Let's Find Out - The Maths of Euler: the Greatest Mathematician (part 2) | feat. Decaf-Math ASMR

Episode Date: May 1, 2019

Part 1 (Euler's biography): https://www.youtube.com/watch?v=R6fIoxjMwDE After googling the greatest mathematicians, I noticed Euler consistently at the top. Last time we peered into his life, now we dive into some of his most revolutionary discoveries, in both math and physics. I'm glad our friend Decaf-Math ASMR was available when we got stuck half way through. Consider visiting her channel if you enjoy math and ASMR together. https://www.youtube.com/channel/UCGjS9qk4gcGknMnySmPI0Og Thanks for watching. #ASMR #Math #Science

Sliberts say swordfish? Yeah, I did. We're talking about swordfish. We're talking about how scientists use oilers constant to guess their age based on the certain characteristics of their bodies, length, and width. Such things like that. You go to your office? My brother is really, really good at swordfish, and so we eat a lot of fresh swordfish. while we were eating last night

I was wondering if he knows about how long they live and so his wife actually looked it up and found a scientific paper and she read that they live almost up to

16 years which is kind of like eons fish she handed me the phone because it was a scientific research paper and she saw some equations So, I'm going to show you what they were real quick. So, K is just an empirical, or, it's an empirical constant that is arrived at.

It's kind of the best number that fits in this equation to make it so that this number fits this number fits the actual. fits the actual experimentally observed data. So if we just draw this really quick, so it's just the current age. Kay is a constant that is determined experimentally so that this equation fits the data. This, so it's the kind of the limiting factor. This inside here, because it's a positive number, so this is 2.7.1, but 0, between 0 and 1, less than 1 times a number,

will always give you a number less. You're multiplying it. Three constants. Let's take the natural log, just one. We're studying a mathematician. His mathematics to real-world situation. I don't always, I don't always know if that is always the case. Very abstract, complex processes, such as the famous rhyming vectors. Einstein ended up heavily, fundamentally, to verify his equations of relativity. But they weren't seemingly, practically feature. It's space that doesn't follow the rules that Euclid laid out 2,000 plus years ago. Perfectly on Euclidean space.

If you're at the North Pole and you go in a straight line, so looking from overhead, for instance, if you went this way from the North Pole, you turned exactly 90 degrees, and you started walking that way, you would also, That just means that about this. It's 90 degrees here. Go down and you're at 90 degrees relative to the equator or the line intersecting both these two points.

A triangle with three seemingly straight lines, three 90 degree angles, connecting them. Non-Euclidean analogy. Would that would be that if this was the equator and you're at two different spots and you go straight north

these two lines in flat space the earth was not curved would never intersect but because the earth is curved and remember this is and it means that and this is exaggerated but it was proven in 1919 with a solar eclipse

Getting to, in this video, is that ridiculous number of physics. Maybe he was, and I just didn't realize it, in my little research. It helps give an understanding in a context. Algebra, do him justice here, but I will talk about at least a handful. He introduced and popularized several notational conventions and concepts. that we all know today. Perhaps the most familiar. He introduced the concept

because M and B are traditionally just numbers that we already know the value of, whereas X is meant to represent a constantly changing... I shouldn't use constant. Set of number effect is just a... Sigma summation. It just represents...

Well, actually, especially if you stopped taking math in high school. The gene symbols that you can thank Oiler for that. And believe it or not, he genuinely made it much more clear, indistinct, and easy. Manipulate such symbols. So, that's just the tip of the iceberg, though. Something you stem majors, taking the upper level, mathematics. might recognize from your shortcuts to solving differential equations is oilers famous

times a x and y axes seemingly innocuous conversion you i looked it up you wouldn't be able to we wouldn't have electronics rockets stuff like your your ac thermostat we wouldn't understand and how these things worked and created them. I mean, radiation therapy, modern chemistry even, these might still be waiting to be discovered. So you're probably starting to think, like, all right, this is cool. These are used.

We'll see if she can enlighten us a little bit as to how oil her. Oh, hi, Rich. Oilers formula, huh? Yeah, let's see if we can improve it. I was actually just taking a look at it. Alright, I'll see ya. So let's take a look at this beautiful, beautiful equation. E to the i pi plus 1 equals 0.

