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Lecture 31: GCD, Definition of modular numbers

• Reading: MCS 9.6–9.8

• Review proof of correctness of GCD (this has been added to the lecture 30 notes

• Define $\mathbb{Z}_n$, the set of "modular numbers"

• Review exercises:
  • Prove that $\equiv_m$ is an equivalence relation
  • Prove $[a] = [b]$ if and only if $rem(a,m) = rem(b,m)$.

Modular numbers

The next several lectures will explore the arithmetic of remainders. Usually these results are presented as a set of equations about congruence mod $m$ or about remainders when divided by $m$ (MCS does both).

Instead, we will raise the level of abstraction a bit. We will define a new kind of object (the modular number), and redefine operations like $+$ and $\cdot$ for these objects. This is not a new process: you already have several kinds of things you know how to add and multiply: natural numbers, integer, rationals, reals, complex numbers, vectors, matrices, and random variables, to name a few. Each of these kinds of objects has a different algorithm for doing arithmetic on them; but because they all have a common interface, you have built up lots of intuition about how to manipulate them.

Definition: $a$ is congruent to $b$ (mod $m$), (written $a \equiv b~(mod~m)$ or $a \equiv_m b$). if $m|b-a$.

Note: It is easy to misinterpret this as $a \equiv (b~mod~m)$; this interpretation leads to confusion. Think of the "mod $m$" as a big note on the side of your equations or proofs, not as part of your equations.

Note: $\equiv_m$ is an equivalence relation (proof left as a review exercise).

Definition: The set $\mathbb{Z}_n$ of modular numbers is defined by $\mathbb{Z}_n = \mathbb{Z}/\equiv_m$.

Recall that $\mathbb{Z}/\equiv_m$ is the set of equivalence classes of integers by the relation $\equiv_m$: $\mathbb{Z}_{m} = \{\dots, [-2]_m, [-1]_m, [0]_m, [1]_m, [2]_m, \dots\}$, where $[a]_m = \{b \mid b \equiv_m a\}$. When the $m$ is clear from context, we will simply write $[a]$.

Note that $[-1] = [m-1]$ (because $m|m-1 -(-1)$ so $-1 \equiv_m m-1$), and $[-2] = [m-2]$, and $[m] = [0]$ and $[m+1] = [1]$, etc. In general, $[a] = [rem(a,m)]$, so $\mathbb{Z}_m$ can always be written as

$$\mathbb{Z}_m = \{[0]_m, [1]_m, [2]_m, \dots, [m-1]_m\}$$

Key facts: the following are equivalent:

1. $[a] = [b]$ (mod $m$)
2. $a \equiv b$ (mod $m$)
3. $m | b-a$
4. $rem(a,m) = rem(b,m)$

This follows from the definitions, with the exception of the equivalence of (3) and (4). To see that (3) implies (4), assume $m | b - a$. If we write $a = q_am + r_a$ and $b = q_bm + r_b$, we see that $km = (q_b - q_a)m + r_b - r_a$. This means that $r_b - r_a$ is a multiple of $m$. Since $r_b$ and $r_a$ are both less than $m$, we have $-m \lt r_b - r_a \lt m$; since $0$ is the only multiple of $m$ satisfying this property, $r_b - r_a = 0$.