problem solving and active learning in over 60 courses in math, science, and computer science, Brilliant is a great place to start learning a ton of concepts that you, as a listener of Breaking Math, may be interested in. So if you're interested, you can go to brilliant.org-breakingmath. The first 200 Breaking Math listeners can get 20% off the annual subscription. Now, back to the episode.
00:21:00
Speaker
So let's say we're doing three stones, two stones, right? Let's start with the one color set. If I add a stone to the right part of your one color set, you have one, right? No, right? At another stone, you have two. At another stone. So you have three, but you take those out and then you add two to the next one over. Yeah, the one to the left.
00:21:22
Speaker
So now we have two stones and no stones, right? So we got two, zero. Yeah. And then I add a stone to the right. And now we have, so two, one, we add another stone and we have two. And then, so we'd have two, three, but we'd have to take those out and then add them over, but we're adding two to the next one over. So we got four, zero.
00:21:39
Speaker
Yeah, so we have four stones in the second slot from the right. And then in the rightmost spot, we have no stones. However, four is greater than equal to three, right? So because we're doing three to two, we have at least three stones to grab, right? Right. So what happens when we take, what happens when we take three stones from four stones?
00:21:59
Speaker
So if you take three from four, then what I'm guessing is, because we know that four in this base would be represented as two and then one, what I'm guessing is that we move, so we take all but one stone out of that second place, and then we have two, one, zero.
00:22:19
Speaker
and that is actually correct yep actually it's correct no but yeah it's correct so two one zero is the next number and here's this here's what's interesting is that two one zero in base three three halves would be two times one half times one half which is a two times two point two five or four point five plus one times three halves which is one point five plus zero which is four point five plus one point five which is six so it actually works as base three halves
00:22:48
Speaker
I think it just broke my brain.
00:22:50
Speaker
Oh, I know this broke my brain when I found it too. And for an explanation of why, oh, and by the way, if you heard sounds because we moved into a new place and things are still moving around, but we'll be able to provide more regular episodes for you now, which here in the background is a dog known as Math Pig.
Podcast Updates and Sponsor Mention
00:23:09
Speaker
But yeah, for the reason why we might do that on a problem episode. If not, it's left as always as an exercise to who? The reader. Yeah. Or I guess the listener in this case.
00:23:19
Speaker
So yeah, that is how to, and you could do that with any fraction greater than 1. It's distinct from the other notation because in base 3 halves in the other notation, we only have digits 0 and 1. However, in this one, we have 0, 1, and 2, right? And the difference is the numerator, and actually there's a theorem that states that basically
00:23:43
Speaker
any fractional number can be represented an infinite number of ways with digits if you don't limit the digits in the bases.
00:23:53
Speaker
Alright, so infinite bases, super quick. Everything is super quick. But no, infinite bases, the word that works is just by never carrying, right? So you have one on the, you just keep counting. If you just keep adding one to something, you only have one digit ever. If you multiply them by like 10, like one zero in this base already, then you could get up to that. And the only reason we mentioned this is because Pascal's triangle, do you want to explain Pascal's triangle really quick?
00:24:19
Speaker
So, about the binomial theorem, I'm teeming with a lot of news. Now, maybe... I am very well equated to with matters mathematical. I understand equations both simple and quadratical. About binomial theorems, I'm teeming with a lot of news. A lot of news.
00:24:47
Speaker
So about the binomial theorem, I'm teeming with a lot of news. Pretty much you start with 1, and then the next row is 1-1.
00:24:58
Speaker
So on the piece of paper, you can write the number one. And then if you draw two diagonal lines from the one down to the sides, you just write ones on each of those. And the one you write beneath each of those two ones.
00:25:13
Speaker
Right, so it's like you have one and then you have one one and then so you have on the left side you'll have one and then in the middle you'll have so that's between two digits you add those they're just ones so one two one and now yeah because two is one plus one which is the above it
00:25:32
Speaker
So then you have 1, 2, 1. And the next row would be 1. And then 1 plus 2 is 3. Then 2 plus 1 is 3. And then you end with 1. Yeah, so 1, 3, 3, 1. Next is 1, 4, 6, 4, 1. 1, 5, 10, 10, 5, 1, et cetera. All right, so how is the Pascal's triangle related to the binomial theorem?
00:25:55
Speaker
So we all know that if we have something that's just a plus b, we're adding two things together. So that's just to the first power. So that's just the one one row of the triangle. But let's say we're doing a plus b all in parentheses squared, and then we know that's a squared. So that's the one for the coefficient is one times a squared. Yeah.
00:26:21
Speaker
Right. Plus 2ab. 2 times ab. Plus b squared. Yeah. And then the next one, the numbers without the r would be a. So let's say we're doing a plus b cubed. The a and b combinations, the variable combinations we're going to do, we're going to be a cubed, then a squared b, then an ab squared, then b cubed. You notice that the power of a goes down each time and the power of b goes up each time. And then the coefficients are,
00:26:51
Speaker
So we'd have 1, 3, 3, 1. And that's the fourth row down. Third row down, fourth row total of the Pascal's triangle. Right. So it would be, you know, a cubed plus, and then we'd have 3 a squared b plus 3 a b squared plus b cubed.
00:27:09
Speaker
Yeah, exactly, and so on. And why do we mention all this? We mention this because 11 or 11 in an infinite base to the nth power is the n plus first row of Pascal's triangle. So for example, yeah, 11 is basically like, basically you do this, just multiply things together, like multiply 11 by 11 multiple times and multiplying by 11, but just never carry your digits.