So simple yet so complex, huh? So one thing I'm going to start with is I'm going to move this plus 1 over to the other side. Just using a little bit of algebra. So e to the i pi then equals negative minus. minus 1 and negative 1, right? So we get e to the i pi equals negative 1. And the reason I wanted to do that is because I'll realize that that is a special case of this formula, e to the i x equals cosine x plus i sine x.

I see here that I have e to the i x and then I have e to the i pi, which kind of hints at the fact that so if I plug in pi for x in this entire equation, plus I want it, don't worry, I'll just do all the math today, but cosine pi is negative 1, is the x coordinate because it's cosine, the x coordinate of the point that corresponds to the angle of the point. And then sign pi is actually 0 because it's the y coordinate, so 0 times i is still going to be 0, and negative 1 plus 0 is negative 1.

So E to the i pi equals negative 1. So this isn't really a... So the goal will be to simply prove this thing. So there are many ways to actually prove this. So today I'm just going to go over two of my favorite, cute and fun. They both rely heavily on calculus and calculus concepts. So I kind of feel like this is where I'm going to lose people or people who are really interested or loctucus, might find this fascinating.

But either way, I hope that this is relaxing and fun. So about the Taylor series already, so this is the first way. I'm going to expand this, this, and this. And so either the I, x, cosine x, and sine x. and with Taylor series you can basically just expand a function into like around a point. So the Taylor series about a point is F of X is F of A. So F of the value of the function at the point A plus F prime of A times X minus A. And those two terms make the linear approximation.

But you can keep going. So plus the second derivative at A times X minus A. x minus a squared over two factorial plus dutaband a function. And specifically, if you set A equals zero, you get the McCorin series. So, and I already wrote the Macoran series for E of the X, cosine x, and sine x. And sort of intuitively, we're just plugging in each of these functions into this thing. So intuitively, E to the X, we know that the derivative of U of X is E to the X, and so the second derivative is U.S. And it's always going to be E to the X.

But then at A equals 0, so E to the A, so E to the 0, is 1. So all of these coefficients are going to be 1, and then everything else goes away. So X squared over 2 factorial x cubed over 3.5. For cosine and sine, it's actually not too bad. It's actually really cool, because if you were really, Remember the derivatives of these trig functions, for instance, cosine x. The derivative of cosine x is negative sine x. The derivative negative sine x is negative cosine x.

The derivative of negative cosinex is sinex, and the derivative of sinex is back to cosine x. So it's cyclical. So if you plug in A equals zero into those, to those you can notice that all of the sign terms are going to go away. 0 is 0, so negative sign 0 is 0 still. So you're left with like this every other pattern and cosine of 0 is 1 and negative cosine of 0 is going to be negative 1. So you get this negative, positive, negative, positive negative with every other term. So that's why it's all the even. And so sign x has the same cyclical pattern for the derivatives and but it's just it's the opposite of the cosine where you're doing the the odd part.

So with that, I'm just going to this E to the X. I'm going to start on one side. So I have E to the IX, but I'm actually going to start by writing out the one for E to the X. So I'm just copying this part. And then what I'm going to do is put in instead of X, so anywhere I have X here, I'm going to put I. I could have just written like E to the Y, or not Y, because that's going to be confusing with F of X, but you know, E to something. So if you imagine that every time I have an X here, since I have E to the IX, I'm just going to put in IX instead. Okay, so E to the IX equal.

And now I just have to expand all of these exponents here because I have I X squared. So I'm going to expand that just using our rules for exponents. So ix squared is i squared x squared plus i cubed x cubed over 3 factorial. We actually recently talked about the cycles of these exponents in our last hangout session. So feel free to check that out and kind of review. but we have one, so I'm just going to simplify the i squared and the i cubed parts and the pattern goes 1 i negative 1 because i squared is negative 1 and then i cubed would be that negative 1 times another i so negative i and then 1 and then back to i so it's every 4 so i have 1 plus i x x to the 4th of the 4 factor i so i'm just going to do you know, just one extra term so you can see somewhat of a pattern.

And so that's great and all, but who cares, right? So actually, if we notice which terms have eyes and which terms don't, if we group this, we'll actually see that we'll have a real part and an imaginary part. So let's see what we have here. And so I'm just going to group those together and just move them because I can, because addition is commutative, so we can just regroup. So we have 1 minus x squared over 2 factorial plus. So I think you get the parentheses around that to group those together.

