All About e! 2.718…

Edited by Kimberley Chee.

You’ve used it in high school, you’ve used it in university. But is there more to this “letter” than what meets the eye? Learn all about its unexpected origin story, and properties in this article!

Introduction

Leonhard Euler (1707-1783) was a famous Swiss mathematician, physicist, astronomer, geographer, logician, and engineer. He has made great contributions in many fields of study, but we will focus on his major contributions to maths in this article, namely one of the most famous numbers which were named after him: the Euler’s number. Surprisingly, the number ‘e’ was not first conceived by Euler, but ‘discovered’ by Jacob Bernoulli in 1683⁶. What’s even more surprising is that Bernoulli was actually looking at a relatively different field — economics — when this discovery was made! But we will look more into that in the final section.

Calculating ‘e’

So what’s so special about ‘e’? Well for starters, ‘e’ is the base of natural logarithms (ln), and is also the base of eˣ which is the only function, apart from y=0, that is its own derivative! We will put this to the test later on, and first focus on the definition of ‘e’, which is as follows³:

But before we delve deeper, let’s prove the convergence of this formula, and ensure that this definition of ‘e’ doesn’t go to infinity.

Using the binomial formula, we get:

Using the definition of the combinatorics, we get:

By this clever algebraic manipulation, we can bind each of the terms:

The bounded terms are then removed as all of them are less than one:

The factorial is bounded using the following steps:

Now this sum is less than 3, and we show that this definition is convergent and ‘e’ does indeed have a value. However, to approximate the value, you can use:

which approaches ‘e’ as ‘n’ approaches infinity. But you can, of course, still use the original representation.

Before we move on to show that ‘e’ is the base of the function that is the derivative of itself, we will show an alternative representation:

Now let’s let n=1/k, which gives us:

This is from the fact as n approaches infinity, k approaches 0, and 1/n=k and 1/k=n

The k’s can simply be replaced with n and therefore you get:

This is another way to represent ‘e’.

Deriving the Derivative of eˣ

We know that ‘e’ is special as it is the base of the only function which is the derivative of itself, namely f(x)=eˣ. We will be using the definition of a derivative as a basis, and delta x instead of h, (both symbols are equivalent)¹.

Factoring out eˣ

This limit is quite difficult to simplify, so let’s try to change the equation into a simpler expression, or one we know from before instead. Firstly, let ‘n’ equal the numerator:

Then, find the expression for the denominator:

After taking the natural logarithm (log), we are left with:

Also, because the limit is from delta x approaches 0, the new expression will be as ln(n+1) approaches 0, or when n approaches 0.

Therefore we get this expression:

Which is equivalent to:

And using log rules, we get:

If you look closely, the expression below looks similar to the alternate definition we derived earlier, so we are motivated to do the following manipulation:

However, the limit is equal to ‘e’, which we have derived earlier. Therefore, we are left with:

And as we know it, ln(e) is equal to 1, so we get:

In conclusion, we have successfully obtained

as we wanted. Therefore, ‘e’ is the base of the exponential function which is also equal to its derivative. How cool is that?

Proving the irrationality of e

This proof will use the partial sum of ‘e’. Earlier, we have shown that the the sum of ‘e’ indeed converges, and can be represented as the following⁴:

We will then split the infinite sum so the partial sum can be used:

Let Sₙ be the partial sum of:

Thus, it is obvious that:

By using factorisation, we get:

At this step, we will try to bound each of the terms in the infinite sum, and a good way of doing that will be to find the bigger value in each of the terms, and break them down. We also did a similar method to this in proving the convergence of ‘e’.

The infinite series on the right is simply a geometric sum which we can find. The ‘r’ value is 1/(n+1), which is clearly less than one, so we can apply the formula. (The ‘s’ used here is not related to the one in the partial sum. Don’t get confused!)

As the first term of the infinite series is 1, and the common ratio is 1/(n+1), by substituting the values, we obtain:

Bringing it back to the original formula, we get:

Therefore, leaving us with:

Which is equivalent to

In particular, these manipulations have been done to show that the term in the middle is bounded and is between, and not equal to 0 and 1/n. This will set up our proof by contradiction. Note that we have not made any assumptions to reach the conclusion above yet, and instead, this is where our proof of contradiction begins.

Firstly, let’s assume the opposite, and that ‘e’ is a rational number. Let:

such that ‘p’ and ‘q’ are natural numbers and co-prime. Now, let’s allow ‘n’ to be any value larger than ‘q’ (Remember, our definition of the partial sum allows this!). From that, n>1, thus 1/n <1. Therefore, we can see that:

Now this means that the term in the middle is not an integer! Let’s investigate that odd term a little further by using algebraic manipulation:

However, by looking at each of the terms, we observe the following: Our original assumption states that ‘n’ is any number larger than q. So, by the definition of ‘n!’, ‘n!’ contains ‘q’ as a factor, so the first term is an integer. The terms on the right are also integers, as also using the definition of a factorial, ‘n!’, contains all the numbers below or equal to ‘n’ as factors. Hence, we can conclude that ‘n!’, ‘n!/2’, ‘n!/3’, and all the way to ‘n!/n!’, are all integers! Therefore, the expression e-Sₙ is an integer, but upon recapping the first section, we have shown that it is not! So, our original assumption that ‘e’ is a rational number is false, and ‘e’ is therefore irrational!

