e, π and the Exponential Function

Throughout mathematics and its applications, we often encounter the numbers e and π. But what do they actually mean, what makes them so prevalent, and how are they related?

Both numbers are deeply intertwined with the exponential function, denoted exp, which can be described simply as “the function which is its own derivative”.

(Or, in slightly less simple but more accurate terms – exp is the only function f:\mathbb{R}\to\mathbb{R} which is differentiable everywhere and satisfies f'(x)=f(x) for every x\in\mathbb{R}, and f(0)=1. You can also use \mathbb{C} instead of \mathbb{R}).

Another way to say this, is that exp is a solution to the simple differential equation y'=y. As such, it is a building block for solutions to differential equations of all kinds.

Differential equations describe how the change in some quantity relates to the quantity itself. They describe how the universe works at all levels – from the most microscopic and fundamental, such as

  • Electromagnetism (Maxwell’s equations),
  • Gravity (Einstein’s field equations),
  • Quantum mechanics (Schrödinger equation),

to the macrosopic –

  • The motion of springs, pendulums, projectiles and planets,
  • Waves – be it sea waves, sound waves or radio waves,
  • Electronic circuits,
  • Radioactive decay,
  • Structural integrity of buildings,
  • Rockets and space launches,
  • The growth of populations, be it humans, animals, bacteria in a petri dish, viruses in a human host, or people sick with COVID-19,
  • Financial dynamics, like money in a bank account, stock prices, the revenues of a company, or the exchange rate of currencies such as Bitcoin,
  • Adoption of new technologies,
  • Social phenomena, like memes and viral videos,
  • And much more – including purely abstract mathematical concepts which have no direct ties to phenomena in the physical universe.

So it is no surprise that the function which is the building block for solving differential equations comes up very often. In fact, some dub it “the most important function in mathematics”.

Because the function is so important, we want to know more about it. One question of interest is – what is the value of \exp(1)? This is useful, because one of the properties of exp (which we can prove using the definition we started with) is that \exp(x+y)=\exp(x)\exp(y). Using this, we can show that \exp(n)=\exp(1)^n for every integer n (where taking a power is a simple repeated multiplication). In other words, knowing the value of the function at 1 allows us to find its value for every integer. So we give the value of \exp(1) a name. The name we choose is e.

That’s what e is – the value of the exponential function at 1. The importance of e can be understood by understanding the importance of the exponential function, which itself can be understood by understanding the importance of differential equations. That understanding can come from some experience with their applications; the examples I gave above might help.

In fact, if we extend a bit the definition of taking a power, we will find that for every real number x, we have \exp(x) = e^x, not just for integer x. This is why the exponential function is often written e^x instead of \exp(x).

The exponential function is also where π comes from. If we look at it as a complex function, we find that it is periodic – there is a specific number p\in\mathbb{C} such that for every z\in\mathbb{C}, we have \exp(z+p)=\exp(z) (which is the smallest with this property). This number happens to be purely imaginary, so if we divide it by 2i, we get a real number. This real number is what we call π.

This way of looking at π – as the period of the most important function in mathematics (divided by 2i) – is much more fundamental, and better explains why π comes up so often, than definitions based on the girth of arbitrary geometric shapes we might scribble.

The Blind Men and the Elephant. [the BIG picture] | by Sophia Tepe ...

It’s also noteworthy that the exponential function is reminiscent of the blind men and the elephant. It behaves differently and seems to be a different thing if we look at it from different perspectives. If we look at the positive real axis, it is rapidly growing. On the negative real axis, it is rapidly shrinking. On the imaginary axis it is neither growing nor shrinking – it is periodic, repeating the same values in a cycle.

Which nature of the exponential function comes to light, depends on the specific differential equation we use it to solve. That’s why some of the applications I mentioned exhibit growth or decay, and some exhibit rotation and cycles.

In fact, the well-known periodic functions sin and cos can be seen as projections of what the exponential function does along the imaginary axis.

We’ve defined e as the value of the function at 1 – a real number, and we’ve defined π using the period of the function along the imaginary numbers. It should come as no surprise, then, that e often comes up in applications dealing with growth and decay, and π often comes up in applications dealing with cycles and circularity. They are two sides of the same coin.

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