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P5: All Your Base Are Belong to Us (Fractional Base Proof)
416 Plays4 years ago
Join Sofia and Gabriel on this problem episode where we explore "base 3-to-2" — a base system we explored on the last podcast — and how it relates to "base 3/2" from last episode.
[Featuring: Sofía Baca; Gabriel Hesch]
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Transcript
Introduction & Patreon Plug
00:00:06
Speaker
I'm Sophia. And I'm Gabriel. And you're listening to a Breaking Math Problem episode all about bases and specifically fractional bases. Before we continue, we'd like to go and really quickly plug our Patreon page. We love all of our Patreons. We love your support. Thank you so much. There's several there's several tiers at our Patreon page if you'd like to support us.
00:00:24
Speaker
Oh yeah, and that's patreon.com slash Breaking Math. Any tier will get you free ad-free, I mean not free, the $1 ad or more ad-free episodes of Breaking Math and also the outlines for the episodes.
Merchandise & Pricing Humor
00:00:36
Speaker
We also have a $35 tier where you could get the poster we have on Tensorus. But if you want, you could get that from our Facebook poster store at facebook.com slash Breaking Math podcast and just click on store.
00:00:47
Speaker
Yeah, it's nice that we have that poster or rather our merchandise is available through multiple outlets. I think with Patreon, it requires a monthly contribution. And the reason why it requires that is actually built in. It's not anything that we did, but on Facebook, it's a one-time purchase. So there's that. Yeah, we did not realize that for a very long time. Yes, and it is $15.15. And what's the significance of that dollar amount? I believe it's pi to the E. Yeah, we'll have to. The E to the pi.
00:01:14
Speaker
One of those, we're having fun with crazy numbers.
Social Media & Website Content
00:01:18
Speaker
When you talk about the most significant numbers in math and physics, there's E and there's pi, and I suppose zero, one, and infinity, but if you take anything... Yeah, it's E to the E. Okay, if you take anything to the zero power, then it just turns into one, or infinity, then it just turns into infinity, so yeah. It's E to the E. Now, if you want to follow us, we are also available on social media, Facebook, at, of course,
00:01:42
Speaker
facebook.com slash breaking math podcast as well as twitter and it's just at breaking math pod also there's a website uh the website is interesting it's uh breaking math podcast dot a pp at the website you'll find our first three um is it out no we didn't outline for our first episode but we haven't well we had yeah we had an outline for our first episode
00:02:07
Speaker
We also have One on Reality is More than Complex. We have the papers available that were made by Alex Alanis. Yep, who I do another podcast as well, or rather a YouTube show
Episode Theme & Nostalgic Memes
00:02:16
Speaker
with. And the website also has some really cool applets that we always talk about, like the Hypercube applet, as well as the Foyer series applet. And what else? There's a Game of Life on a Klein bottle. That's pretty cool.
00:02:30
Speaker
Yeah, yeah, it is very cool. Okay, so today's episode is all your bases are belong to us. No, actually, it's just called all your bases. What is that from the whole all your bases are belong to us? That does a blast from the past. Is that a 10 year old? All your base are belong to us is like a meme from early 2000s. It was some RPG.
00:02:55
Speaker
One of the things that I love this day and age is we have the older millennials who were born in the early 80s and we still understand memes and we have memes from the internet, you know, the early days of the internet. And of course the generation, you know, before us, including baby boomers, were well into their adulthood. Yeah, that is a blast from the past meme back in the day. No, it's current, just, I'm just kidding. No, it's not current anyway.
00:03:23
Speaker
Yep, pretty soon. I'm sure there'll be a meme museum officer. I'm sure that thing exists already, right?
Exploring Base Three to Two
00:03:29
Speaker
I mean there's a know your meme calm
00:03:32
Speaker
Yeah, and I'm sure that they've got stats on memes as well, including things like frequency of use over time and some memes last longer than others. Yeah, all that actually on Know Your Meme. That would be, oh man, how about a math of Know Your Meme and see if there's any, you know, if one can extrapolate some really rigorous academic data from Know Your Meme. That'd be a great episode, I think.
