Friday, May 18, 2018

It's all the same to me.

Mathematics is the subject, probably more than any other, that we associate with precision. And yet a particularly powerful tool in mathematics consists of a fruitful failure to distinguish between objects.
The basic idea is that we have a set of objects, but rather than thinking of them all as separate, we partition the set up into a collection of subsets which, between them, cover the entire set, and no two of which have any element in common. These subsets are called equivalence classes, and an element of an equivalence class is a representative. The set of equivalence classes itself is called a quotient space.
And most of the time, it's the quotient space that's the interesting thing, not the equivalence classes themselves. It can be all too easy to lose sight of this if one is buried in the detail of showing that a given relation between elements of a set does this job of partitioning into equivalence classes, and so is an equivalence relation.

Some Examples

A simple example is to partition up positive integers, as normally expressed in base ten, into classes of length; 1 digit numbers, 2 digit number, 3 digit numbers, and so on.
But this isn't very interesting. There's not much useful structure in this quotient space for us to play with. For example, if we add together two 1 digit numbers, sometimes the answer is 1 digit long, and sometimes it is 2 digits long.
On the other hand there are others, just as familiar, but more interesting.

Parity

Partition up integers (all of them this time, not just positive ones) into odd and even. This time things are very different. The quotient space consists of just two classes, which we can call ODD and EVEN. If we add together two odd numbers, the answer is even, whatever odd numbers we start with. Similarly if we add an odd and an even, or multiply.
The key here is that we do the arithmetic on two representatives, but no matter which representatives we pick, the answer is in the same equivalence class. This means that the quotient space \(\{\textrm{ODD},\textrm{EVEN}\}\) inherits operations of addition and multiplication from the set of all integers, \(\mathbb{Z}\).
There's another example, just as familiar, but the source of well-known problems.

Fractions

Two pairs of integers, \((m,n)\) and \((M,N)\) (where \(n\) and \(N\) are non-zero) are equivalent if \(mN=Mn\). This partitions the set of all pairs of integers (where the second integer is not zero) in the same way as being odd or even partitions the set of all integers. Secretly I'm thinking of the first pair as the fraction \(\frac{m}{n}\) and the second pair as the fraction\(\frac{M}{N}\).
This relationship - this equivalence relation - captures exactly the notion that the ratio \(m:n\) is the same as the ratio \(M:N\), in other words that that the two fractions represent the same rational number. We can think of each equivalence class as a rational number, and a pair of integers, i.e. a fraction, in the equivalence class, as a representative.
Then the usual arithmetic of fractions gives a way of adding and multiplying these equivalence classes, so that the equivalence class of the result doesn't depend on the choice of representatives of the classes being added or multiplied. In other words the rules for doing arithmetic with (and simplifying) fractions gives the arithmetic of rational numbers.
I can't shake the suspicion that the core of why schoolchildren have so much difficulty getting to grips with fractions is this idea of different representatives for the same rational number. It's a subtle idea, and it takes quite a lot of getting used to.

Going further

These ideas can be extended in various ways, some more familiar than others.

Congruence arithmetic

In the case of odd and even numbers, although it isn't the way we tend to think about it, we're regarding two numbers as equivalent if their difference is a multiple of \(2\).
Of course, there's nothing special about \(2\). We could pick any number to form the equivalence classes and it would work in just the same way. If we used \(3\), then sensible representatives are \(0,1,2\); if we used \(4\) then \(0,1,2,3\), and so on. If we are being careful, we use a notation such as \([0]\) to denot the equivalence class that \(0\) lives in. To save wear and tear on our fingers and keyboards, we can rather naughtily just use these representative numbers to denote the equivalence classes.
If we obtain our equivalence classes by using \(n\), then we call the resulting quotient space \(\mathbb{Z}_n\).
Actually, there is something a bit special about \(2\). If we multiply together two elements of \(\mathbb{Z}_2\), then the result is only \(0\) if one of the elements is. This is also the case with \(\mathbb{Z}_3\).
But with \(\mathbb{Z}_4\), we see that \(2 \times 2 = 0\).
OK, so from what I've written so far, it might be \(4\) that's special.
Really, what's special - or at least, well-behaved - is \(\mathbb{Z}_p\), where \(p\) is a prime. In this case, a product can only be zero if one of the multiplicands is zero. In fact, if we work with a prime, then we get a structure that is actually better behaved than \(\mathbb{Z}\), the set of all integers. We can divide by any non-zero element, in the sense that if \(a\) is not \(0\), then there is a unique solution to \(ax=1\); and multiplying by this number does just the job of dividing by \(a\). In the jargon, \(\mathbb{Z}_p\) is a field, which basically means that has a well-behaved addition, subtraction, multiplication and division.
This results in both fun and profit: the resulting number systems are fun to play with, and form the mathematical basis for some of the contemporary forms of cryptography.

