Be rational — get real!

Why all numbers are imaginary

In mathematics, imaginary numbers are all multiples of i with a real number. They are called imaginary because they are the solution to the square root of any negative real number, which doesn’t exist.

I will explain why many other types of numbers are actually imaginary, which means they don’t really exist or, in other words, we can only imagine them.

“God made the integers, all else is the work of man” — Leopold Kronecker

For all numbers, we will analyze their most natural representation in the real world. You will probably disagree with some points I say, and that’s fine. This article is more a thought I want to present than anything like a proof or similar; especially because it dives deep into philosophy. Please bear that in mind while reading and please let me know what your thoughts are and on which points you agree or disagree on!

The Beginning

Let’s start at the beginning. The natural numbers; or in other words, those numbers that we can naturally observe and that are deeply rooted in our language and counting. I can for example say “I have no (0) sheep” or “I have five (5) sheep” or “this is my second (2) sheep” and so on.

Operations

Additionally to numbers, we also have operations that represent things from the natural world; either arithmetically or geometrically.

Addition: “I have 2 sheep and add 3 sheep so that now I have 2+3=5 sheep.“ or “I have a 5 units long street and add another 5 units so that now the street is 5+5=10 units long.”

Multiplication: “I buy 3 chocolate bars that each cost 2 units. I have to pay 3*2=6 units.” or “the area of a rectangle with sides of lengths 4 units and 5 units is 4*5 = 20 area units.”

Subtraction: “I take away 3 sheep from 5 sheep so that then I have 5–3 = 2 sheep.” or “I cut 20 units away from a 70 units long piece of wood so that now the piece is only 70–20 = 50 units long.”

Division: “I have 9 sheep and 3 friends and want to give each of them equally many sheep, so I give each of them 9/3 = 3 sheep.” or “I cut a 100 units long piece of wood into 5 equal pieces so that each of those pieces has a length of 20 units.”

These are the very basics that probably feel pretty obvious to you. However, we want to extract some lessons. One of the most intuitive is that addition and multiplication make something bigger/more. Addition and multiplication represent when we take smaller things and make bigger things out of them — essentially building. On the other hand, subtraction and division make something smaller/less. They represent taking big things and splitting them into smaller pieces — destructing.

The other thing we notice is that whether we add 2 sheep to 3 sheep or 3 sheep to 2 is just a matter of perspective. But, in essence, they represent exactly the same thing. Similarly, when we rotate the 4-by-5 rectangle into an 5-by-4 rectangle, the area obviously doesn’t change. This concept we call commutativity (swapability/moveability).

But the most important thing is that some operations are simply not possible. You are physically not able to take 6 sheep from a farm with only 5 sheep. And you cannot divide 5 whole sheep onto 2 people fairly (meaning equally).

Negative Numbers

And this leads us to the truth, which is that all negative numbers are essentially man-made:

• -10 °C are simply 263,15 °K (ignore the comma for now)
• -5000 Euros on your bank account really just means you are 5000 Euros in debt
• and a force of -10 N really just means it is equal but exactly opposite to a force of 10 N. We only use the man-made properties of negative numbers to model that these forces cancel each other out if we apply (add) them to an object. Which is exactly what we will experience in real life

To put it way too abstractly

In some ways, you could say that a negative number is simply a subtraction that couldn’t be resolved yet. If you have -1 sheep that isn’t actually possible, it just means that if you get another sheep you can pay your debt or alternatively resolve the transaction that would bring you down to -1 sheep mathematically.

So, similarly to how 1 is defined as 0+1, 2 is defined as 1+1, etc. -1 is defined as 0–1, -2 is defined as -1–1, etc.

For a computer scientist, this fact is fascinating, as this means that all numerical algorithms only require the operations increment and decrement (and loops and equality operators)

Negative times Negative

This math exchange question revolves all around a proof for the fact that two negative numbers definitely result in a positive number. I think it makes sense to again come from the most practical side, which we already did the whole time from the start. Someone else getting rid of your 5 debts worth 5000 € each is the same as giving you 25.000 €: Getting rid of something naturally means subtracting — so something negative, and being 5000 € in debt is also a negative, but together they add 25.000 € — so something positive — to your total balance.

Rational numbers

Rational numbers, like natural numbers, come very naturally to us. They are symbolized in two ways: either as fractions or as (decimal) point numbers. And they represent two things:

Arithmetically — Acting out a division into equal pieces: Cutting a cake in 8 pieces so that each piece of cakes is an eighth of a cake. Eating 3 pieces of cake is then eating 3*1/8 = 3/8 of a cake.

Geometrically — Choosing smaller units: A 3.8 Meter long board is actually just 38 Decimeter or 380 Centimeter long.

So in essence we observe that all rational numbers are simply natural numbers, but this we essentially already knew from the proof for the equal mightiness of N and Q.

0/0

The biggest question is what 0/0 is. Some will say the answer is not defined. But the answer actually is very simple: it isn’t a single number but a huge range of numbers which essentially comes down to the set of all defined numbers. Ask yourself: How many people do I need if I have 0 sweets and I want to give every person 0 sweets?

Well, really, the number of people is totally irrelevant. You could have 10 or 20 or 50 or 1,000,000,000,000 people; everyone would just get 0 sweets and everything would be fine apart from some sad faces.

Infinite rationals

Now you might be thinking of infinite (periodic) rationals, e.g. 1/7. When we try to convert this fraction into a decimal number, we get a periodically appearing set of decimal places. How can that be a natural number?

Well, what you have to see is that the real problem lies within the representation of our point numbers. We decided to use the decimal system, but this leads to some problems when trying to fit fractions into decimal numbers. If we just used a different number system that shares a divisor with 7, e.g. a seven-based number system where 1/7 is 0.1, then everything would work out perfectly. And we can always find a “good” number system for any finite set of rational numbers (a closed system).

