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79: 1 2 3 (Counting)
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Join Sofia Baca and Nerd Forensics co-host Jacob Urban as they discuss all things counting!
Counting is the first arithmetic concept we learn, and we typically learn to do so during early childhood. Counting is the basis of arithmetic. Before people could manipulate numbers, numbers had to exist. Counting was first done on the body, before it was done on apparatuses outside the body such as clay tablets and hard drives. However, counting has become an invaluable tool in mathematics itself, as became apparent when counting started to be examined analytically. How did counting begin? What is the study of combinatorics? And what can be counted? All of this and more, on this episode of Breaking Math.
This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License (full text: https://creativecommons.org/licenses/by-sa/4.0/)
[Featuring: Sofia Baca; Jacob Urban]
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Transcript
Introduction to Counting and Early Methods
00:00:00
Speaker
Counting is the first arithmetic concept that we learn, and we typically learn during early childhood. Counting is the basis of arithmetic. Before people could manipulate numbers, numbers had to exist. Counting was the way that numbers first came about. Counting was first done on the body, using digits and locations on the body such as shoulders and knees, before it was done on apparatuses outside the body such as clay tablets, abacuses, and hard drives.
00:00:25
Speaker
However, counting has become an invaluable tool in mathematics itself, as became apparent when counting started to be examined analytically. So how did counting begin? What is the study of combinatorics? And what can be counted?
Guest Introduction: Jacob Ervin
00:00:38
Speaker
All of this and more on this episode of Breaking Math. Episode 79. 1, 2, 3.
00:00:50
Speaker
I'm Sophia and this is Breaking Math. With me I have on Jacob Ervin for the Nerd Forensics podcast. Welcome, Jacob. Hey, how's it going? Nerd Forensics is a more explicit podcast on the same network as us, Santa Fe Trail Media. If you're listening to this, you could probably listen to that.
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00:01:09
Speaker
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00:01:14
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00:01:35
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00:01:51
Speaker
I recommend you try it out. I use it for like, you know, creative slumps and it works if you do it multiple days in a row. And yeah, I hope you like it if you try it. And now back to the episode.
Ancient Counting Methods
00:02:04
Speaker
All right. So today we're going to go over the history counting. We're going to talk about combinatorics. We're going to talk about some topics related to that. And then we're going to talk about what you can and can't count even as you go to infinity. So Jacob, you ready?
00:02:20
Speaker
Yeah, I want to know what I can't count. All right, so the history of counting goes back to, I mean, it's kind of intertwined with the history of just representing numbers by themselves. You could kind of date it back to the earliest times. There's this now kind of defunct theory that society started with only the concept of one, two, and many.
00:02:44
Speaker
But that's not too far off. A lot of times, early societies don't have many different words for different numbers. So they have a word for one, a word for two, and then they'll say four is two plus two, three is two plus one, and then they'll have a word for five, which is usually something related to hand or something like that. There's places where locations on the body are counting, so you go from one upset fingers to the other, and then you use your shoulders and elbows and things like that.
00:03:14
Speaker
So you remember a number like that. And it wasn't long before people started grouping things together, usually by tens or twenties because it's a number of fingers or fingers and toes that we have.
Babylonian and Mayan Numeral Systems
00:03:26
Speaker
That is to say counting was instrumental in early times too because it was also, there was also a time where things weren't recorded on
00:03:37
Speaker
paper or yeah exactly like that yeah there wasn't a standard way of recording these there wasn't like an alphabet and like you know all that stuff it was more like a knot sometimes in fact khipu comes from this a knotted record was sometimes used to record things an example of using knots for counting was actually the Pueblo Revolt where Pope in 1680 were a Pope united all the different various Pueblos around New Mexico yeah
00:04:05
Speaker
He had a knotted string with him that had the countdown. Do you know about that part? Yes, yes, I do actually yeah, and yeah, so everybody would copy the amount of knots that are left as he went from place to place and on zero day they Repelled yeah And it was a relatively successful revolt actually. I mean it's a good system. What do it is?
00:04:28
Speaker
the knots and the rope like. Oh, I thought you meant what happened. I thought you meant as a result of the Pueblo Revolta. I was like, it's still kind of bad. Oh, no, no, no, that didn't end great.
00:04:38
Speaker
That didn't degrade, it ended better than it started, but not great. Yeah, it ended a little better than it started, but it didn't degrade. It was still colonialism, which is never good. Yeah. So, yeah, the idea of a place value system, you know, like ones, tens, hundreds, et cetera, you know, just having digits in general. Yeah, yeah. Appears about 4,000 years ago in ancient Babylon.
00:05:02
Speaker
are called Babylonian numerals in their base 60 sideways but within each numeral sideways wedges meant 10 and up and down wedges meant one so that's just a quick history of that what's up that just seems like a real inefficient way of doing it actually it's pretty efficient if you if you understand it
00:05:24
Speaker
Oh yeah, exactly. Because that's the thing is that they're not transferring from tens and hundreds to that system. They're already counting in that system. So it's like if you read ancient Babylonian literature, like Epic of Gilgamesh, instead of saying like hundreds and thousands, they'll say like sixties and thirty six hundreds and things like that, you know, because that's how their numbers worked. I know because Bay 60, which is funny.
