e (Euler's Number) is seriously everywhere | The strange times it shows up and why it's so important

this video was sponsored by brilliant 2.71828 1 and so on this is e or Euler's number and I remember when I learned this in high school I thought when would that even show up in the real world but now I realize the better question is when doesn't that show up or at least that feels like a fair question now mathematically he is just what you get when you calculate 1 plus 1 over a really big number for that same really big number and as that number gets bigger and bigger you get Oilers number now if you're in high school and just learn about this it's probably because you're in the compounding interest part of the curriculum which is where I'm gonna begin so let me first try to intuitively explain what this seemingly random number really is imagine you put $1.00 in a bank that pays out 100 percent interest per year that means after one year you'll have two dollars but that's only if the interest compounds once a year if instead it compounds twice per year and that means you get 50 percent after 6 months and another 50% after 6 more months so after that first 6 months you'd have a dollar and 50 Cent's and after another 6 months you get 50% of that which would leave you with two dollars and 25 cents a little more than before if instead the interest compounded four times per year then you get 25% every three months the first payment would leave you with a dollar twenty-five then 25% of that leaves you with a dollar 56 another 25% gives you 195 and the final 25% gives you about two dollars and 43 cents so notice that the final value keeps going up if the interest compounded daily we'd end up with a little more than two point seven one dollars and as we keep going up to compounding every our second microsecond nanosecond and so on the amount in your bank would become $1 after a year now that's what we learned in high school at the minimum but now let's see the other times the eeesh owes up across math and science in ways that usually have nothing to do with compounding interest and I'm gonna start with some more random examples the first one being probability let's say there's a one-in-a-million chance of winning the lottery if you then to play the lottery a million times your odds of losing every time is about over e or 36.8% well approximately and this will always be the case for games with small probability the reason for this is because a one-in-a-million chance of winning means you have a point nine nine nine nine nine nine chance of losing or one minus one over a million raise that to the one million and those are your odds of losing every single time and as this value gets very large since we have a minus sign this time the whole thing will approach one over e the next step imagine a party where everyone brings an umbrella after the party's over people are in a rush and everyone grabs a random umbrella from the stack not caring if it's theirs or not the probability that no one left with their same umbrella is again about one over e if the number of guests is pretty large another way to look at this is take a fresh deck of cards that's in proper order and shuffle it if you do so randomly there's about a 1 over e chance that no card is in its original spot next up let's say you interview a hundred people for a secretary job and you judge them all by something like typing speed that can be given a value now after each interview you can either hire them or never see them again and you're only going to pick one person so how would you pick the optimal candidate because with each interview even if the person has a fast typing speed you never know if there's someone better that you just haven't interviewed yet well the math says if you have 100 interviews lined up first interview a hundred over a of them or about 37 people however you do not pick any of those first 37 you just keep track of who is the best you will then pick the next person who is better than any of those first 37 people and not only does a show up in the number of people you interview before making a decision but if you follow this algorithm you have a 1 over e chance of picking the best one and this probability stays the same no matter how many people you have to interview so even if you have a million interviews lined up just interview a million over Y of them and then pick the next best person and you'll have a 1 over e chance of picking the best one of those 1 million now here's a random one if you take a stick of length 10 and chop it up into two pieces you get lengths of 5 and 5 whose product is 25 if you instead divided it into 3 pieces you get lengths of 3.3 3 for each whose product is 37.0 37 if you cut the stick up into 4 pieces each length is 2.5 and the product will be 39 point zero 6 to 5 and lastly if you make 5 divisions every pieces length 2 and the product is 32 notice the maximum product happens when there are 4 divisions which is also in the length of each piece is closest to e that will in fact always be the case as in the maximize the product cut up whatever length you have in two equal pieces such that each piece is as close to e as possible for the calculus people out there to prove this you can find that the maximum of some length divided by x times itself x times occurs at x equals that length over e now if you're saying wow i really don't like how often a shows up in these random examples then you're not gonna like this next one imagine a function that is X to the X to the X and so on forever or the infinite tetration now this will diverge for most values of course like if you