Motivating Number Systems

As people learn about extensions of natural numbers, they sometimes complain about them not being real:

• If numbers are used for counting, why do we need negative numbers? Is it really meaningful to have -3 apples?
• If a number can’t even be written down fully, then what’s the point? Does it even really exist? You can’t really have $\pi$ apples.
• If numbers help us calculate physical quantities, why are there imaginary numbers? You surely can’t have $i$ apples¹?

Let’s talk about each one.

Negatives

Let’s start with the natural numbers ($0, 1, 2…$). The most basic operations on these numbers are addition and subtraction. Even if we accept that the primary value of numbers is to count things, addition and subtraction are useful so you can combine pre-counted values. If someone removes one apple from a bag with 12 apples, you don’t want to recount the apples. You just want to subtract 1: $12 - 1 = 11$. There are 11 apples left in the bag.

Addition and subtraction behave differently on the natural numbers though. Addition over the natural numbers is “closed”. If you add any two natural numbers, the result will always be another natural number.

Subtraction is not closed.

If you subtract a larger natural number from a smaller one ($3 - 5$), the result is not a natural number! This means that if you are only working in the natural numbers, you are never allowed to subtract a larger number from a smaller one. This breaks a lot of “rules” of numbers we take for granted.

For example, here is the associative rule:

$$ (a + b) - c = a + (b - c) $$

The associative rule says that it doesn’t matter if you add $a$ and $b$ first or if you subtract $c$ first. As long as you do all the operations, the end result will be the same.

But if we are working only in the natural numbers, we can’t do this anymore. If $b$ is smaller than $c$ then $b - c$ doesn’t exist in the natural numbers! Even if $a + b - c$ is a natural number, we can’t regroup the operation because $b - c$ might be invalid.

Restrictions like this make simple mechanics of algebra cumbersome because things that look equivalent are no longer equivalent. So to help our mathematical machinery, we extend the natural numbers to include the numbers that can be reached by subtraction. These numbers together make the integers which are closed under subtraction.

Sometimes the real-world application has an obvious use for negative numbers (e.g. money, distance, time). But even if we didn’t have those cases, negative numbers would still be useful to allow us to do elementary math more simply.

Irrational

From the integers, the next level up takes you to the rationals which are all numbers that can be represented as a fraction ($\frac a b$) of two integers. The next extension from that is the irrationals, which are numbers that cannot be represented as fractions.

Similar to subtraction, we could make an argument about rational numbers not being “closed” under the “root” operation. You can take the square root of many rational numbers where the result is not a rational number. $\sqrt{2}$ is a famous example.

But this doesn’t provide justification for most irrational numbers because most irrational numbers cannot even be represented via roots of rational numbers².

The main motivation for irrational numbers comes from limits.

Example limit

Imagine a sequence defined like this:

$$ p = \{1, 1.4, 1.41, 1.4142, … \}$$

Each element of the sequence can be identified by calculating $\sqrt 2$ to increasing precision. This gives us a sequence where each element is a rational number. But if we take the limit of the sequence, we see that the sequence is approaching a number that is not rational (namely $\sqrt 2$).

Similar to how subtraction is not “closed” over the natural numbers, limits are not closed over the rationals. Rational numbers can define a sequence that approach irrational numbers.

Compactness

The fact that rational numbers can approach numbers that don’t exist in the rationals leads to a fundamental observation of rational numbers. The rational numbers are “sparse”, if you keep zooming in on the number line, there are many numbers that are not covered by the rational numbers. The hole only gets filled when you extend the set to include the irrational numbers. This property is called compactness.

The compactness of the Real Numbers (the set of numbers that include rationals and irrationals) is important to all of calculus. It is how we get continuity of functions. It is how we get derivatives and integrals. Even the simple-sounding Intermediate Value Theorem , that states the equivalent of “if you go from point A to point B then you must go through all the points in between”, requires compactness.