00:05:48
Speaker
Yeah, it sounds funny to us. Oh, it sounds real funny. But I imagine if we said if we said to them, we're like one call, one call, one colon 40. Yeah, they'd be like, what? Yeah, exactly.
Evolution of Arabic Numerals
00:06:02
Speaker
So yeah, completely independently in the Americas, in Mexico and Guatemala, the Mayans came up with the base 20 system because of fingers and toes. And yeah, so like, for example, 100 in their system would actually be 400 because it's ones, 20s, 400s, 8,000s, 160,000s, 3,200,000s, et cetera.
00:06:29
Speaker
And the Mayan dates are written like this. And what's weird is that the Mayans have two years, one that's accurate and one that's 360 days. It's a ceremonial year. So I thought that was kind of interesting. It just floats around the year. But they do that because 360 is divisible by 20. It's 1820s. That's actually why in writing dates, instead of all the bases being 20, the second one from the top,
00:06:58
Speaker
Uh, no, the second one from the bottom, because it's written from bottom to top, like ones, twenties, four hundreds. Yeah. Uh, the second one from the bottom, uh, in the ceremonial dates is actually base 18. Yeah. Which makes it is hard to calculate stuff in that in ceremonial notation, but honestly, the normal base 20 is really, really actually easy to calculate stuff. Um, Oh, I'm sure it seems like it would be fairly easy for us to.
00:07:23
Speaker
It seems like it would be fairly easy for us to translate our system over to that. And I'll teach everyone how to draw my numerals right now real quick. So if you draw kind of a lemon shape, roughly a lemon, and then you draw a curve on the bottom of the lemon going up and down so that you have kind of a circle on the bottom of the lemon shape, almost like an eye looking down. Yeah, yeah, I see it. It's like a half circle.
00:07:53
Speaker
Yeah, so you have a half circle and then kind of like this arch shape, right? Yeah, yeah. If you divide the arch shape into three pieces, that's one of the ways to write Mayan zero. It's a shell. Ah, interesting. Do you want to guess how to write the number one? No, but you can show me.
00:08:16
Speaker
It's just a dot. Yep, just a dot. And you put it on the shell? No, no. No, the shell is just a... Yeah, the shell would be a zero. You know what? I've seen Mayan... Numerals? Yeah, Mayan numerals before. Oh, yeah. Also, I keep calling them Mayan numerals. They're called Maya numerals. I keep forgetting. Maya? Yeah, Maya numerals. Yeah. Yeah.
00:08:35
Speaker
Yeah, I've seen my enumerals before and like it goes like dots up to a point and then doesn't start doing dashes. Oh, yeah, because it does lines one dot two dots three dots four dots in a row. Yeah. And then five is just a line. Yeah. Line six is a line with a dot above it. Seven is a line with two dots above it next to each other.
00:08:59
Speaker
Yeah. And just keeps progressing like that. Yeah. And all the way until 19, which is three lines all on top of one another. So almost like stacked on top of each other, like pieces of paper. Um, with three, I mean, with four dots across the top. So that's night. That's a night. That's the numeral 19. Yes. Which makes sense because it's just like, it's a progression.
00:09:24
Speaker
Yeah, and what's fascinating too is that I feel like it's a much easier way to write the numbers down than the system that we have currently.
Introduction to Combinatorics
00:09:35
Speaker
The Arabic numerals. Because actually they did start out more representational, but they got more and more abstract as they got a tier up. So for example, in the original Hindu numerals, one is kind of a stroke.
00:09:48
Speaker
that kind of almost like an eye that slanted it and that's curved a little bit towards the right. Two is that with a little half circle, the bottom part of a half circle. Like a quarter of a circle hanging off the top of it. Yeah, it kind of looks like a scythe. Yeah, yeah, that's going to the right side. And then three is like that.
00:10:10
Speaker
and just adds another little hook essentially. Yeah but what gets different is that four is kind of a down facing fish. Yeah it looks like the Jesus fish but it's facing downwards. If you've ever seen one of those decals that has a very simplified fish shape that's similar to one of the early representations for the number four and then five is that upside down.
00:10:34
Speaker
And I'm not comparing the worthiness of number systems here. I just say my preference personally is because I like the neatness of it, is the mind numeral system.
00:10:46
Speaker
No, I mean, I get it. I get it. Like I I like the the numeral system that we use in the West on an everyday basis up to like up to like you start getting into like millions and billions and those types of numbers with like just an obscene amount of zeros behind it.
00:11:08
Speaker
Oh, yeah. It goes trillions, quadrillions. Yeah. Yeah. Now, I understand that a lot of times they'll just do like tenths to the whatever power. Oh, yeah. That's more common. Yeah. More mostly people use those big words for articles. Exactly. Yeah. Sixteen quadrillion quadrillion trillion. Yeah. No. Any any like person who's actually like doing math at that, like with numbers at that scale or doing them to like the power of.
