A numbers party

Imagine you are throwing a fancy party. You’ve arranged everything in great detail; from the music and the location, to the food. The last thing left to do is invite some esteemed guests. To give your party some quality, you want to invite the five most important numbers. Who should you invite? Well, to start off with you of course invite the numbers 1 and 0. The number 1 is the first number ever invented and could therefore be seen as the mother of all mathematics. The number 0 might be seen as the mother of algebra, leading the way to the introduction of relative numbers (i.e. negative numbers). Its invention made it possible to solve the simplest equations like 1 = x + 1.

All right, now that you have invited the Two Mothers, who’s next? Definitely, the number π should be invited. Everyone with some knowledge of maths knows this number and it actually is much older than 0. In the ancient world, π was already used. It depicts the beautiful relationship between the circumference of a circle and its diameter, making it the symbol of Geometry. The number is an irrational number and can be proven to even be transcendental, which somehow means that it lies outside the scope of classical algebra. Thus, π makes a great addition to your party!

The next invitation is reserved for the number e, also known as Euler’s number. This number is the symbol of calculus. It can be defined in many ways, but maybe the most natural one comes from looking at functions that are equal to their own derivatives, like the function e^x (actually all other such functions are multiples of this). Euler’s number might be most famous for its application in Dynamics and specifically growth equations. Therefore, it will come back a lot if you want to study systems that grow (or shrink) over time and that might be a good reason to invite e.

Finally, our last invitation goes to the imaginary unit i. In some way, this number finishes the search for solutions of equations, since every polynomial equation has solutions in the complex numbers. (I hope you know this, if you’ve taken my course!) However, i is actually most famous for its use in Physics. It is essential for the study of waves and electricity. One might even argue that without the invention of i, you would not have a working cell phone right now. So, in some way, this imaginary number is just as “real” as the previously introduced ones. Apart from that, you always want a mysterious figure at your party to make it interesting, so i is the perfect candidate for this.

There we are! We’ve invited the five most important numbers to our party! However, as soon as you send out the invitations, you start worrying. These numbers come from such different areas of mathematics and such different time periods in the development of science, will they get along? Nothing ruins a party like a bunch of numbers that will avoid each other the whole evening. Fortunately, Euler is also at the party (remember Euler from the Euler’s number?) and he unifies the five numbers with a beautiful equation known as Euler’s Identity:

e^iπ + 1 = 0

All of the five numbers have found their own spot in this equation and happily get along. Before you read on, take a moment to appreciate the beauty of this equation. It combines all of the previously introduced numbers from vastly different areas of mathematics into an amazingly simple and elegant equation. Using only 7 symbols, it unifies Geometry with Dynamics and the Ancient world with Modern Physics. In fact, this equation is so satisfying, that a group of mathematicians chose it as the most beautiful equation in mathematics.

Euler’s Identity was proven in the 18th century by making use of complicated infinite sums. I will spare you the details here, but what I do want to do is give you the intuitive proof of why this is true. Note that it is far from obvious that this identity holds. What should e^iπ even mean? When we write x³ we mean that we multiply x by itself 3 times, but how can we multiply a number iπ times?

It shows that mathematics is the language of the universe

Let’s look back at our introduction of the function e^x. It arose as a function that is equal to its own derivative. The derivative of a function at a point can be seen as the speed at which the function changes at that point. So, if we would have an object moving along a line, whose position is given by e^t at every time t, then its speed is also given by the function e^t. How about an object whose position is given by e^2t? Well, the derivative of this function is 2e^2t and therefore the speed of this object at time t is twice its position. If we keep following this pattern, then an object whose position is given e^it should have speed ie^it. What does this mean? Well, to make sense out of this we can no longer think of objects moving along a line, we have to turn to the complex plane. Complex numbers can be seen as points in the complex plane or equivalently vectors in the complex plane. If you have taken my course, you know that multiplication by i amounts to rotating a vector by 90^∘ in the plane. If the object at time t has position e^it then its speed at that time is ie^it: the vector that is exactly 90^∘ rotated from the position. So the velocity of this object is always precisely perpendicular to its position. What kind of motion does this describe? Do we know an object that has this behavior? Yes we do: the moon! The velocity of the moon is always perpendicular to its position relative to the earth. What motion does it make? Indeed, a circle! So the object that we are describing is moving in a circle around the origin. At time t=0 it has position e^it  = e⁰ = 1 (any non-zero number raised to the power 0 is 1), so the circle it moves in has radius 1. The object moves with speed 1 (the speed is the modulus of the velocity and since we are moving along the unit circle the modulus is always 1), thus after a time t it has travelled a distance t along the unit circle. What happens at time t = π? Well, the object has by then moved a distance π along a circle with radius 1. The circumference of the whole circle is 2π, so the object will have moved exactly half a circle. It thus ends up at the point exactly opposite the point 1 on the unit circle, which is the point -1. This proves that e^iπ = -1, or equivalently e^iπ + 1 = 0. □

Well, after having heard this whole story you are happy, your party is a huge success and all the numbers that came are happy to be there. You let out a sigh of relief while you ponder about the beauty of maths. The fact that these different numbers, from vastly diverse areas of mathematics and various distinct time periods, combine so beautifully into one equation is not a mere coincidence of a made-up system. It shows something deeper. It shows that mathematics is the language of the universe.