Lecture 31: GCD, Definition of modular numbers

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, \(r_b - r_a = 0\).