And then for the I terms, I'm going to group those together, but I'm just going to go ahead and factor out an I, just plus. So I'm going to factor out that I. And do we see where we're going with this? This is so cool. This 1-9x squared over 2 factorial, blah, blah, blah. It's actually the expansion for cosine x. So basically, this is cosine x.

We still have our i, but this inside part is sine x, so plus i, sine x. So we have cosine x plus. So another really cute way to do it. is also based on calculus but not quite so complex. If we let, if arbitrary, you can kind of see this mimicking this, but why we do this is because if we take the derivative, the first derivative, you get zero.

So the first part and the second part of the product will actually cancel to give you zero. And why that's significant is if we zero, always zero, then same everywhere so I can pick any arbitrary point. So conveniently, I am going to pick x equals 0 and see what we get. So f0 is, if I just plug in 0 for all the xes, I get e to the negative i times 0. So e.0 times cost 0 plus i sine 0. and E to the 0 is 1, cosine 0 is 1, sine 0 times I is still 0, so 1 times 1 is 1

So f of 0 is 1, which from our derivative here we see that f of x equals 1. Always. Since we picked f of x like this, algebraically, if you are you, if you are, we picked f of x like this, If I move this first part over here, I get 1 over E to the negative IX. E to the positive IX. And we don't have to worry about dividing by zero because E to the stuff. Either this stuff is never going to be zero. So we get E to the I, like the positive IX.

Isn't that cute? I like this one too. It's really cute. So that just comes from, you know, recognizing that the derivative of this. It's really cool. But you have to kind of be clever in setting what your F of X is. And I'm not sure that I would be able to just do that derivative in my head and realize, oh, wow, it's zero. Because you have to do the product wall.

You have to do the chain wall. So that's kind of a mess. But I hope that was helpful. I hope this was fun. It's definitely fun for me. And I will see you around soon. say hello to Molly and the manager for me. I'm sending you all my love.

Take care. And if you ever need some math help, just give me a ring, and I will see you around. Bye. No, I never really realized there's so many different. It looks like he had a... Looks like this thing is the focal point. Builder series, our series, physicists,

areas of delved into integration with a prototype of a field called modern complex analysis. And he did calculus of variations. Well, handedly introduced a new field of study, analytic number theory too, by fusing disparate branches of math. And in the process, created the theory of hypergeometric series, as well as the Q series, hyperbolic trigonometric functions. This guy was clearly the Elon Musk. Didn't stop there. He made enduring, enduring contributions to the fields of number theory,

graph theory, applied mathematics, physics, astronomy, logic, and even music. We're interested in delving a little deeper. Into number theory, graph theory, physics, astronomy, logic, and music. different sections that I'll briefly talk a number

theory. He developed some of Pierre de Fermas. Not his last theorem, but some of his ideas and disproved some of his less famous conjectures. He contributed significantly to the theory of perfect

which had fascinated mathematicians a positive integer that is equal to the sum of its proper positive divisors. And what that means is that it's, you take a number. Let's do a simple, not for example. Six, not specific to as Euclid. But Euclid also proved a formulation rule where he had a formula that worked for,

even that work to give even perfect numbers so for instance let's see his formula was Q even is a what's called a a mercene thousands of years ago discover it's gonna pop out perfect numbers that whose divisors other than itself add up to itself and this is one of those weird examples of consistency in mathematics that we don't necessarily see in nature yet. Like rhyming tensors, a genius like Einstein may come along one day. Revolutionize our understanding of the physical, maybe mental, maybe they're one in the same world. So, as an example, 7 is a Mercing prime. When we plug 7 into this, we get 6.

We can show that 7 can be derived from this formula. We do, instead of this, see, we do. Let's use the prime number 3. Well, 2 to the power 3 is 8, minus 1. one is indeed seven. Nothing really spectacular. There are three. We get this. 31 Q is after a little bit of tedious adding up, factoring and then adding up 32 divided by that quantity divided by two. Plug this into Google, maybe Wolfram Alpha, but the factors of 496, 300 years ago.