A simple way to observe the importance is by looking at the function $y = x^2$. We know that $1^2 = 1$ and $2^2 = 4$. Just from the graph we can see that the function should cover all values between $y = 1$ and $y = 4$. Even if we restrict ourselves to rational y-values, we can come up with a problem. When $y = 2$, we notice that $x$ has to be $\sqrt 2$ which is not rational! So if we only allowed for rational x and y values, our simple function $y = x^2$ would have “holes” in it (since there would be no x-value such that $y = 2$)! This is why calculus needs irrational numbers.

Even though real-world applications don’t need final results that are irrational (rarely do we have enough precision to worry about more than a handful of digits), the mechanics of calculations greatly benefit from being able to work in the real numbers. This is very similar to how even if the application doesn’t need negative numbers, the existence of them allows for a much richer foundation for the mathematical system.

Imaginary

I basically wrote this article for the imaginary numbers. Negative numbers and irrationals occasionally get some hate but imaginary numbers are the hated champions. This is not even new. Alice in Wonderland was written as the 1800s version of a diss track against imaginary numbers (and other abstract math). Imagine being so mad at a number system that you end up writing a timeless tale.

Imaginary numbers have two things going against them:

• Poor branding. Can’t call yourself imaginary and ask people to take you seriously. Descartes literally named it that to make fun of it.
• Arbitrary explanation. Students get told never to take a square root of a negative number and then suddenly one day are told “hey we did that thing anyways, here you go”

The branding is hard to solve. Gauss tried “lateral” numbers but it didn’t really stick. But we can try to improve the arbitrary explanation.

If we look at history of mathematics, a large chunk of it has been focused on solving polynomial equations like:

$$ ax^n + bx^{n-1} + … + c = 0$$

The popular quadratic formula that is taught in high school is the solution to the polynomial equation where $n$ is 2. We have similar, but more complex, formulas when $n = 3$ and $n = 4$ and we know that a formula can’t be written down for anything higher.

In the quadratic formula, we have this square root term: $\sqrt{b^2 - 4ac}$. If the part under the square root is negative, then we say that there are no real solutions and move on. The cubic formula is more complicated though. When Cardano’s formula was discovered, it had the weird property that in the calculation you had to use imaginary numbers (square root of negative numbers) even when the final result was a real number! This is very similar to how calculations with positive numbers may need to use negative numbers even when final result is positive.

This was the historical motivation for studying imaginary numbers. Initially, many mathematicians struggled to go from a “mechanical” trick to seeing broader implications of imaginary numbers. Reframing imaginary numbers as being “orthogonal” to real numbers (like the x-y plane) provided a geometric understanding of complex numbers. This allowed for a lot of trigonometry to be studied using complex numbers. Since then, complex numbers have become essential parts of many math and science fields.

Other

Some people find imaginary numbers frustrating because it feels “made up”. It feels like some authority figure decided to do something that wasn’t allowed and now we have to respect it for some reason.

While the previous section provides some historical context, it doesn’t really address this feeling of arbitrariness. One could argue “if I can just decide to take the square root of a negative number, then why can’t I do other things that are not allowed like dividing a number by 0”.

The true answer to this question is simply that you can! Mathematics is the study of rules. You define a system with whatever rules you want and then you study what those rules imply. What do those rules support and when do those rules break down. And you take the learnings and tweak the rules to see if you can get other fun results. Defining the rules is as much part of the game as figuring out what the rules result in.

So if you wanted to define a numerical system that allowed division with 0, you are totally allowed to do it. You just have to reverify which things still work in the new system and which break down. If we define that division by 0 is always infinity ($\frac a 0 = \infty$), then we end up with the projectively extended real number line. It’s a mathematical system that literally defines a system to support division by 0. As a result of supporting division by 0, the system comes with some quirks. Not all numbers in this system are comparable - the “number line” is more like a circle where everything comes before and after each other. Additionally, while this system allows for division by 0, it has to introduce restrictions on even more operations by limiting how 0 and infinity interact. Since this system doesn’t seem to provide any major advantages and doesn’t have major applications, it is not widely taught. But if we did have a reason to use such a system, it would be just as valid as the negative numbers or the imaginary numbers.

If the rebel in you thinks this is stupid and wants to come up with even stupider systems to prove how silly this is, then congratulations – you are a mathematician! Next step is to piss off someone enough to write a modern day Alice in Wonderland.