Everything Everywhere Daily: History, Science, Geography & More - e: Euler's Number

Starting point is 00:00:00 There are an infinite number of numbers, but some numbers are more important than others. One number, which just might be the most important number, lies hidden in a wide variety of things in the natural world. It can be found in everything from the mathematics of radioactive decay to population growth and even compound interest. The number turns out to have a central role in calculus and mathematics most elegant equation. Learn more about E, also known as Euler's number, on this episode of Everything Everywhere Daily. What if your perceptions about the past were wrong? ThruLine is a podcast that takes you back in time

Starting point is 00:00:50 to uncover the parts of the story that may have gone unnoticed. It effectively turned day into night. And how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. If you were to make a list of the most important, or at least the most interesting numbers, most of those numbers would be pretty simple to grasp. One is an extremely important number, and it's the most fundamental number there is. Likewise, zero is an important number that's easy to comprehend.

Starting point is 00:01:25 It took several thousand years for the idea of zero to catch on, but it eventually did, and today everybody can grasp the concept of zero. Other important numbers aren't quite as obvious, but they can still be grasped easily. The square root of two is important. Its existence supposedly caused the destruction of the cult of Pythagorean, but you can grasp the idea quickly enough by looking at a simple right triangle. Likewise, Pi is very important, but it can be represented geometrically by using the ratio of a circle's circumference and diameter. The number I takes a bit of imagination, excuse the pun, but it represents nothing more than the square root of negative 1. But E is different. E is more subtle. You can't

Starting point is 00:02:07 geometrically express E in the form of simple shapes. But if you look closely, you look closely, enough, you can find E almost everywhere. E is an irrational number, which means it can be expressed in decimal notation as an infinite string of numbers that never repeats. Unlike the square of two, it can't be expressed algebraically, so it's considered to be a transcendental number as well. The value of E, when written out to 15 decimal places, is 2.7-1828-18-28-48-45-95-90-45. Knowing the value of E is easy enough, but why in the world is this collection of seemingly random digits so special? Well, it really starts with compound interest. You may have thought that a discussion of a mathematical constant would have started with something in nature,

Starting point is 00:02:59 but this actually starts with interest payments. Unless you think that interest isn't important, there is a quote attributed to Albert Einstein, which may or may not be true, where he supposedly said that compound interest is the most powerful force in the universe. In 1683, the famous Swiss mathematician Jacob Bernoulli was working on a problem with compound interest. The problem he was working on was actually pretty simple to understand. Let's say, just to make the math easy, you have $1 in an account that earns 100% interest per year. If you compound the interest annually, then at the end of one year, you will have a total of $2, the initial $1, and then the 100% interest, which is another dollar.

Starting point is 00:03:43 But what if you were to compound the interest more frequently than annually? Would that change the total amount you made? And the answer is yes. Now let's assume you had two compounding periods every six months. At six months, you get 50%, and at 12 months you get another 50%. How much would you have at the end of the year? At six months, you'd have $1.50, and at 12 months, you'd earn 50% on the $1.50, and you'd have a total of $2.25.

Starting point is 00:04:14 You made more money by changing the compounding period, but not the interest rate. But now let's take this further. Let's assume you compound the interest quarterly. You make 25% interest every three months. At the end of a year, you wind up with $2.44. What about compounding monthly? You'd get $2.61 at the end of the year. And how about daily?

Starting point is 00:04:38 you'd have $2.71. That 2.71 should sound familiar. What Bernoulli realized was that as the compounding period got smaller and smaller, the total amount you would make didn't grow to infinity. It actually approached a limit. And that limit is the number E. It turns out that other mathematicians had stumbled upon E earlier, but they just didn't realize its significance.

Starting point is 00:05:06 The first person who stumbled upon it was, John Napier. In 1618, he worked on a table of logarithms. A logarithm is basically the opposite of an exponent. They're incredibly handy for doing calculations, but they can be very difficult to calculate by hand. For example, 10 raised to the power of 2 is 100. 10 raised to the power of 3 is 1,000. But what power must you raise 10 to get to a value of, say, 50? Making these calculations, Napier stumbled upon E without really knowing its significance. For this reason, E is sometimes known as Napier's constant. However, we're just getting started. The same Jacob Bernoulli, who worked on compound interest, also worked on statistics and probability. And guess what? The number E turns up there as well.

Starting point is 00:05:56 Let's say you play a game where you have a one in N chance of winning. If you play the game n times, what are the odds that you don't win a single time? Let's make this a bit more concrete. A roulette table with a single zero has 37 slots. Assuming a ball can fall into each slot equally, what are the odds that you wouldn't win once if you played 37 times? It turns out that you'd have a 36.3% chance of not winning once. Or as it's usually expressed, 0.363. It turns out that number is very close to 1 over E.