00:11:35
Speaker
Oh, yeah. And most likely, if you're doing those numbers, you'll actually more likely be a physicist. Yeah. Because mathematicians don't usually use anything greater than the number eight. No, I'm just kidding. That's not true. But it's a joke that mathematicians use, that if you're using any number greater than eight, you're not doing real math. That's pretty good.
00:11:56
Speaker
And I didn't actually plan this, but I was actually going to say I'm using a borrowed computer to edit this right now. But on my original computer, I have some merchandise that might be coming up pretty soon. It's a poster on counting. I didn't plan it for this, but I thought to mention it.
00:12:13
Speaker
So, yeah, so that's kind of like the history of counting and the ability to count. You know, using language makes it easier to keep track of large groups of things. And it turns out that the easiest way to keep track of groups of things, mathematically speaking, is the place value system. And it's using computers too, binaries just base two. Exactly, yeah.
00:12:36
Speaker
Alrighty, so now we're going to talk about combinatorics. So what is combinatorics? Combinatorics is like, let's say I had a few questions that might be combinatoric questions are, how many different ways can you scramble a Rubik's Cube? Or how many ways could you shuffle a group of 20 people? Or if you had 10 different choices and you could choose three things, how many different groups of choices can you make?
00:13:04
Speaker
Yeah, yeah, it's a it's a grouping. It's like a how many grouping systems are available within this set of things. Yeah, and perhaps, yeah, and perhaps more succinctly, it's about counting hypothetical objects.
00:13:20
Speaker
Yeah. And, uh, you know, you, because you have a high level description of this object, like Rubik's cube has such and such mathematical properties, or, you know, and like, um, the idea of counting has this or that, and you build from there. So, okay. So I was looking this up a little bit before we started. And like, one of the simpler ones that I saw was, uh, the number sequence of one, two, three, how many ways can you order that?
00:13:46
Speaker
Oh yeah, that's actually a great example. Um, so, uh, yeah, so one, two, so let's, let's, it's six. Yeah, it's six, but let's, let's, let's say we didn't know what, that it was six already. So we'd start as one, two, three, right? One, two, three. But, uh, and then, so one of the things that we can do with those is we could swap the last two, right? Yep. So one, three, two, one, three, two. All right. So now we can't swap anymore. We're at a dead end. So now let's rotate the original one. So we started one, two, three. Let's change that to two, three, one, right? Yep. Two, three, one.
00:14:15
Speaker
And then two and three because you swapped the last two. Two, one, three. And then three, one, two. Three, one, two. And then three, two, one. And then three, two, one. And notice that you have three groups of two. Yep. And that's not coincidental because if you did it with four objects, so one, two, three, four. If you swap the last two, you get one, two, four, three. And then you're at a dead end. So now you rotate the last three. So you get one, three, four, two. Then you swap the last two. And then you get one, three.
00:14:43
Speaker
one two one three four two one two one four two three this is something that's going to be hard to keep track of yeah exactly like paper oh yeah but exactly you notice that it's this time it's four groups of three well it's actually no it's actually four groups of three groups of twos so it's four times three times two so that'd be 24
00:15:04
Speaker
four times, three times. Yeah, exactly. Yeah. So, and that's actually the number, the way to write that down is four factorial, as you learned on the internet earlier, which is just four with an exclamation point after it.
00:15:18
Speaker
Yeah. Which, yeah, if there's 24 answers to that solution, Sophia, there's no way you're going to remember all of those.
Binomial Coefficients and Pascal's Triangle
00:15:26
Speaker
Oh yeah. I wasn't going to keep, I was going to stop after like six, but yeah. Yeah. One way that you can prove this, um, that this is the amount is that you can improve it kind of recursively. So you can say that, um, for like, let's say we have N.
00:15:41
Speaker
different objects, right? Well, actually, let's say we have one different objects. One factorial is one, right? Yeah, so we've proven it for one. So now we have, let's say we have n different objects and there's n factorial ways that we can mix them up. Let's say that that's what we know. Let's take that as given.
00:15:59
Speaker
So let's see if we can prove that this applies for n plus 1 too. So let's say we have a group of n plus 1 items, right? So that's like a group of 1 plus another n items. So how many ways can this start?
00:16:12
Speaker
Uh, it has to be n plus one items. Exactly. Yeah. Yeah. There's n plus one ways that this can start. And for each way that this starts, there are, there's a, there's n left over, right? Yes. And those have to be, um, you have to go through all of those permutations permutations too, right? Yes. So what, and we already assume that the permutation for a number n objects is what? Uh, it's the n factorial. Yeah, exactly. And factorial.
00:16:38
Speaker
Yeah, n factorial. So now we have n factorial times n plus one, right? Yes. What do you think that equals? At this point, n plus one or one n plus one factorial. Well, n plus one doesn't make much sense because you have these giant numbers times a big number. So you see a bigger number. But it'd be n factorial one.
