Category Archives: Mathematics

The Unabashed Expanse of Ordinal numbers

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Let me take you on a magical journey to explore one of the most fundamental and elegant, yet underappreciated, concept in mathematics – the ordinal numbers.

This cannot be done without a few words on what they are and and why we need them, but my focus will be on trying to instill a sense of awe at just how many of them there are – something that I have always found beautiful.

A language primer

The concepts of ordinal numbers – and their close cousins, the cardinal numbers – were not invented by mathematicians. They were originally a language construct, which we are all familiar with in our spoken language of choice, be it English, Hebrew or whatever.

A cardinal numeral is a way to express how many things there are. “One”, “two”, “three”, “four”, “fourty-two”, “seventy-eight hundred and twenty-five” are all cardinal numerals.

An ordinal numeral describes the position of something in a list of ordered items. “First”, “second”, “third” and “fourth” are ordinal numerals.

Back to math

In mathematics, cardinal numbers are basically the same as their natural language counterpart – they describe the size of a set, how many elements there are. 0, 1, 2, 3, etc. – known as “natural numbers” or “non-negative integers” – are cardinal numbers. Where this gets interesting – and where they truly diverge from the linguistic concept – is with infinite sets. Their size is measured with infinite cardinal numbers, which have names such as \aleph_0 or \beth_1.

Ordinal numbers are silghtly more complicated. They represent the different fundamental ways in which the elements of a set can be ordered, known also as “order types”. I’ll show you what I mean, but I should warn you that they are not terribly interesting in the realm of finite sets. For every given finite size for a set, there is exactly one fundamental way to order it – so each finite cardinal number corresonds to exactly one finite ordinal number. Because finite cardinal and ordinal numbers are basically the same, it may not be clear why we even introduced ordinal numbers as a new concept. This should become clearer when we explore infinite numbers.

So let’s take a set with, say, 4 elements – \{A,B,C,D\}. There are many ways to order it (24, to be exact, if you remember your combinatorics) – for example, A<C<B<D, or C<D<B<A, or D<A<B<C. But all these orderings are fundamentally the same. They all follow the pattern – “One item, then another, then another, then another”. Different orderings have different items in each position, but they all follow the same basic structure. This structure is represented by the ordinal number 4 (so called because it is the only way to order sets of size 4).

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e, π and the Exponential Function

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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.