Complex Numbers

We can use the same ideas to construct the complex numbers without inventing a new kind of 'imaginary' number whose square is negative.
The starting point is to consider the set of all real polynomials, and then to consider two polynomials as equivalent if their difference is a multiple of \(x^2+1\).
Then one (and the useful) representative for any polynomial is the remainder when it is divided by \(x^2+1\), so the representatives are of the form \(a+bx\) where \(a,b\) are real numbers. But then \(x^2 = 1\times(x^2+1)-1=-1\), where I misuse '\(=\)' to mean 'is equivalent to'. Then the arithmetic of these objects looks just like normal complex arithmetic with \(i^2=-1\), except that I have an \(x\) where you might expect to see an \(i\).
To phrase that slightly differently, the complex numbers, \(\mathbb{C}\), are the quotient space obtained by regarding two polynomials as equivalent if they differ by a multiple of \(x^2+1\).
This doesn't give us any algebra we didn't have before, but it does show that there is a perfectly respectable way of constructing the complex numbers out of non-problematic objects. We don't have to postulate some new kind of number whose square is negative, and then worry about whether we have accidentally made mathematics inconsistent. Well, at least we can be sure that complex arithmetic is no more inconsistent than real arithmetic.
And just as there is nothing special about \(2\), there is nothing special about \(1+x^2\); we could use any polynomial here.
In fact, the similarity to \(\mathbb{Z}_n\) is profound.
Just as with the integers, if the polynomial we work with cannot be expressed as a product of lower degree polynomials, we can add, subtract and multiply any pair of polynomials, and also divide by any non-zero one.
This idea lets us construct a whole new collection of algebraic structures, including in particular the finite fields, which are important in aspects of statistical experiment design and error correcting code.

The wild blue yonder

It doesn't stop here. Examples of naturally occurring partitions and quotient spaces abound thoughout mathematics, in geometry, algebra and analysis.

Linear algebra

If \(V\) is a vector space, and \(S\) is a subspace of \(V\), then we can think of two vectors \(u\) and \(v\) as equivalent if \(u-v \in S\). This relationship partitions \(V\) up into subsets that look like copies of \(S\) obtained by translation away from the origin, called cosets. The quotient space is again a vector space.
This turns out to be an important construction in analysis, where \(V\) and \(S\) are spaces of functions.

Group theory

If \(G\) is a group and \(H\) is a subgroup of \(G\), the we can say that two elements \(g_1,g_2 \in G\) are equivalent if \(g_1g_2^{-1} \in H\), and again this gives us a partition of \(G\) into cosets. This time it isn't automatic that the quotient space gets a well-defined multiplication from \(G\); \(H\) has to satisfy a certain condition, called being a normal subgroup.
In the case of Abelian groups, this lets us break a big group down into a kind of tower of other groups, in a way analogous to the prime decomposition of an integer.
It is also important in Galois theory, which relates the structure of certain groups to the solubility of polynomials.

Geometry

We start off with the \(x-y\) plane, and consider two points to be equivalent if their \(x\) and \(y\) coordinates differ by integer amounts. This time the quotient space is a new surface, which we can visualise as a unit square with opposite edges glued together: a familiar structure in certain video games, where leaving the screen to the left, or top, brings you back in at the right, or bottom. In fact, this quotient space is a torus.
Constructing surfaces as quotient spaces is a powerful tool in the attempt to classify all the possible surfaces. And it also suggest the question of when two surfaces should be regarded as 'really the same': another partitioning of all surfaces described in different ways into equivalence classes of 'the same surface, but a different description'.

And so on

This barely scratches the surface. The more mathematics, and the more mathematical structures, you meet, then more you come across interesting quotient spaces.
Googling for 'quotient' plus the thing you are interested in (group, surface, vector space, and many more) will probably return something cool, so don't waste any more time here, go and look.

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