Many would claim that organization is a basic aspect of being human–but organization in mathematics seems to be essential. The most basic thing organized in math are the numbers we use as a part of our tool kit. Just like the various ways to organize tools, there too exists ways to organize the numbers we use. We organize our normal numbers into natural numbers, integers, rational numbers, irrational numbers, real numbers and complex numbers. These categories exist to measure the depth and difficulty it requires to conceptualize each class.

Notice in the diagram how at the top, we start at just the general idea of numbers, but we slowly funnel “all” the numbers into real and complex. That is, all numbers are either real or complex and will only filter into one of the two. This keeps going on until we reach our most fundamental group of numbers, the natural numbers!

The organization is done by us to do whatever we see fit to use it for. The German mathematician Leopold Kronecker went even as far as to say “The natural numbers are the work of God. All of the rest is the work of mankind.” Yet just like the tool kit analogy we used, there are different ways to categorize the numbers. New categories can be created with specifications that overlap several of the old categories previously stated. Algebra is usually the gateway to the abstract side of mathematics, so it is fitting that it is our gateway to a new classification of numbers that may not be the most intuitive one. This classification is the algebraic numbers. BUT, what if a number is not algebraic? We will see in the following text how algebra is used to define the algebraic numbers–but what does this imply about numbers that are not algebraic? Are they beyond algebra?

Lets start with the algebraic operations (addition, subtraction, multiplication, division, raising to a rational exponent) and the natural numbers given to us by “God”. From them we can subtract to get the other half of the integers. Then we can divide them by each other to get the rational numbers. Then, finally, we run into irrational numbers by square rooting non-perfect square integers. Things can even get a little complex when we start taking the square root of negative integers.

The formal definition of an algebraic number is any root of a nonzero polynomial equation in the form

where the components are integers.

We can see how all the basic number types fit into the algebraic numbers. All of our integers and thus natural numbers would satisfy equations of the form:

Since each “a” can be any integer you want to solve the equation. Now we move on to the rational numbers. Let’s remind ourselves that all rational numbers can be expressed as fractions. Thus, when making an equation, we can make our 2 “” components in the following equation the denominator and numerator of any fraction we could possibly think of:

Irrational numbers are a change of pace, to say the least. In grade school, we were taught that irrational numbers were “non-repeating, not terminating decimals.” You can see the conundrum in defining them all as algebraic, but that doesn’t mean some aren’t! Perhaps our simplest way of creating an irrational number is simply taking the square root of a non-perfect square number. So, let’s make an equation that we would solve in such a way!:

Even the complex numbers fit into this framework with equations such as:

Many of the numbers commonly used in mathematics can clearly be seen to be algebraic numbers. But what happens when a number isn’t algebraic–when suddenly our construction methods fall short. That is when we come across a group of numbers that transcend algebra.

From what we know about algebraic numbers, what can we say about these transcendental numbers? The complex number plane contains numbers that are very hard to conceptualize, so let’s stick to the real numbers for now. Well, clearly the rational numbers are out–it was clear to see how all rational numbers were algebraic. Thus, it follows that if a transcendental number is real, then it must be irrational. Here’s a visualization:

This problem child of classifications dates as far back as antiquity to the problem of circle squaring. In that problem, the Greek mathematicians attempted to construct a square equal in area to a circle using only a straightedge and compass–a method comparable to the modern definition of algebraic numbers where rational number coefficients in polynomial equations are used. It took quite a while to resolve this problem, and it turns out this problem is related to the transcendental numbers!

Leibniz, in his study of the irrationals, demonstrated that the sin(x) function cannot be constructed algebraically from merely x, and this gave the building blocks to Liouville to construct a method for creating numbers not computeable through algebraic methods. However, it wasn’t until Charles Hermite that transcendental numbers were shown to also be naturally resulting. Hermite showed this by demonstrating that the number could not be algebraic, for if it were, then the components would have to be strictly between 0 and 1 (a contradiction since the components must be integers).

Carl Louis Ferdinand von Lindemann took the idea even further by demonstrating that raised to any algebraic number is also transcendental. But this helps us out even more as von Lindemann would show by using Euler’s formula:

That formula shows a power of that is an algebraic number. That means the exponent cannot be algebraic since that violates von Lindemann’s proof. Well, we showed earlier that must be algebraic, so we are naturally led to the conclusion that must be transcendental! you say–a number that is probably the superstar of mathematics? Well, this brings back our circle squaring problem. If we are trying to create a square with the area of a circle using only ruler and compass, that is the equivalent of using an algebraic polynomial to solve for the area of a circle. But now we know that isn’t possible because it would involve setting an algebraic polynomial equal to !

The term transcendental is a quite fitting term considering our knowledge of the numbers themselves. Hardly anything is even known about their interactions. The list of known transcendental numbers is actually quite small. SO WHAT NOW?!

We started our discussion by talking about organization. Many of our sciences seem to be in the business of organizing and categorizing, from astronomy to biology to chemistry. Math is just like one of those sciences in its attempt to classify the numbers. But unlike those other disciplines, we mathematicians have a much harder time classifying things. We have to rigorously prove a number classifies into a group–doing so seemingly involves delving into the abstract more so than not. Can you say the classifications in other disciplines go beyond observation?

Carl Friedrich Gauss is considered by many to be the greatest mathematician. That’s a tall feat considering people like Euler and Newton existed. But the extent and depth of Gauss’s contributions speak for themselves. According to Gauss “Mathematics is the queen of sciences…She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.” Some might say Math is the Queen because she is involved in all other sciences, but I think her true beauty is revealed by her ability to conceptualize and rigorously prove the unintuitive and seemingly impossible quantities of life. However, Math is a difficult queen.