Add, subtract, multiply and divide. From small we have learned that these are the fundamental operations on which all the mathematics of numbers are built. We also learn that the division has something a little special, such as that it cannot be divided by zero, but that is small details, isn’t it? The idea of ”adding” or “multiplying” is so basic that, no doubt, we will find them almost anywhere.
When we learn a little more, we realize that mathematics is not only about numbers. In the zoo there are other animals, such as vectors, or matrices, which are a bit more peculiar. Everyone can be added or subtracted, but multiplying matrices is quite different from what we do with numbers, and vectors cannot even be multiplied, or at least not to give another vector. The division is even more problematic, because in the world of matrices there is not even talk of “dividing,” but of “calculating the inverse.” If you don’t know what I’m talking about, don’t worry. What I want to get to is: what is happening? Were multiplication and division not very simple notions that we should find everywhere?
Dissecting the multiplication
They are, but you have to be careful. Actually, the intuitive notion of multiplication is to “make something bigger,” and that of division, “make it smaller.” That is achieved in the world of vectors and matrices simply by multiplying by a digit. On the other hand, the multiplication between matrices is “I will combine two matrices to get a third one”, and that is what, in general, we do not usually do with vectors. The trap is that in the world of numbers “making a number bigger” is “combining two numbers to get a third (greater)”, so our numerical intuition has mixed these two notions.
But, since we have found this distinction, let’s examine it a little closer. This “combine two things to form a third of the same type” is it really a multiplication or is it something else? The sum, for example, follows that same pattern. Actually, we call all the operations that follow that pattern internal operations. So that we can properly call them multiplications we need one more element: the distributive property. That is, when multiplication is combined with addition, it does so in the same way as numbers: a⨯ (b + c) = a⨯b + a⨯c. In other words: multiplication is an internal operation further of the sum. We will rarely call something “multiplication” if we cannot also add.
In mathematics there is a name for the sets in which we have, at the same time, a sum and a multiplication: they are called algebras (Yes, it is the same name as “Algebra”, one of the branches of mathematics, but this is another meaning). There are many different algebras: the algebra of the matrices, for example, or the algebra of the numbers of all the life. All of them are sets whose elements can be added and also multiplied. But when the thing gets interesting is when we want too divide: that is, if we want to reverse multiplication. That can no longer be done in so many ways. In fact, surprisingly, it can only be done in four ways: it’s four o’clock division algebras.
The four ways to divide
This result was demonstrated one hundred years ago by the German mathematician Adolf Hurwitz, who did not publish it in life, and then, independently, by the Austrian Johann Radon. Essentially, what it says is that we can imagine very complicated, truly rare objects, as Martian as we want, but if we want to be able to add them, subtract them, multiply them and divide them there are only four ways to do it. Depending on the properties of those objects, and the properties of their multiplication, we will have to use one or the other.
The first “division style” is the one that sounds the most: that of real numbers. It has all the properties to which we are accustomed, and one in particular that makes it special: the objects that we are multiplying and dividing can be ordered. If we take two numbers, one will always be bigger and the other smaller. As long as we have a set that can be ordered, this will be the only style of multiplication and division that we can define, although the opposite is not true: not all ordered sets can have a well-defined division.
The second division algebra will sound to you if you have done science careers: it is that of complex numbers. It is, in a nutshell, two copies of the real numbers, but we give one of them a somewhat peculiar property: when multiplying by themselves they give a negative number. Every complex number is built with a part real, whose square is positive, and another imaginary, whose square is negative. These parts can be added or subtracted, but they will always be different, because they have antagonistic properties: positive square or negative square. That is why the complexes need two complete copies of the real ones, and therefore their dimension is 2.
It is notable that complexes are constructed by changing one of the properties of multiplication: adding invented numbers whose square is negative. This already indicates that we are facing a new way of multiplying.
Quaternions and other fantastic animals
With the third division algebra the curves begin to arrive. This time we need four copies of the real ones to build it. Of these, one will have positive squares, and the other three, negative: it is like taking the complex numbers and adding two more copies of the imaginary numbers. In addition, we will modify the multiplication again: the three types of imaginary numbers will not respect the commutative property. You know, the old saying “the order of the factors does not alter the product”. Well, for those three copies it will be a bit different: we will have to a⨯b = -b⨯a. Once this is done, we can add, subtract, multiply and divide. Welcome to the algebra of the quaternions.
Quaternions may seem quite Martian: they have dimension 4, three imaginary units and their multiplication is noncommutative. However, when we look closely, it turns out that rotations in three-dimensional space can be understood with quaternions. Applying several rotations in a row is equivalent to multiplying quaternions in the proper way, and therefore they are used regularly in 3D design programs. Ironies be a division algebra.
The last guests of today are genuinely exotic beings: octonions, the largest of division algebras. They are formed by the royals and seven copies of imaginary numbers. I will not drive you crazy with the properties of its multiplication, but suffice it to say the following: if quaternions had lost commutative property, octonions lose their associative, that is: expressions like a⨯b⨯c It is meaningless. For that expression to mean something we have to say in what order we are going to multiply the three octonions: (a⨯b) ⨯c It is a perfectly acceptable expression, and a⨯ (b⨯c) also, but in octonions they may have different values.
In a world accustomed to associativity, in which we hardly know how to think without associative property, octonions are a Rare avis. There have been several attempts to relate them to certain properties of fundamental physics, but they are all very speculative. They do have very stimulating connections with deep branches of mathematics, such as exceptional Lie groups or sporadic finite groups, which we may talk about another day. But who is surprised: the octoniones have, after all, a very special multiplication: one of the four into which it is possible to divide. The only thing they couldn’t be is boring.
DON’T HAVE IT
- Usually, in mathematics the sign of multiplication is omitted, which we have represented here by “⨯”. In the scientific literature you will never find a cross when you have to multiply.
- The algebras of division are four, but each one of them can take many forms: sometimes you can be looking at the same “type of multiplication” and not realize it because it looks so different.
- There are algebras where division is possible, but only partially: some elements can be used as divisors and others not. Those spaces, which are many more than four, are not division algebras, but their multiplication may have common properties with some of these four.
- Dixon, Geoffrey M. Algebras Division: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Springer-Verlag, 1994.
- Sanderson, Grant (3blue1brown). What are quaternions and how do you visualize them? A story of four dimensions. Youtube video. Sep 6 2008
- Baez, John C. The octonions Bulletin of the American Mathematical Society, vol. 39, pp. 145-205 (2002)
- Furey, Cohl. Charge quantization from a number operator. Physics Letters B, vol. 742, pp. 195-199 (2015)