00:03:56
Speaker
Nonetheless, today's episode, of course, is based on that old meme. All your base are belong to us. Yeah. And basically, in our last episode, it's episode negative two zero one. We talked about this base called base three to two.
00:04:12
Speaker
And base 3 to 2 is, to explain what that means just really quick, and if this doesn't make sense, just listen to the episode. It means that usually we have base 10 to 1, where we count up to 10, and then when we reach 10 in our digit spot, we augment the place next to it by 1. So like 19 goes to 20, 99 goes to 10, 9, which goes to 100, things like that.
00:04:36
Speaker
Of course, I was not on that episode. I was kicked out. You guys were. Yeah, I'm just kidding. No, yeah. So like three to two is like... So it's counting three to two. It's one, two, and then 20. Two, one, two, two, two, one, zero. And then it just keeps going like that where you replace three with two.
00:05:02
Speaker
And the problem that we're going to do, if I may be so bold as to go ahead and read the problem out loud, why is it true that the three to two base is base three over two? So this will be a little bit of a short episode. And the reason why is because the problem isn't not that big, but it is one of the problems from the last episode that seemed a little unintuitive. Yeah, it's still a very significant one. Again, this is talking about a base conversion, but a very significant one. We are comparing two bases. Yeah, which turned out to be the same base. Spoiler alert.
00:05:30
Speaker
Yes, exactly. Now the bases are, I don't know if you want to say this or not, but the bases are three to the two and the three halves. But I want to make sure if you listen to the last episode, they'll understand why three to the two, technically speaking, from its definition is not necessarily three halves. Yeah, and it's not base three halves necessarily because it uses a different set of digits, zero, one and two versus just zero and one.
00:05:52
Speaker
However, because it's a type of base three halves, that's one thing that we're proving in this episode. Yeah, so this is a really nuanced thing in math. Math is full of nuanced things that shed light on lots of other areas of mathematics.
Mathematical Proofs & Induction
00:06:05
Speaker
So we are interested in the nuanced proofs and in the subtle details that make math beautiful. For example, A in base three to two is two, one, two.
00:06:16
Speaker
But if you take 2 and multiply by 1.5 times 1.5, you add 1 times 1.5 and then add 2 to that, you get 8. So for some reason they're the same number and I just thought it'd be fun to explore that on a problem episode. So, induction, we've done this many times on the show and I believe in problem episodes before too.
00:06:33
Speaker
Okay, this is a really, really good example of induction. I love this because induction is used very, very frequently. So again, to say what it is that we're about to prove, the proof with dominoes is that if you can topple the first domino, then all of the dominoes following that will topple as well. So basically the way that it works is that you topple the first domino, right? Yes. And then that one falls over. So we know that the first domino falls. Yes.
00:07:00
Speaker
So let's assume every domino up to a certain domino has fallen. Okay. The next domino will inevitably fall. Yes. So therefore, based on those two facts, we know that every domino has fallen. Okay. That's the inductive step. Yes. And it's weirdly an entire step in its own right. It's kind of like the parallel postulate where it's something that it's kind of like, why do we need to have a rule just for this?
00:07:23
Speaker
Yeah, because in proofs, you have some things that seem completely unnecessary, but they absolutely are necessary. Things like the identity, like i equals i. There's some things that seem rather obvious, but you still have to state them if you're going to be mathematically rigorous. Yeah, but not only that in this situation, but it's like induction itself cannot be proven using anything else.
00:07:46
Speaker
How do you like that? That's crazy. You're just going to have to accept it. One of the things you have to accept, that's also pretty crazy as well.
00:07:59
Speaker
So first inductive step is that we have to first prove the first step, right? Yes. So one is one in base three halves, right? Yes. And we're done because one is also one in base three over two. Yes. So that's all we need to do with that step. Okay. Very good. So now suppose we have some number, right? It's in base three over three to two and base three halves, and they're the same number, right? Yes. And then we augment it. Okay. If the last digit is a zero or a one, then it just gets augmented to one or a two, right? Yes.
00:08:29
Speaker
However, if we need to resolve the digit, then we take three from the first place and augment two in the next place, right? Well, we can also generalize this step because sometimes you have to do that more than once. So let's say that we're in the nth place of the digit, right? So let's say we have our two digits, right? So we have two digits, we have D, and right next to it is E.