00:16:58
Speaker
N plus 1 factorial. N plus 1 factorial, yeah. Like in parentheses and then factorial afterwards. Because if you think about it, N plus 1 factorial is 1 times 2 times 3 times all the way to N. Yeah. And that's actually how factorial can be defined, as N factorial is equal to N times N minus 1 factorial. And then you need to find 1 factorial is 1.
00:17:18
Speaker
which implies that zero factorial is one because one factorial divided by one is equal to zero factorial, so yeah. Because you can't have actual zero in this. Well, not even that, it's just the math just shakes out that way. Oh, okay. Because like if you have one factorial, so two factorial divided by two is one factorial, right?
Magic Squares and Ancient Cultures
00:17:37
Speaker
Yes. Which means that one factorial divided by one is zero factorial. But... And we've defined one factorial as one, so one divided by one is one. Exactly, yes.
00:17:46
Speaker
And then you actually do have negative factorials, but that'll really go. That'll get some more into the gamma function and that'll be its own episode. Oh, and just a quick mention, the history of combinatorics goes all the way back almost 5,000 years to 2800 BC in China. Are you familiar with magic squares? That sounds familiar. So it's like a, it's like a square where. Well, let me try.
00:18:10
Speaker
All right, so read the I wrote. What did I write for you? So from top down, left to right, it's a sequence of numbers. Yeah, it's like it's almost like a matrix, right? Like a three by three. Yeah, yeah. So there's two seven six nine five one and then four three eight on the last row. All right. So add up the first row. What's the sum of the first row?
00:18:34
Speaker
So two, nine and four or no, that's the wrong way. Two, seven, six. Which way do you want me? Oh, that's a sideways. Yeah. Rose. So two, seven, six is for 15. How about the second row? That is also 15. Third row. That is also 15 diagonally.
00:18:55
Speaker
15, 15, no. Up and down. Yes. 15, 15. It keeps making 15. Yeah. That's a magic square where no matter which way you add it up, down, left, right, it's equal to the same number. Yeah. It's always equal to 15. That's pretty cool. And those are developed in China in, um, like I said, 2,800 BC and combinatorics was designed to study the properties of these objects. Like, you know, counting how many possible magic squares have this in that property. Yeah. Interesting.
00:19:27
Speaker
Alright, so now we're gonna talk about a few different things in combinatorics. So we're actually gonna come back to confirm mutations in a second. But first, I'm gonna ask you a question. If I shook your hand, how many handshakes would that be? How many times are you gonna like
00:19:41
Speaker
No, no, no. If I just shook your hand once, how many handshakes would it be in my life? No, no, no. Oh, just okay. I see what you're framing it as now. Uh, that would just be the one handshake. Exactly. And if we have three people and we all shake hands, how many handshakes are going to be there? All three people shake hands.
00:20:00
Speaker
Oh yeah. So like, let's say we have person a prison B, person C. Uh, so that would be six. No, no, because A and B, that would mean A and B would shake hands twice until it beans. So A and B shake hands, right? Once B and C shake hands to, and A and C shake hands. So three. Yeah. So three handshakes. Okay. Okay. How about four people? Um, I'll let me diagram this out. Hold on.
00:20:26
Speaker
So what I've drawn here is two circles connected by a line. And the second thing that I've drawn right next to it is three circles, each connected by a line. So it looks like a triangle with circles at the vertices. Now I got five. I want to draw it out to double check. So now we have four. So that would be four dots, right?
00:20:49
Speaker
So what Jacob's drawn is, uh, is he's connected all the dots. So it kind of looks like a square with an X in it with circles at all four corners. Kind of like when an icon doesn't load correctly. Yes. So I messed up. It was six. Yeah. Six. How about five? So write down the number of connections under each one. So the first one would be one.
00:21:11
Speaker
Second is two. Third one is six, right? And just based on those two numbers, one, two, and six, do you want to guess what the fourth one will be? I was never good at these. Oh yeah, no worries. I'm just asking if you have any guesses. 12. All righty. And what's your rationale for that? Well, I figured at least six is going to multiply it by itself at least once. All right. So let's check it out. Let's do, so draw five circles.
00:21:41
Speaker
Uh, it would actually be a little bit easier. Um, if you made a pentagram. Yeah. Cause you'll have lines overlapping. Yeah. I made a star. Yep. And exactly. And how many it's a pentagram, right? Yeah. Yeah. It's like seven. Right. Uh, let's see. So you have one, two, three, four, five on the outside and then five on the inside. So that's 10. Yep.
00:22:07
Speaker
Like I said, I was terrible at these things. That takes a lot of practice to be good at those. It's honestly stupid that they put them on IQ tests. Yeah. I was always terrible at these. Oh yeah. I was only good at them because I've peddled puzzle books as a kid and I learned because anyway, I could say a lot about IQ, but anyway, so it's 10, right? Yeah. So let me ask you something. What's the difference between one and two? Like how many, what's what three minus one? Two. Yeah. So right. Two between those.