plug it in 5 for X this is obviously going to go to infinity but it will converge for any X between 1 over e to the e and the e througt of e inclusive which are about point zero six six and one point four four four seven respectively so if you plug in something like the square root of 2 or one point four one four since it's within those bounds it will converge to a certain number in this case it actually converges to two but if you plug in something slightly out of the bounds like one point four five then it just goes to infinity now moving on this next example came up on an old Putnam exam which is a university level math competition that consists of what I think most would consider extremely difficult math problems to set this up we first pick a random number between 0 & 1 let's say 0.68 2 in this case then pick another will say point 1 4 5 and we keep going until the sum exceeds 1 so right now the sum is point eight to seven as when we need to keep going if our next number is 0.394 our sum is now greater than one and we stop notice it took three numbers for the sum to exceed one now if you did this with truly random numbers over and over turns out on average the amount of numbers you'd write down before the sum exceeds one is e and the competition problem was to prove that this was the case e to the X also has in my opinion a very weird property that's rate of change is always itself as in if you were in a hypothetical rocket whose position was modeled by e to the T where T is time then your position velocity acceleration and so on will always match which means when you're at let's say the 127 meter mark your velocity would be 127 meters per second your acceleration would be 127 meters per second squared and so on there's another interesting property when finding the area under each of the X as the area from negative infinity to 1 is e to the 1 if we go to 2 the area is e to the second and this would apply to any number the majority of you who've taken calculus I know this is obvious but still interesting to think about now probably the biggest reason that e is so applicable is because of this identity well it's actually where this identity comes from which is something known as Euler's formula I've explained where this formula comes from in a previous video so I'm not gonna do that again but with the formula you can see where that identity on the thumbnail comes from just plug in PI for X and you get e to the PI I equals cosine of PI plus I sine of PI cosine of PI is negative 1 and sine of pi is 0 and thus we have our answer with this formula you can show some really weird math with imaginary numbers by the way I'm not gonna show her these come from but if any of you guys have a graphing calculator nearby go ahead and plug in any of these which look like they should give you a calculator error or something like that but in fact they all equal real numbers that include E and PI now going back what Oilers formula really does is relate an exponential function with E as the base to sinusoidal functions and a typical sinusoidal function is seen in way too many applications to even name from signal analysis to quantum mechanics circuits a simple mass on a spring and so on these all involves some sort of oscillation which is why you can expect to see Euler's formula I mean if you just look at any Circuit's textbook once you get to the alternating current section all voltage inputs are sinusoidal so when capacitors and resistors change the amplitude and phase of that input signal it's harder to use traditional precalc math to represent the system so instead we use Euler's formula to analyze everything like seriously it's everywhere in these text books or just look at an electromagnetism textbook since electromagnetic waves including radio waves visible light x-rays and more are sinusoidal we use Euler's formula to represent them and here you can see basically Euler's formula with some more complexity in the exponent and one of the most groundbreaking discoveries in math was that any function can be represented as a summation of sine and cosine functions as an if you add enough of them up usually an infinite amount you can create any other function that you want in regards to audio and music the Fourier transform gives us the ability to analyze frequencies that make up a certain sound signal and this is used for auto tuning removing unwanted frequencies or noise from a signal amplifying specific notes and more the Fourier transform is also useful in quantum mechanics astronomy optics and so on but signal analysis is definitely one of its most famous uses and of course if we look at the underlying math it involves Euler's formula as some more examples of these real world applications when you let a block on a spring oscillate in some fluid it will oscillate back and forth with a slightly lower amplitude every time the envelope of this equation that kind of squeezes it to zero has an exponential decay equation which includes e to some negative constant times time within it or when you take a pie out of the oven it will eventually cool to the temperature of the room it's in but it cools down much faster at first and then that temperature equation kind of flattens out over time asymptotically approaching room temperature this equation is that same exponential decay that has an e to the power of some negative value times time all right now time for the weirdest part of this video here we can see the graph of x squared times e to the minus X the area under it from to infinity is exactly - which can also be written as 2 factorial now we'll graph X to the third times e to the