What is a Number Really?

Fundamental building blocks of numbers and an exploration into these beautiful atoms

Every time a new number was introduced to this world, it was met with criticism, mysticism and denial.

Even famous numbers like 0 and -1 started out as freaks among other numbers.

Stars of the show like π and √2 have been deemed “not of this world" in the past.

But why?

And can we find new ones?

Throughout the history of mathematics, numbers have undergone a fantastic evolution starting with the positive integers that were used to count stuff, and then on to the fractions which seemed like the right kind of measurement tool when beaking bread into pieces.

Much later, the Greeks discovered that not all numbers can be expressed as fractions.

This started a crisis so huge that had the internet been around at that time, a fierce online debate would have gone viral for sure.

When these mystical irrational numbers first appeared about 500 BC, they were not even considered to be numbers and they were quite paradoxical to people.

It still hurts many people to think about the randomness in π’s digits and that no matter how much tech and energy humans are gonna be able to master in the future, we will never have a computer powerful enough to store the number √5.

Yet our brain is able to understand this number without any trouble! This is the power of mathematics!

When the negative numbers appeared they too were not seen as actual numbers but more like a necessary evil in order to calculate more efficiently, dealing with debt for instance.

Zero was an important Indian invention, that revolutionized our number system and of course gave us our beloved additive identity.

People also found this number quite paradoxical since they saw it as describing nothing, but somehow it was still something!

I am pretty sure that they had an equivalent of “WTF” in the 600s!

In Renaissance Europe, yet another bizarre number appeared as a necessity for solving cubic equations.

This number is now called “the imaginary unit” and is denoted i in most modern math textbooks.

It has the property i² = -1.

This sparked the era of complex numbers.

A complex number is a number of the form a + bi where i is the imaginary unit and a and b are real numbers.

They were seen as an embarrassing but necessary tool and were illegal to use for many years until they were put on a solid mathematical foundation much later.

Nowadays, complex numbers are very real in the sense that they are applied to many real problems like integrals of real functions, number theory, radar technology and quantum physics to name a few.

Fundamental Numbers?

I am telling you all of the above because I want to prepare you for what’s coming in this article.

I want to introduce you to four new numbers with some new properties.

We will append them to our real numbers and see how they interact with each other.

It will reveal many interesting things among them a new kind of Euler formula which in its usual form has been called the most beautiful formula in the World.

Well, we will see a new version of that using our new numbers.

At the end of the article, I will show you what these “numbers” really are mathematically speaking putting them on a more solid footing.

Until then, bear with me and enjoy the adventure into this alien world.

Let’s define them.

We will call our new numbers α, β, γ and ε.

The first property of these numbers is that α + ε = 1.

Secondly, they are independent of each other meaning that we can’t combine a subset of them and get one of the others in return.

We can define how they multiply with each other in the following table.

The first thing to note here is that they do not commute in general.

For instance γβ = ε but βγ = α.

However, all these new numbers commute with the real numbers by convention.

This is already wildly different from anything we are used to and this will in some sense complicate matters, but as you will see, it will also enrich the theory.

The above table should be read as follows: the numbers below the multiplication symbol are left factors of the product and the numbers to the right of the symbol are right factors.

The set of numbers we would like to consider are the numbers we get by formally appending the four numbers to our real numbers.

That is, we consider numbers of the form rα + sε + tβ + xγ where r, s, t and x are real numbers.

We will also assume that the distributive law holds in our new numbers i.e. a(b+c) = ab + ac.

This assumption will be justified at the end of the article.

Note that the set of real numbers is a subset of this bigger set because for each real number r, we have rα + r ε = r (α + ε) = r.

We have already seen that α + ε = 1.

Let’s try to calculate (γ - β)².

We have (γ - β) (γ -β) = γ² - γ β - β γ +β² = - ε - α = - (α + ε) = -1.

So (γ - β)² = -1. But wait… Where have we seen this before?

Oh, that’s right. Then γ - β = i.

So in fact the set of complex numbers is a subset of this bigger set too.

We can actually build the numbers 1 and i from these fundamental numbers that we just introduced!

Beautiful Revelations in This New World

Let us calculate a few fascinating results.

Let us define the exponential function using Taylor series to get the following result. I will let it speak for itself.

Very nice.

By symmetry, we quickly see that e^{ε x} = α + ε e^x. By taking logarithms, we obtain x^α = ε + α x and x^ε = α+ εx.

When you have said A…

Well, needless to say, we have implicitly created some new numbers that we haven't talked about yet.

We know that α + ε = 1 and γ - β = i but what is α - ε and γ + β?

First things first. Let’s give them some names. Specifically, let’s denote them by g = γ + β and h = α - ε.

By inspection we have g² = h² = 1. We also have, for example, ig = h.

You can find more examples yourself.

One amazing result is a calculation revealing the Euler formula for the numbers g and h.

Recall that:

By using Taylor series again, we obtain

By the same property, we get a similar result for h. To sum up, we have the two formulas:

This is of course only the beginning.

How about doing calculus with these numbers? We know that the calculus of complex functions reveals intrinsic beauty and effectiveness in many areas of mathematics.

What would happen if we tried to define derivatives for functions taking linear combinations of i, g and h?

There is a lot more to explore than I have put forward here. But the above will be a separate article because this one becomes too long.

So What Are These Numbers Really?

It turns out that one can understand numbers from many perspectives.

For example, the imaginary unit i can be understood geometrically as a rotation by 90 degrees.

But we can also understand the complex numbers as a mathematical field containing the real numbers as a subfield.

More generally, we could view it as a special kind of ring. The integers and the rational numbers are also rings. Commutative rings.

In the same way, 2×2 matrices with integer entries also constitute a ring, but this ring is not commutative.

In fact, we can show that the above-mentioned ring is isomorphic to the ring generated by α, β, γ and ε, and in that sense, what we have done in this article is nothing more than matrix multiplication in disguise.

Conclusion

Why shouldn’t we allow these exotic numbers into our other numbers?

Recall that the negative numbers were once seen as devilish beasts too!

Well, it might be because the non-commutativity of these ring elements makes it a little hard to apply them.

But that is just until we are done with this research, right?

Please think about this and tell me if you find something of interest. Then I will do the same…