00:22:35
Speaker
How about between, uh, between, uh, three and six, how about between six and 10? Yeah. One, two, three, four. So what do you think the next number is going to be five? So what do you think the next number is going to be? Five. Uh, five. Yeah. So what's, uh, so what, what, what does that make? Five. Yeah. That would be 15.
00:22:57
Speaker
Yeah, exactly. So basically, do you know the general formula now? Yeah, I'm getting it. Yeah, it's 1 plus 2. It's basically, if it's 5, it's 1 plus 2 plus 3 plus 4 plus 5. Yeah. A really good way to kind of... I mean, 2 would just be 1. 3 would be 1 plus 2.
00:23:17
Speaker
Four four different people shaking hands would be one plus two plus three and so on So if you have like n people shaking hands, it's n minus one plus n minus two plus all the way to one, right? Yeah, and One way you could prove this is that let's say we have the first person, right? Then they shake everybody else's hand that the first person to shake everybody's hand They've done n minus two handshakes. I mean n minus one handshakes, right? Yes. Yeah
00:23:45
Speaker
So how about now let's say one of the people shakes hands with everybody they already shook hands with one of the people so he's gone he's out of the pool for that person to start shaking hands with and actually anybody else to start shaking hands with so this person he shakes hands with n-2 people right and so on yeah that makes sense the next person would be n-3
00:24:06
Speaker
Uh, yeah. And so on. Yeah. Until you go all to get all the way to the last person who has only one unique handshake. Exactly. Uh, that's why when, you know, you have a group of people clinking glasses cheering, it could be so many clinks when you just have a few people. Yeah. So now we're going to talk about the n choose K function is sometimes called also just the numbers in them called binomial coefficients. So let's say.
00:24:32
Speaker
Let's say we had a collection of Snow White and the Seven Dwarves, right? Yes. So that's eight people total. Yes. How many ways can we select three people out of those eight?
00:24:46
Speaker
three unique sets or just three people or just sets of three people, just a three, three, just one set of three. How many unique ways can we select three people? So how many unique groups of three people can we select from this other group of 10 people or eight people? I mean, eight. Yeah. So think about it this way. It's a lot. It's like 20 some odd.
00:25:13
Speaker
So the way that we actually calculate this is called the, we use binomial coefficients. They're actually found also in the Sierpinski triangle, I mean, in the Pascal's triangle. I'm just gonna mention Pascal's triangle briefly because it makes sense in this context. So you write a one, right? So then we write two ones below that. What's one plus one? Two.
00:25:36
Speaker
two and then one plus nothing so it's an infinite amount of nothing on that side is one one so we have three rows the first is one second is one one third is one two one and it's out going outward like a triangle right yes the next one is one one three three one and then one four six four one
00:26:01
Speaker
These are actually pretty useful in several ways. For example, if you have x plus 1 quantity to the fourth, that's equal to x to the fourth plus 4x cubed plus 6x squared plus 4x plus 1.
00:26:17
Speaker
So it's useful for things like that, but it's also useful because you use binomial coefficients to calculate an arbitrary thing given the row and column. So what the binomial coefficient is, it's n factorial. So n over k, it looks like this. It's a parentheses with an n above a k with a little bit of space between them.
00:26:42
Speaker
And that is equal to n factorial divided by k factorial divided by n minus k factorial. So here, remember n is 8, right? Yes. And k is 3, right?
00:26:58
Speaker
Yes. So write that out. So n factorial would be what? 8 factorial, right? So you've written 8 factorial over 3 factorial times 8 minus 3 factorial. 8 minus 3 is? 5. So write that again, but replace that with 5 factorial. All right. So what is 8 factorial?
00:27:19
Speaker
It's 1 times 2 times 3 times, et cetera, right? So write that out. So that's equal to 1 times 2 times 3, et cetera, all the way to 8, right? And Jacob has written 1 times 2 times 3 times 4 times 5 times 6 all the way to 8, all divided by 1 times 2 times 3 times 1 times, and you put in parentheses, but you don't need it necessarily, 1 times 2 times 3 times 4 times 5. Now start crossing things off from the top and bottom that are equal. So cancel things out.
00:27:48
Speaker
So let me continue this for you just because it's a lot of fiddling. Yeah. So you start with 1 times 2 times 3 times 4 times 5 times 6 times 7 times 8 times... Oh, not times anything. All over 1 times 2 times 3 times 1 times 2 times 3 times 4 times 5. 1 times anything is just the thing itself. We could just cancel out all the 1's already.
00:28:15
Speaker
So then we have 2, 3, 4, 5 on the bottom and the top. So we're left with 6 times 7 times 8 over 2 times 3. So we know 8 divided by 2 is what? 4. So we replace the 8 on the top by 4 and get rid of the 2 on the bottom. So the only thing we have on the bottom is the 3. So one of the things on the top is a 6. So 6 divided by 3 is? 2.