Euler’s Formula

Next, we will look at a proof for Euler's formula. It relates to and deepens our understanding of the imaginary plane. This formula also derives one of the most famous identities called Euler’s identity². Firstly, let’s consider the function:

We take the derivative and apply the product rule, treating ‘i’ as a normal constant.

Let’s attempt to factorize the equation: multiply out the i’s on both equations and the negative from the left part. Remember that i × i = -1, so the equation simplifies to:

By factoring, we get:

almost magically! The bracket on the right cancels out,

and we finally find that the function is actually equal to 0!

But let’s take a step back to see what that means. The derivative of a function, otherwise known as the rate of change, equals to 0. That means that the function doesn’t change its value, or is a constant for all values of theta (θ). So, by substituting any value of theta, we are able to find the constant. After some testing, we notice that although all values of theta are equivalent, by letting theta = 0, a lot of the terms cancel out:

Therefore, sin 0 = 0, and all of the imaginary values are removed in the end.

Don’t you find this amazing? A function that includes many different symbols and functions, especially the imaginary number ‘i’, just simplifies to one in actuality! And because all the values of the function are equal, we can conclude that:

which brings us the to final formula:

, that proves the validity of Euler's formula. A special subcase of the formula would be theta = pi (π) in terms of radian (rad), so let’s see what would happen then:

Because sin(π)=0 and cos(π) =-1, the imaginary number ‘i’, is removed from the right, and the expression simplifies to:

And there you have it! We have arrived at Euler's identity. You should note that this identity is special, and even a little magical as it uses 5 of the most special numbers, namely e, i, π, 1, and 0. But what makes these 5 particular numbers so unique? Well, ‘i’ is used to extend the real plane to the imaginary one⁵; π is the ratio of the diameter to the circumference; 1 is the multiplicative identity; 0 is the additive identity; and ‘e’ is special for all the reasons and more from the topics in this article.

Origins

Finally, let’s take a look at the origins of this number, which as aforementioned, was actually rooted in the field of economics! The thought experiment began like this: A bank account starts at $1, and pays 100% compound interest a year. The account would end up with $2 at the end. However, if the account were to begin at $1 and instead pay 50% compound interest every half year, it would be $1.50 in the first half, and $2.25 in the second half. Similarly, if we were doing 25% every quarter year, we would have roughly $2.44. The question would be: how high can this number go?

If we generalise the problem into mathematical notation, we get the following:

where the ‘1’ on the left represents our starting point of 1 dollar, and the power of ‘n’ represents the compound interest. And as we know, this representation is actually ‘e’ when n approaches infinity! What began as ordinary compound interest calculation in the field of economics, resulted in the origination of a deep mathematical constant!

Conclusion

Euler’s number is wonderfully complex and deep, with many special properties. From its origins in economics, to its irrational and transcendental properties, its relations to calculus and the imaginary plane, and the list goes on! It is safe to say that there is much more to learn about this number as it continues to push our curiosity beyond the limits.

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References:

1. Khan Academy. (n.d.). Proof: The derivative of 𝑒ˣ is 𝑒ˣ. Khan Academy. https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-7/a/proof-the-derivative-of-is#:~:text=e%20x%20e%5Ex%20e,is%20the%20derivative%20of%20itself!&text=(Well%2C%20actually%2C%20f%20(,very%20interesting%20function...)

2. Mathema Education. (2013, November 13). Euler's Formula - Proof WITHOUT Taylor Series. YouTube. https://www.youtube.com/watch?v=N3Xxj4xNjIo&t=170s

3. mathlove. (2015, June 22). Prove limit converges in definition of e. Stack Exchange. https://math.stackexchange.com/questions/1334832/prove-limit-converges-in-definition-of-e

4. UMKC. (n.d.). Proof - A Proof That "e" is Irrational. YouTube. https://www.youtube.com/watch?app=desktop&v=FtIL7nGgDNM

5. Wikipedia. (n.d.). Imaginary unit. Wikipedia. https://en.wikipedia.org/wiki/Imaginary_unit#:~:text=The%20imaginary%20unit%20or%20unit,numbers%2C%20using%20addition%20and%20multiplication.

6. Wikipedia. (n.d.). Jacob Bernoulli. Wikipedia. https://en.wikipedia.org/wiki/Jacob_Bernoulli

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