00:08:50
Speaker
Yes. E is a higher place than D. Yes. D represents, in this case, since it's in base three halves and base three to two, right? It represents D times three halves to the N for power, for sub N, right? And then E represents E times three halves to the N plus first power.
00:09:06
Speaker
Now our inductive step is that our resolving step is that we have to subtract 3 from d, right? Yes. So d minus 3, let's call that d prime. So d prime would represent d minus 3 times 3 halves to the nth power.
00:09:21
Speaker
Now we add two to E resulting in some number E prime where E prime represents E plus two to the three halves I mean times three halves to the n plus one.
Equivalence & Properties of Bases
00:09:33
Speaker
And I will point out in mathematics you have a lot of ambiguities and in this case E does not necessarily mean the number E it just means a number following D.
00:09:42
Speaker
Oh yeah, in math, any symbol means like 80 things. Yes. Yeah, exactly. So I didn't want to confuse this. So now before we resolved it, we have D plus E, right? Yes. And we're going to call that the digit and what they represent, I mean. So what D plus E represent together is D times three halves to the nth plus E times three halves to the n plus first power. But D prime plus E prime is D minus three times three halves to the n plus E plus two times three halves to the n plus one.
00:10:11
Speaker
Yes, by the way, I will say this is a lot easier to look at a script than to just hear this with the audible part. Yeah, so just following along with this, and I think the bigger thing in here is just to see how it shakes out, really. Yes. So when we factor it down, we get a mess of algebra. Are you all ready for this? Brace yourselves. Here it comes. D times 3 halves to the n minus 3 times 3 halves to the n plus E times 3 halves to the n plus 2 times 3 halves to the n plus 1.
00:10:40
Speaker
Whoo, goodness made it through it. And all we have to do is subtract that D prime plus E prime, what they represent from D plus E to see if there's been any difference. And it turns out that there's actually no difference between the two. Yeah, exactly. Zero is this difference between D prime plus E prime and D plus E. Oh, okay. Very good. Very good.
00:11:00
Speaker
So therefore, since 1 is 1 in base 3 to 2 and base 3 halves, and if we have any number that's already in base 3 to 2 and base 3 halves, if we augment it by 1, we'll get the same number back. Since we know those two things, we know that base 3 to 2 is the type of base 3 halves where you use the symbols less than 3, 0, 1, and 2. Now the reason why we bring this up is because
00:11:22
Speaker
Bases have a lot of interesting properties. I mean, you can think of them as being long polynomials, and we have this proof. There's another proof out there showing that any real number can be written in infinite number of different ways using integers.
Debate: Does 0.9 Repeating Equal 1?
00:11:39
Speaker
If you don't limit the kind of digits you have, I mean, technically in base 10, we could have like, I mean, what is it in the Hobbit or something? They have their 11D first birthday? Yes.
00:11:49
Speaker
Yeah. So you can do this stuff. It's a little bit of abuse of a notation, but it turns out that in this base three to two system, like negative numbers get really weird. Like it turns out that negative numbers go on forever. Wait a minute. Don't negative numbers go on forever in our base 10? Oh, sorry. I meant like not, not there's an infinite amount, but like negative two has an infinitely long representation.
00:12:09
Speaker
Okay. So negative two, when you say infinite, are you saying it's irrational? No, it just has an infinite number of digits in it. It's like, it's like, okay. So for example, 0.1 in this type of base, if it's base 10, 10 to one would be like 9, 9, 9, 9, 9, 9, 9, 9, 9 from infinity all the way to the first digit. Okay. Okay. Got it. It's really weird.
00:12:27
Speaker
Yeah, by the way, do you remember that there was a friend of the show from the earlier episodes, Zach, remember Zach from the physics? Oh yeah. He does not agree that 0.9 repeated to infinity does equal to one. He said he will go to his grave fighting that. Well, I don't agree that he'll die, but, you know. Yeah. Okay. There you go. Oh, shots fired. I'll have to hear. Oh yeah. Yeah. Wait for his response.