Counting Infinite Sets
00:28:40
Speaker
So we replace that with a two so we're left with two times seven times four and that's equal to let's see 56 So there's 56 ways to select three people out of a group of eight So does that make sense now? Yeah. Yeah. Yeah that makes sense So now we're gonna talk about permutations a little bit Jacob write the letters a through e in a random order
00:29:10
Speaker
So you've written a C B D E. I mean, D B D B A C D B E. So below that, right. A B C D E. So, um, right there. Um, the first thing you've written is called a permutation that, you know, the, the second one. And we talked about that earlier, right? But with numbers. Yeah. And what's interesting is that permutations all have cycles within them. So to show what I mean, um, a just goes to a, right?
00:29:39
Speaker
Let's say the first row is what you start with and the second row is what you end with. Let's say you start with the second row and then you end with the first row.
00:29:50
Speaker
So what I've written in parentheses are two rows of numbers, two rows of letters, A, B, C, D, E, and A, C, D, B, E. And that's actually how you write down the permutation that you've described. So now I've just written A, B, C, D, E in a circle, right? Yes. A transforms into what? Oh, A is always A. Yeah, so we could just write a circle like that. B transforms into? C. C transforms into? D.
00:30:20
Speaker
D transforms into B and E transforms into itself. Yeah. And see how we have three cycles, two of length one and one of length three. Yes. And it turns out with any permutation, there's no way to do a permutation that doesn't have a cycle in it. Okay. Yeah. Yeah. I get what you're saying now.
00:30:38
Speaker
And just for fun, we're gonna talk about the, so there's an episode of Futurama. There's a machine that switches heads, right? Brains, I think. Yeah, brains, yeah, not heads. You got switches, brains, consciousnesses. Yeah. You could swap robots and humans and stuff, but they get all these people, but the only thing is that you can't swap back. You could swap with a new person each time, but not an older person. Yeah. So the Globetrotters do is they write this equation on the board.
00:31:05
Speaker
And so in the in the episode that board actually contained a valid proof of how to do this. So the proof is show that the problem basically would be show that you can with at least one additional person without I mean within with at least two additional people swap any group of scrambled people back to its original self. Yes.
00:31:31
Speaker
All right, and so the way that works is we start by assuming that we just have... So we consider a simple permutation. Let me write it down. So whatever written down here, you can see how it's kind of just the bottom cycle is the top cycle rotated and we have a bunch that are in the same order, right? Yes. All right, so we're just going to consider that simple permutation.
00:31:54
Speaker
So if we have two additional people, right? X and Y. What we can do is we swap X with the first person, then X with the second person, all the way with X to some random person, right? Some random person could be the K-th person, the second person, but has to be less than or equal to K, right? Yes, yes. And then you swap Y with the person next to that, and you swap that all the way to K.
00:32:22
Speaker
And then you swap X with the I plus the first one, so I is just the number that we've chosen. And then we swap Y with I, and which can swap, which swaps X and Y. So what we keep doing is, so that's how to solve that problem, right? Do you have any intuition as how to solve the general problem, given what I told you earlier about how every permutation has a certain number of cycles?
00:32:49
Speaker
Not, not exact. So what we do is permutations can be applied to themselves. So we do know as we just assume this, we do. So that was what I was doing. Yeah, exactly. You, you, you divide the last part and you, you play it to there and then you divide the last part if you need to, et cetera. Yeah. And at the end you have all the permutations. Uh, and then if the globe trotters are swapped, uh, you swap them back. Exactly.
00:33:16
Speaker
But it's that's the original proof. There is actually a simpler proof that some I'm not simpler, but a proof that somebody came up with that only needs one globetrotter. No, I mean, it kind of makes sense if you guess if you put him in the rotation at the right point. Yeah, I feel like it must be a really long sequence of moves, but perhaps not. Yeah, I'm not exactly. Yeah.
00:33:41
Speaker
It's interesting though that it exists. And so, so I mean, honestly, like if somebody out there could solve the most efficient way to do this, that would actually probably be like, you know, something you could publish. All right. Just in case we ever have the technology.
00:33:58
Speaker
So partial permutation just really quickly is like so let's say that that in when we were selecting the Snow White and the Seven Dwarves let's say that the order that we selected a matter so not just the which three they are but which order we select them in yeah the the formula for that would just be n divided by k so 8 factorial divided by 3 factorial which turns out to be which is 6728 ways to select three people in a unique order from those eight
00:34:27
Speaker
and you were asking me to do that in my head. Oh, no, I was going to. You were asking me to do that in my head, Sophia. I I plead the knot.
00:34:39
Speaker
All right. So now we're going to talk real quick about. OK, so now let's talk about the problem. We have eight people, right? Yes. How many ways can we select groups of people from that eight? Any number of groups. So including zero. So all of them could be a group. Sleepy Snow White and don't be a group. I mean, I feel like you have to multiply everything.
00:35:07
Speaker
Well, it turns out that, so think about it in terms of binary. If you had binary numbers, so 0, 0, 0, 0, 0, 0, 0, right, would be like none of the people. Yes. Like 0, 1, 0, 1, 0, 1 would be like such and such person. Like if you had them in an order, right? Yeah. So it would be the number of binary numbers. So it turns out that, so let's go back to powers of 10 for a second, right? If I have two digits of power 10, how many numbers can you write with two digits?