So what Euler did on top of this, this is all what Euclid had already proved, proved that all even perfect numbers are related to Mercing primes. By multiplying a Mercing prime, by a number of one less power, a prime exponent that created the mercy. prime. So written simply every even perfect number can be found by this equation to the power of

n. That quantity, 2 to the power of end. Please forgive me. But 1772 Euler had proved that to the power of 31 minus 1. No, sorry, 2 billion, 147 million, 480,000.480,000.000. 480,000.000. 480,000. thousand and six mercing prime we plugged it into this formula we would get the even perfect number of 0 580s zero scientific notation here because it's such a large number times 10 to mercine this one remain the largest number 67 so that's just one one a lot incidentally had asked about the graph theory and so here we are so Euler came up what's called the seven bridges of Konigsburg it's considered to be the first 1535 he presented the solution to the problem of we have seven total bridges connecting it and we represent that with this node or this dot right here similarly the This node has one, two, three to the bottom by two, which is also connected to the right-hand bit.

I'm not sure if the river ever can be represented like that. So we, uh, the whole point of the problem was to show that logistically you can't take any path that will lead you over one bridge only once. and return to your starting point. So we can try it to go here and not go on that bridge. Or if I did, I would end up here. And I've already went four bridges. He essentially founded what's called graph theory. Logical refutation.

There's no such thing studying Uranus successfully, mind you. Many of it, he also applied these techniques to celestial problems. Mind you, there's a paper I looked up that literally was titled, Euler the Blind Astronomer, which is a pretty bold occupation to be a key figure in when it relies so heavily upon. So he, his astronomy's work in astronomy, he applied those, some of his mathematics to celestial. bodies and orbits. His accomplishments include determining with great accuracy, the seven bridge number of the orbits, the eccentricity of the orbit, which means how much it deviates from a circle. It's how elliptic it is. His equations were riding on the tailcoats of Newton's, very recently in his time established

confirmed equations that ridiculously helped sailors, voyagers, adventures, expeditioners. Anybody voyaging across the ocean in its minor deviation, small inaccuracies, if the theory would lead to huge

errors. And if you're looking for a little mile, 30 miles, you know, maybe 30 miles, would have been within the sights of some of the better telescopes, theory of light in the optics, which was when the prevailing in the 40s paper helped ensure that the wave light, and if this observed it, and this is another example of extrapolation, eventually being able to correlate, not to find, but describe very accurately, a real world what's called

Invisite flow The zero viscosity Which means resistance Of itself to Yeah really resistance Of itself To begin motion Which means resistance

to motion really Just think of the exact opposite Of a cold, thick maple syrup relatively features, it will actually climb up outside of it. So this is, it's really crazy, it's just a fluid dripping down the outside of a container that's nowhere today. Super fluid. The physical aspects of water that was observable, and he took it to its extreme.

It's mathematically logical extreme and said, If the amount of viscosity that syrup or water has was eliminated with this superfluid look like his equation's small but interesting contribution to the field of logic that Euler made. And it's the Venn diagram that we know. It's like we had. So Euler is credited with using closed curves to represent syllogistic reasoning. And what this means essentially is that in a Venn diagram is actually a variance of what Euler is purported to have invented to illustrate logic and reasoning. So for instance, diagram where this channel might be, we have a separate.

subject. We have a set, a classification, ASMR. Inside that, you have many different types of things stimuli that produce more trigger ASMR tribes, if not the originator. And then lastly, the last thing, I left a nice little spot for, is his contribution to to music theory. Hoyler's approach with music, of course, is mainly mathematical. His writings on music are not particularly numerous.

A few hundred pages in his total production. About 30,000 pages. Possible divisions of the octave using the prime numbers 3 and 5 and describes 18 such. genres. With a general definition, 2M, the sum, as long as the sounds are perceptible. So he's really having a scientific theory, octaves in the sense of perceptible patterns of waves,

of frequencies that we hear. Yeah, I'm really not understanding this at all, but essentially he says, you have your octave and then you have a third divided into their thirons and their fifth there because I'd be long if I'd be lying if I'd try to explain any further than that because that really doesn't even make sense to me

but essentially it was some interesting diagram in what really caught my eye about it so if you are familiar with Music theory? Maybe you can help us out. Really, really make sense to anybody familiar with music theory. It's way out of my scope. It's a about does it that we can clearly see from his discoveries in this odd realm we call mathematics. Maybe because of his closeness to the limits,

he was a devout Christian and a believer in biblical, in errant, who wrote apologetics and argued forcefully against the prominent atheists. Leave the Bible to be inspired, and I have no doubt that he acted out this belief with each tireless day in the living sea of mathematics with its clear medium of logic. Perhaps he even dared to dive deeper than those that hadn't been stricken with blindness, consequently foregoing the paralysis to those.