The physical world

The funny thing is that you can actually observe the equivalence of Q to N in the real world in quantum mechanics but also more generally in physics/engineering.

• There is something called the Planck length. It is the smallest possible unit of space in which the laws of physics make sense and can be applied. It doesn’t really make sense to operate on a much smaller scale.
• One of the most important and also most underrated discoveries of the 20th century was that energy is quantized. This means that energy is always transmitted in small packets of a minimal load. If that wasn’t the case, we would be grilled by high energy waves from the sun.
• Mass is, at least on a chemical level, also quantized. Taking the greatest common divisor of the masses of all atoms gives us a unit with which we can represent all possible masses of all possible matter exactly.
• The SI unit system actually redefined the meter so that the speed of light is exactly 299 792 458 m/s. This allows any speed to be described as a fraction of the speed of light.
• To become more realistic, we should think of measuring accuracy. It doesn’t really make sense to have a unit that is much smaller than what we can measure. If you can only weigh up to the 100th of a kg, your unit should be that, and then you will see that any measured value is simply a natural number.

Infinity

Infinity simply means endless. So if something is infinite, it doesn’t have an end. In reality, however, everything has an end and there is always something greater than any created thing. We cannot really imagine something that doesn’t ever end. To come to the point, infinity is beyond imaginary.

To handle infinity, we use the limit notation. It says what if we went on a long and endless travel towards infinity, what would happen on that journey. When thinking about this problem, you recognize that absolute values simply don’t have an effect on “infinity” and therefore inf+5=infand inf*5=inf. You also notice that infinity and 0 are in some way — a very informal way— multiplicative inverses of one another. Meaning that lim [x->inf] (1/x) comes closer and closer to zero and lim [x->0] (1/x) goes towards infinity (we assume -inf=inf just like -0=0).

The Real Numbers

Let’s just summarize, what we have understood already:

The natural numbers, their operations and their limitations come very naturally to the human mind and our understanding of the world around us. Negative numbers describe that something is missing and fractions/rational numbers describe dividing something into pieces, which is also very natural to us.

In contrast, the real numbers are nothing like that. Even though they are called real, in some sense, they are really unreal. This is very intuitive to a computer scientist, who knows that a computer can never store a really real — meaning irrational — number; it would take infinite space which doesn’t exist.

Many mathematicians will tell you the story of how numbers were continuously created to be solutions to mathematical expressions that “didn’t have a solution”. They will tell you how the whole numbers were created to make subtraction always possible (mathematically) and rational numbers were made to make division always possible. Real numbers were created to describe the square roots of some natural numbers (like 2), π or e. And then finally, the complex/imaginary numbers were created to describe the roots of negative numbers (like √-1).

This narrative is beautiful, but it often ignores reality.

• You can’t cut 2 meters from a board that is only 1 meter long.
• You can’t divide one sheep into several pieces (without turning it into pieces of sheep meat).
• There is no number thats square is 2 exactly, you can’t draw a perfect circle, and you can never reach infinity.
• Also, if you think about what a square root is geometrically: The square root of a number is the length of the sides of a square with an area of that number. But since when can areas be negative?

So what are the real numbers that all mathematicians work with every day. For that, we have to look at the units that we briefly introduced when talking about the rational numbers, and we should look at the proof for the irrationality of √2.

Units

We first need to assume that every number only makes sense if it is assigned to some sort of unit. This could be anything that the operations you do make sense on. The unit could even be something highly abstract.

What we then discovered is that for any fixed set of rational numbers given in some unit (e.g. some sensor values) we can always find a new unit so that all the values can be described as natural numbers in that unit. And we can do this as long as we need to when adding new numbers that are even more fractioned.

The irrationality of √2

The proof for the irrationality of √2 is a very simple proof by contradiction and if we express it in our units theory it goes like this:

• Take any unit with √2 as a rational number in that unit. Now whatever other smaller (fractional) unit you take, √2 isn’t a natural number in that unit, but instead it is at least a half fraction.

Random note to think about: Let’s look at the problem from another perspective: We imagine that we have a square with an area of 2sqm. Now we define a new unit x^2=1/8sqm and our square is 16x² big. So that the squares sides lengths are exactly 4x big. Now the problem is obviously that we can’t convert meters into x exactly — only approximately, but is it nonetheless a valid unit?

So now lets follow this idea and let us look at what happens when we take our unit on a long infinite journey towards 0 (e.g. by constantly dividing the unit by 10). What will happen then?

• We add more and more numbers -> towards infinity
• We add more and more decimal places -> towards infinity
• The natural numbers associated with our decimals grow -> towards infinity.

And suddenly we have the real numbers in front of us. And suddenly everything makes sense.

• It makes sense that real numbers have infinitely many decimal places (but not really)
• It makes sense that the real numbers are mightier than the natural numbers. They are literally endlessly mightier because — don’t take this to literally — they start where the natural numbers end: at infinity.
• It makes sense that a real point has dimensions of 0 but an arbitrary amount of infinitely many number of points in a range together have positive dimensions (0/0=inf/inf=0*inf=?)
• It makes sense that between any two numbers there are infinitly many more numbers
• It makes sense that all irrational numbers are infinite sums of rational numbers (Taylor series, calculus)

Why the real numbers are still imaginary

Because real numbers are “infinite” and therefore must be imaginary as everything infinite is simply imagined and cannot be perceived, experienced or observed. A perfect point only exists in our heads, a perfect circle, even maybe just a perfectly straight line.

Signining Out,

Rashid Harvey