00:35:34
Speaker
90 100. Can you do 100? Yeah. Cause you could go zero zero through 99. Oh yeah. Yeah. Oh, okay. Well, okay. So here's the thing. Oh, you thought they had to be different in my, well, in my head, I wasn't counting zero one as being two digit numbers. Oh, I get you. I was like, one is like, that's a single digit number. Two's a single. Okay. So I get you. So that would have been correct. But yeah, I mean, including zero. No, no, that makes way. Yeah. Yeah. You can get to have a hundred.
00:36:03
Speaker
And if you had four digits, it would be 10,000, right? Yes. And so it's basically 10 to the number of digits, right? Yeah. So in base two, it would be two to the number of digits, right? Yes. So that'd be equal to two to the N.
00:36:16
Speaker
So two to the eighth, and this? Yep, which is equal to 256 different ways that you can select some group of these dwarfs from each other. And what's interesting about that is that that means that if you add up the number of ways that you can select zero, so n factorial divided by zero factorial times n factorial, plus n factorial over one factorial times n minus one factorial,
00:36:44
Speaker
all the way up, you get two to the end. Which I think is kind of a fascinating thing. Yes, that is interesting.
00:36:54
Speaker
All right. So now we're going to talk about counting to infinity. So exactly. Uh, so we could actually conceive of an infinite set of objects, right? So let's say we're all the counting. So let's say we have the numbers like, you know, zero through infinity, zero, one, two, three, all the, all the whole numbers, not, not like 0.5 or 1.7, right? So like just whole numbers. So, uh, so yeah, let's say we have all the numbers from zero to the whole numbers, right?
00:37:21
Speaker
Yes, yes. Would there would there be fewer even numbers than there are whole numbers like fewer even integers and there are integers? I mean, yes, because if there if we're combining the evens and odds.
00:37:38
Speaker
then. So that's the, so let's, so, okay. Are you, are, oh, I'm sorry. You're saying home. I'm wait. No, no, no. That you understood what I meant. Okay. I did understand. So yeah. Yeah. So let's, let's go about seeing if that's, if that's right. Right. I mean, just if you're including the odd numbers, it makes sense to me that'd be more than just the even. Oh yeah. Right. That that's what that's my intuition too. Honestly. Yeah. Um, so what's interesting. So let's say it's infinity we're dealing with. Oh yeah. Exactly. So things sometimes get weird.
00:38:07
Speaker
So let's say we have a set 1, 2, 3 and set A, B, C, right? If I drew a line between A, 1 and A, B and 2 and C and 3, that's called a complete bijection because it means that each thing has exactly one thing assigned to it.
00:38:23
Speaker
And that means that these have the same number of items in the set, right? That's actually a way of mathematically showing that two things have the same number of items is if you could create a function that transforms one completely into the other. So let me ask you this, what's one times two? Two. Three times two. Six. Four times two.
00:38:50
Speaker
8. And those are all even numbers, right? Yes. So that is that in that function, can 8 divided by 2 be anything but 4? Wait, I'm sorry, can 8 be anything but 4? Can 8 divided by 2 be anything but 4? No. Yeah, exactly. So it's a it's a complete bijection, right? Each thing has exactly one thing and vice versa. So so what that means is that bizarrely, the number of even numbers and the number of whole numbers are actually exactly the same.
00:39:19
Speaker
Yeah. Because we're dealing with infinity and things get weird. Yeah. And it turns out that all countable infinite sets have the same number of objects. Yes. So that includes fractions. So there's the same number of fractions as there are numbers divisible by 10. Yes. They're the same number of fractions as there are whole numbers.
00:39:40
Speaker
Yeah. Isn't it crazy because they could, you could order it. So you do one divided by one and then two divided by, and then one divided by two and then two divided by two and then one divided by three, two divided by three, three divided by three. Then one divided by four through four divided by four and then so on. And I'm, I'm including that that's a really simple way to do it. There's better ways to do it, but, um, I am blanking on what those are right now. What can't you count?
00:40:04
Speaker
That's what I want to know. I want to know what you can't count. Are there rules against counting? Well, here's the rule to counting. The rule to counting in mathematics is that you have to have a set that can contain everything. So the set, the set can be infinitely large, but if it's a set of, it has to be a set of discrete objects to be countable. Okay.
00:40:28
Speaker
So, uh, let's consider the real numbers. So including things like PI square to two, any number that's like, you know, act like the number, a bunch of numbers and then decimal and then a bunch of numbers. Right. So do you think that this can be counted number? Is it like,
00:40:45
Speaker
is because we talked about fractions earlier. Yeah. And since decimals are just fractions, it makes sense that, uh, that the number of debt of, uh, real numbers, are you asking me, do you think I can count? Do I think we can count all the decimals out there? Well, all the decimal numbers out there. Cause remember, remember we can count all fractions. Well,
00:41:03
Speaker
No, no, not realistically, mathematically. Yeah. Mathematically. Yeah. Yeah. Because remember you found, yeah, we counted. So let's go. Let's see if that's true. Yeah. Yeah. Mathematically. Yes. Realistically. No way to count all the decimals. And what's your reasoning? Well, they go on forever. Oh, yeah. But that you can automatically because, you know, it's part of the set and therefore it or you can make it part of a set.