Starting point is 00:06:34 In fact, the larger N gets, the closer the odds get to 1 over E. But this isn't even the good stuff. The real power and significance of E has to do with something called the exponential function. The exponential function is pretty simple. It's just the number E raised to some power of X. If you remember back to my episode on what calculus is, a function is just a mathematical relationship where you put something in and you get something out. The exponential function has unique properties.

Starting point is 00:07:05 Exceptional properties, actually. Many mathematicians consider it to be the most important function in all of mathematics. Why? Again, if you remember back to my episode on what calculus is, the derivative of a function calculates the rate of change of a function at any point. The exponential function is the only function that is the derivative of itself. And that means when x equals 1, the slope of the line at that point is 1. When x equals 5, the slope of the line at that point is 1.

Starting point is 00:07:35 When x equals 5, the slope of the number of it. with the line at that point is 5. If the exponential function is the derivative of itself, then it also must be the integral of itself. That means the area under the curve at a given point is the same as the value of the point. At x equals 1, the area is 1. At x equals 5, the area under the curve is 5.

Starting point is 00:07:58 Logarithms are the inverse of exponents, and the inverse of the exponential function is called the natural log function. It's usually written in, the notation LN, which is a special way of indicating that something is a logarithm in base E. The fact that the exponential function is the derivative and integral of itself means that you will find the number E all over the place. I began the episode by talking about compound interest, which is really just about growth, in that particular case, the growth of money. However,

Starting point is 00:08:30 growth can apply to anything, not just money. When algae grows in a pond, it will do so according to a formula based on E. If you're looking at populations of animals consisting of anything from rabbits to humans, it will be based on the same formula based on the number E. E is also involved with the reverse of growth, decay, in particular, radioactive decay. The half-life of a radioactive isotope is determined by an equation with the natural logarithm of E at its heart. If you're familiar with statistics, you've probably seen a bell curve. It's a statistical distribution. that is shaped like a bell. Many things in the world are distributed on a bell curve.

Starting point is 00:09:11 Well, it turns out that the equation describing a bell curve is based on E. I could go over many, many equations and formulas that are all based on the number E, the exponent function or the natural logarithm function. Suffice it to say, the number E can be found all over the place in science and mathematics, and it is very, very important. Now I want to go back to something I mentioned at the start of the episode. the number E is often known as Oilers Constant, and I haven't yet mentioned,

Starting point is 00:09:40 Leonard Euler. Euler was unquestionably one of the greatest mathematicians in history. In the 18th century, he conducted studies on the number E and found many remarkable things. For starters, he was the one who associated the number with the letter E. And believe it or not, he didn't name it after himself. Supposedly, he just selected the letter because it wasn't being used for anything else at the time.

Starting point is 00:10:05 One of the things he discovered was a way to quickly calculate E using an infinite sum. If you remember back to my episode on combinatorics, a factorial is just the number you get when you multiply up all the numbers to it and including it together. So 5 factorial would be 1 times 2 times 3 times 4 times 5. What Euler discovered is that you can calculate E by adding together the following. 1 over 0 factorial plus 1 over 1 factorial plus 1 over 2. two factorial plus one over three factorial plus one over four factorial and on and on and on. As factorials get big very quickly, the corresponding fractions get small very quickly, which means that

Starting point is 00:10:49 you can calculate a useful version of the number without very much effort. But perhaps the greatest discovery he made is what's considered to be the most beautiful and elegant equation in all of mathematics. In a very simple equation, he managed to tie together five of the most important mathematical constants. The number one, which is kind of self-obvious. The number zero on which I've done a previous episode. The number pi on which I've done a previous episode. The number I on which I've done a previous episode. And finally, E, which is the subject of this episode.

Starting point is 00:11:24 His equation is simply, E raised to I times pi, plus one equals zero. It's known as Oilers identity or Oilers' formula. Keith Devlin, a mathematics professor at Stanford University, once said about it, quote, like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. Oilers identity captures the beauty of the number E

Starting point is 00:12:01 and how it's bound into so much of nature. And that's why it is probably the most important constant in all of mathematics. The executive producer of Everything Everywhere Daily is Charles Daniel. The associate producers are Peter Bennett and Cameron Kiever. I wanted to give a big thanks to everyone who supports the show on Patreon. Your support helps me put out a new show every day. And if you're interested in Everything Everywhere Daily merchandise,

Starting point is 00:12:29 Patreon is currently the only place where it's available. And if you'd like to talk to other listeners of the show and get notified to future episodes and projects, please join my Facebook group or Discord server. Links to everything are in the show notes.

Everything Everywhere Daily: History, Science, Geography & More - e: Euler's Number

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