00:27:37
Speaker
Never carry over from one digit to the other and you'll see that this works
00:27:41
Speaker
Yeah. So I just want to make sure I actually understand that. So we know that, but wait, doesn't that work just for base 10? Like 11 squared is 121, 11 cubed is 1331. Yeah. It actually devolves at a certain point because it's one, one, two, one, one, three, three, one, one, four, six, four, one, or 14,641. But then when you go again, it starts carrying over. So you have 161,051 and that's not good eats. I see where you're going now.
00:28:10
Speaker
Yeah, so that's pretty much why that works, because just adding the number to the next number. OK, so what I'm curious about, though, is what decides what digits go in what place in an infinite power? Oh, everything just goes in the last digit if you're doing integers. All the numbers are infinite if they have any more than one digit, or infinitesimal if they're to the right. Meryl has questions. I mean, I'm mostly just trying to wrap my head around this.
00:28:35
Speaker
And it doesn't make much sense outside of just like, I mean, infinite base is just another way of saying, don't ever carry. But if you start off with a few digits, like 11 in base, infinite would be like infinity plus one, which is kind of silly. Yeah. I'm just trying to figure out if you've never carry, how do you get more digits?
00:28:51
Speaker
Oh, by multiplying. Because if you when you multiply, you know, the Oh, oh, I'm finally starting to get it. Yeah, because you get the last digit, which is just the last two, and then you go to the right and then you just keep going. And then if you multiply any two, if you multiply two numbers that have a certain amount of digits, the product has the number of digits combined, at least, or in base infinity, of course, at most.
00:29:20
Speaker
Okay, this is starting to click. This is what I get for being a computer engineer, and the only bases you know are 2, 10, and 16. Maybe 8. Basic bases. Yeah, because I mean, this is of course incredibly important to the human race, all this research about complex bases. But no. So okay, let's see. Complex bases, finally.
Complex Bases and Episode Conclusion
00:29:43
Speaker
But now we're going to finally talk about the very strange concept of complex bases. And there's already complex bases out there, which basically the base is a complex number. A complex number is a square root of 1 times some number plus some other number. So like negative 1 plus or minus i. Because i is the square root of negative... That was math pig. Because i is the square root of negative 1.
00:30:09
Speaker
So now let's talk about quadr imaginary base for just a second. It's used sometimes in dragon fractals and things like that. It uses digit 0 and 1, but moving on. Breaking math has an exclusive version of a complex basis that I haven't found anywhere online.
00:30:26
Speaker
It uses Gaussian integers as digits. So basically Gaussian integers are like 3 plus 2i or 2 plus 5i, but not like 2.3 plus 3.8i because of the decimals. So it's the same definition as a complex number. a plus bi is just that a and b are integers. Exactly. And the magnitude of the digits is not more than the magnitude of the base in this system. That's how it works.
00:30:48
Speaker
And the digits, basically it's the same algorithm that we used, the first non-integer algorithm. And just a quick, we'll talk about this on a problem episode if we have time, or maybe we'll write a paper about it, we'll let you know. But Pi plus Ei, as promised in the intro.
00:31:06
Speaker
In base, i plus 1 is negative i, 0, 0, 1, point negative 1, negative i, 0, negative 1, negative 1, negative i, i, etc. It goes on forever. You can represent any complex number with no sign and no i. Aside from the stuff that's in the digits themselves.
00:31:30
Speaker
Right. So I guess I'll start it. So I had a question here and I was noticing like in like certain places, we have things like we have both I and negative I, and I'm a little curious how that works. Oh, no, those are just written too closely together.
00:31:48
Speaker
Oh, yes, but so there's still eyes and negatives eyes. Oh, yeah, because I negative I 1 and negative 1 are all less than or equal. I mean, are all less than or equal to the magnitude of 1 plus square root of 1 minus square root of... Okay, so it's all about magnitude and not just, you know,
00:32:08
Speaker
Yeah, and actually this relates to the Gauss circle problem, which relates how many lattice points are in the circle of a given size and how quickly it grows over time, which is not as simple of a problem as you might think. It's kind of a fun one, though.
00:32:20
Speaker
And I'm also curious, do you know that if in this space does every number in the comp like every Gaussian integer, let's say, or I guess complex number since we have a radix point, does each one of those still have a unique representation in the space? Yeah, constrained like this, it absolutely does. They have unique representations. I did a quick back of the envelope proof, but more research is needed.
00:32:47
Speaker
Bases can be simple or complicated. Your life using Base 10 has shown that they could be simple. We have just shown that they could be complicated. We had no reason for doing so, other than it's kind of cool. And sometimes that's all you need to do math. I'm Sophia, and this has been Breaking Math. With me, I head on Meryl. Meryl, thanks for joining me.
00:33:07
Speaker
Hi, I'm Meryl Flaherty. I am a PhD student at UNM in Peer Mathematics. And again, sorry for all the extra noise. It's because of moving studios, everything's half moved, all that. The old place was really hard to record at. Suffice it to say, it was a pretty terrible apartment, but now everything is much better. And we hope to produce regular content again.
00:33:36
Speaker
As again the patreon is patreon.com slash breaking math Facebook is facebook.com is breaking that podcast You can also see us on Twitter at breaking math pod And of course check out the sister of the show to this touring rabbit holes starring Gabriel and Alex Merrill anything you want to plug All right