00:41:28
Speaker
Yeah. So you're saying you can enumerate all the exactly. And you can give them value essentially. All right. So not, we're just going to consider real numbers that are like points, whatever. So zero point, whatever. So including zero point zero, zero point, nine, nine, nine, nine, nine. Right. Yeah. So let's assume, let's say we had a set of like five numbers like that. Right.
00:41:48
Speaker
So for the first, so let's say let's consider the first decimal, the first number, right? Yeah. So point one, let's, what's a number that, so let's, so just select a number that isn't one, let's just do nine minus that, you know, so nine minus zero would be nine and zero, nine minus nine would be zero, right? Yeah. So it's nine minus one? Nine minus one would be eight. How about, okay, now let's go diagonally one. How about three, nine minus three? Would be six.
00:42:20
Speaker
Seven. Nine minus two is seven. And the next one is two again, so seven. And then it doesn't matter what you write for the last number. So is this equal to any of the numbers above? No, it is not.
00:42:35
Speaker
And can you prove actually that no matter how many, let's say we had an arbitrary number of numbers, a finite but arbitrary one, right? Let's say that, let's include zero as a digit always. So even if they terminate, they end with zeros. Can you prove that doing this diagonally will always result in a number that was not in that set? Can I prove that? Yeah, how would you go about reasoning that?
00:43:05
Speaker
Honestly, you would have to go through each set. Well, one thing, instead of having to go through each set. That's the way I would do it. Yeah, but what you could do mathematically is say that the first digit after the decimal point, right, is eight. Yes. Which means that there's at least one that isn't going to be equal to that and is the first one.
00:43:27
Speaker
There's six, which means at least one isn't going to be legal to that end, the second one, and so on. So you could always create a number by going across the diagonal and doing this process. I mean, you can even do that plus one and rotate it. It doesn't matter exactly what process, but that's how you do that. Yeah.
00:43:44
Speaker
And that's called a Cantor's diagonalization argument. So let me keep going with it. Let's say we had a set of all the, let's say we could do a set of all the real numbers, right? Let's just assume and prove that. So if we have this set of real numbers, if it's countable, this set is complete, right? You can't add anything to it that's new, right? Correct. But what if you did a diagonal thing across it? Wouldn't that create a new number?
00:44:12
Speaker
Technically yes, it definitely would and that's the thing is that it since since you can always create a new number It's uncountable. Yeah, because you can't put it into countable So you the way you have to define real numbers is through things like like the way that kauchi did it is by saying like, you know that that they're sequences of fractions and
00:44:33
Speaker
So there's sequence of fractions that trend towards something and it gets smaller and smaller. You have concepts like eventual epsilon closeness, which says that there is a certain point for any epsilon, so no matter how tiny epsilon is, than in the sequence. So let's say we have a number approximating pi.
00:44:53
Speaker
And let's say epsilon is 1 over 1,000. That means that there's going to be a certain point in the sequence that the difference between any of the guesses for pi as we go along will be less than 1 over 1,000 or less than 1 over a million or less than 1 over infinity as we go into infinity. Interesting.
00:45:09
Speaker
So that's how you define real numbers actually, is through that. It's like, you know, they're technically a set, but they're an uncountable set.
Conclusion and Listener Appreciation
00:45:17
Speaker
And then you have problems like, are there any sets that have a number between the countable and uncountables? And I think there's been a solution to that and it's like, it's both depending on how you count or something weird like that. Don't quote me on that, everybody. Let me research that.
00:45:37
Speaker
Counting is a way of making sense of the many different ways in which the world can appear, whether it's the number of apples that can appear in your basket or the number of combinations in a Rubik's Cube. There are even deeper questions to be asked about counting and countability, as we saw with the number of integers and real numbers. One thing that is for sure, though, is that as long as we're measuring anything at all, we're going to have numbers.
00:45:59
Speaker
I'm Sophia and this is Breaking Math. With me I head on Jacob Urban. Jacob, do you want to plug anything? Just, you know, you should give nerd forensics a listen if you're looking for something that is generally humorous. Some of the episodes are a little more intense, but
00:46:21
Speaker
Oh, yeah. I mean, we try to make those humorous too, but it's more of a black humor, right? Yeah. More gallows humor at that point. Um, but you know, we, we try to have fun over there.
00:46:33
Speaker
All righty, cool. And we actually did describe some more stuff about accountability in an episode about infinities earlier, so you could probably give that a listen. But yeah, it's good to be back doing breaking math. We'll keep you updated on the new merchandise. Drink Magic Mind, and I don't know. Any last thoughts?
00:47:01
Speaker
worship your math. I don't know. That could lead to problems.