# Modular Arithmetic and Quotient Sets

There is more than one way of counting. The one we are most familiar with goes like this:

$\displaystyle 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,...$

and so on getting to bigger and bigger numbers. The numbers are infinite of course, so with every new count we will be naming a new different number bigger than the previous one.

Another way, also familiar to us but one we don’t often pause to think about, goes like this:

$\displaystyle 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3,...$

I’m talking about the hours on a clock. This way of counting repeats itself, and there is only a finite set of numbers that it goes over.

If we can do arithmetic with ordinary numbers, so can we with the numbers on a clock. What is $11+2$? In ordinary arithmetic, it is $13$, but on a clock, it is $1$. $1$ is the remainder of $11+2$ when divided by $12$. This kind of arithmetic is called modular arithmetic, and it is often associated with one of the greatest mathematicians of all time, Carl Friedrich Gauss.

If the hands of a clock now point to $5$, after $100$ hours, where will it point? We do the procedure earlier, and get the remainder when $5+100=105$ is divided by $12$. We will then get $9$. It is strange to talk of multiplication when referring to a clock, but we can do multiplication also in the same way if we want to. As for subtraction, we can ask, what is $5$ o’clock minus say, $7$ hours? We don’t say “$-2$ o’clock”. Instead we say that it is $10$ o’clock. So there is a way of keeping the numbers positive: Just keep adding $12$ until we get a positive number less than $12$. This is also similar to the remainder procedure above. Essentially we just add or subtract $12$ until we get a positive number less than or equal to $12$. Later we will change our notation and instead choose non-negative numbers less than $12$.

Division is too complicated to speak about for now. Instead I’ll just try to link what I said with the more formal aspects of mathematics. This set of “numbers on a clock” we will call $\mathbb{Z}/12\mathbb{Z}$. $12\mathbb{Z}$ means to the set of integer multiples of $12$ like $...,-36, -24, -12, 0, 12, 24,36,...$ and so on. $\mathbb{Z}/12\mathbb{Z}$ means that if two numbers differ by any number in the set $12\mathbb{Z}$, we should consider them equivalent. The rule that specifies which numbers are to be considered equivalent to each other is called an equivalence relation.

So $13$ o’clock is equivalent to $1$ o’clock (not using military time here by the way) since they differ by $12$, while $100$ is equivalent to $4$ since they differ by $96$ which is a multiple of $12$. For the purposes of notation, we write $13\sim 1$ and $100\sim 4$. Our equivalence relation in this case can be expressed by writing $n+12\sim n$ for any integer $n$.

All the numbers that are equivalent to each other form an equivalence class. We can think of $\mathbb{Z}/12\mathbb{Z}$ as the set of equivalence classes under the notion of equivalence that we have defined here. We can select “representatives” for every equivalence class for ease of notation; we choose, for convenience, that $\displaystyle 0,1, 2, 3, 4, 5, 6, 7, 8, 9, 10$, and $11$ represent the respective equivalence classes which they belong to. Note that we chose $0$ instead of $12$ to represent the equivalence class which they belong to – while we’re used to saying $12$ o’clock, mathematicians will usually choose $0$ to “represent” all its other buddies that are equivalent to it.

We can think of the process of going from the set of integers $\mathbb{Z}$ to the set of equivalence classes $\mathbb{Z}/12\mathbb{Z}$ as being mediated by a function. A function simply assigns to every element in its domain another element from its range. So here the function assigns to every integer in $\mathbb{Z}$ an equivalence class in $\mathbb{Z}/12\mathbb{Z}$. The set of integers that get sent to the equivalence class of $0$, i.e. the set of integer multiples of $12$, is called the kernel of this function.

$\mathbb{Z}/12\mathbb{Z}$ is an example of a so-called quotient set. The rather confusing terminology comes from the fact that we used the group operation of addition to define our equivalence relation; since group operations often use multiplicative notation, the term quotient set makes sense in that context. In this case since our set also forms a group we refer to it also as a quotient group. If we discuss it together with multiplication, i.e. in the context of its structure as a ring, we can also refer to it as a quotient ring. (See also the previous posts Groups and Rings, Fields, and Ideals).

There are many important examples of quotient sets: $\mathbb{Z}/2\mathbb{Z}$ can be thought of as just $0$ and $1$, reminiscent of “bits” in computer science and engineering. Alternatively, one may think of $\mathbb{Z}/2\mathbb{Z}$ as a set of two equivalence classes; one is made up of all even numbers and the other is made up of all odd numbers. We also have $\mathbb{R}/\mathbb{Z}$, where $\mathbb{R}$ is the real line. $\mathbb{R}/\mathbb{Z}$ can be thought of as the circle; I won’t explain now why but one can have a fairly nice mental exercise trying to figure it out (or just check it out on one of the helpful references listed below).

References:

Equivalence Class on Wikipedia

Quotient Group on Wikipedia

Quotient Ring on Wikipedia

Algebra by Michael Artin