minus X the area under this from 0 to infinity is 6 aka 3 factorial and here's X to the 4th times e to the minus x whose area from 0 to infinity is 24 or 4 factorial so as you can see the exponent here factorial will equal the area under the curve from 0 to infinity another way to write this is the integral of this curve from 0 to infinity equals that exponent we see here factorial and this takes us to the gamma function which is written a little differently but says the same thing and what this is is an extension of the factorial function so if we wanted to find any integer factorial we could just use this formula but note that this equation could have anything in the exponent not just integers so if you ever see something weird like 1/2 factorial is the square root of PI over 2 it comes from this equation in fact if we go back to desmos I've graphed the equation and calculated the area under it from 0 to a big number since infinity wouldn't work as a function of Z so if we set Z equal to 4 for example we get 24 or 4 factorial as expected and if we set Z to 3 we get 6 or 3 factorial but now we can let Z equal 1/2 and it spits out what would be 1/2 factorial which like I said equals the square root of pi over 2 and this is also another way to see why 0 factorial is in fact 1 because when Z is 0 the area under this curve is 1 the gamma function shows up in quantum physics astrophysics fluid dynamics and more but from what I've seen it's typically very advanced like it can be used to calculate the volume of n dimensional hyper spheres or basically spheres and higher dimensions or on a paper regarding the Casimir effect which appears in quantum field theory the gamma function shows up several times in ways that I totally understand and next to the gamma function which itself has Oilers number within it we have Euler's number showing up so yes this does have more applications than you may think but now we're gonna see what's definitely my favorite part of this video we're going to go back to that hyper sphere example because something I found when doing research for this that just did not sit well with me let's say we calculate the volume of a sphere with radius one in every even dimension as then we find the volume of a zero dimensional sphere a two-dimensional sphere four dimensional six and so on forever all with the radius of one and yes in two dimensions for example a sphere is really just a circle and volume really just refers to area in this case it'd be pi but we're just using sphere and volume as general terms because of the higher dimensions anyway if we add up the volumes of all those even dimensional unit spheres forever the summation would equal e to the PI and no I'm not joking but the applications of Oilers numbers still don't stop there like the classic normal distribution or bell curve we see all the time contains Euler's number the Laplace transform that's used to simplify calculations for things like circuits and control systems also include two Euler's number and I could go on for a while but if you want to keep learning about these unique areas of math and their applications like with Euler's formula and imaginary numbers you can continue to do so at brilliant org who I'd like to thank for sponsoring this video brilliant is an educational platform that hosts a wide variety of math and science courses they not only teach you the technical information you need to know for the various subjects they also challenge you constantly along the way with practice problems so you have a real fundamental understanding of what you're learning the unique applications of mathematics is one of my favorite things to learn then discuss on this channel and brilliant is actually where I've gotten a lot of my information from like if you enjoyed this video then their complex algebra course may be something you find interesting this course will take you through way more detail regarding imaginary numbers and why they're important within mathematics and even science or engineering plus you'll learn exactly how Euler's formula applies to physics or engineering regarding signals and various complex systems this kind of math really changed the way I looked at imaginary numbers and why we learn the things that we do brilliant also includes daily challenges that turn learning into a habit these questions range from what happens when you cut a mobius strip in half to probability games within quantum systems and much more to give you a range of topics to look forward to learning plus they now have offline courses for ios and android so you can download some of your favorite courses right to your phone you can learn something new and stay productive whether you're commuting to work or school traveling or just somewhere with terrible internet so if you wanna get started right now and support the channel you can click the link below or go to brilliant out org slash major prep to get 20% off your annual premium subscription and with that I'm gonna end that video there if you guys enjoyed be sure to LIKE and subscribe follow me on Twitter and join the major Facebook group for updates on everything and I'll see you all in the next video

Channel: Zach Star

Views: 740,551

Keywords: majorprep, major prep, eulers number, euler, e (mathematics), e constant, mathematical constants, math constants, applications of e, applications of euler's number, what is euler's number, how is eulers number used, eulers formula, eulers identity, why do we learn eulers number, leonard euler, applied math, math applications

Id: AAir4vcxRPU

Length: 15min 50sec (950 seconds)

Published: Wed May 15 2019