e: Euler’s Number

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Podcast Transcript

There are an infinite number of numbers, but some numbers are more important than others. 

One number, which might just 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 even turns out to have a central role in calculus and mathematics’s most elegant equation.

Learn more about e, also known as Euler’s Number, on this episode of Everything Everywhere Daily.

If you were to make a list of the most important or at least the most interesting numbers, most of them are pretty simple to grasp. 

One is an extremely important number, and one is the most fundamental number there is. 

Likewise, zero is an important number that is easy to comprehend. It took several thousand years for the idea of zero to catch on, but it eventually did, and today, everyone can grasp the concept of zero.

Other important numbers aren’t 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 Pythagoras, 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, but it represents nothing more than the square root of negative one. 

But e is different. E is more subtle. You can’t geometrically express e in the form of simple shapes. 

But e is everywhere if you look closely enough. 

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 root of two, it can’t be expressed algebraically, so it is considered to be a transcendental number. 

The value of e, when written out to fifteen decimal places, is 2.718281828459045….

Knowing the value of e is easy enough, but why in the world is this collection of random digits so special?

It really starts with compound interest. 

You might have thought that a discussion of a mathematical constant would have started with something in nature, but this actually starts with interest payments. 

And lest 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, that you have $1 in an account that earns 100% interest per year. 

If you compounded the interest annually, then at the end of one year, you will have a total of $2. 

But what if you were to compound the interest more frequently than annually? Would that change the total amount you made?

The answer is yes. 

Now, let’s assume that you had two compounding periods every six months. At six months, you get 50%, and at 12 months, you get 50%. How much would you have at the end of the year? 

At six months, you’d have a dollar fifty, and at 12 months, you’d earn 50% on the 1.50, so you’d have a total of $2.25. You’ve made more money by changing the compounding period but not the interest rate.

But let’s take this further. Now, let’s assume you compounded 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 would get $2.61 at the end of the year.

How about daily? 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 approached a limit. 

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. 

The first person who stumbled upon it was John Napier. In 1618, he worked on a table of logarithms. 

A logarithim is basically the opposite of an exponent. They are incredibly hand for doing calculations, but they are difficult themselves to calculate. 

For example, 10 raise the power of 2 is 100. Ten raised to the power of 3 is 1000. 

But what power must you raise 10 to, to get a value of 50? 

In 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. 

The number e turns up there as well. 

Let 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 that number is very close to 1/e.  In fact, the larger n gets, the closer the odds get to 1/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 is just the number e raised to the 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. Exceptional properties, actually. 

Many mathematicians consider it to be the most important function in all of mathematics. 


Again, if you remember back to my episode on what calculus is, the derivative of the 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. 

That means when x=1, the slope of the line at that point is 1. When x=5, the slope of 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 value as the point. A x=1, the area is one. At x=5, the area under the curve is five. 

Logarithms are the inverse of exponents, and the inverse of the exponential function is called the natural log function. It is written using the notation ln, which is a special way of indicating 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, growth can apply to anything, not just money. 

When algae grows in a pond, it will do so according to a formula based on the e. 

If you are looking at a population 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 are familiar with statistics, you have probably seen a bell curve. It is a statistical distribution that is shaped like a bell. Many things in the world are distributed on a bell curve.

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 exponential 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. 

I want to go back to something I mentioned at the start of the episode. The number e is often known as Euler’s constant, and I haven’t yet mentioned Leonhard Euler. 

Euler was unquestionably one of the greatest mathematicians in history. In the 18th century, he conducted studies of the number e and found many remarkable things. 

For starters, he was the one who associated the number with the letter e. 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.

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 all the numbers up to it and include it together. So 5!  Would be 1x2x3x4x5.

What Euler discovered is that you can calculate e by adding together: 

1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …

As factorials get big very quickly, the corresponding fractions get small very quickly, which means you can calculate a useful version of the number without much effort. 

But perhaps the greatest discovery he made is what is 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. 

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. 

His equation is simply ei? + 1 = 0 

It is known as Euler’s identity or Euler’s Formula. 

Keith Devlin, a mathematics professor at Stanford University, said, “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.”

Euler’s Identity captures the beauty of the number e and how it is bound into so much of nature.

It is why it is probably the most important constant in mathematics.