Negative modulus

Question

In the programming world, modulo operations involving negative numbers give different results in different programming languages and this seems to be the only thing that Wikipedia mentions in any of its articles relating to negative numbers and modular arithmetic. It is fairly clear that from a number theory perspective $-13 \equiv 2 \mod 5$. This is because a modulus of $5$ is defined as the set $\{0,1,2,3,4\}$ and having $-13 \equiv -3 \mod 5$ would contradict that because $-3 \not\in \{0,1,2,3,4\}$. My question is then regarding a negative modulus. What definition of a modulus of $-5$ would make the most sense in number theory? One in which $13 \equiv -2 \mod -5$, one in which $13 \equiv 3 \mod -5$, or something else entirely?

Answer 1

Negating the modulus preserves the congruence relation, by $\ m\mid x\color{#c00}\iff -m\mid x,\,$ so

$\quad a\equiv b\pmod{\! m}\iff m\mid a\!-\!b \color{#c00}{\iff} -m\mid a\!-\!b\iff a\equiv b\pmod {\!{-}m}$

Structurally, a congruence is uniquely determined by its kernel, i.e. the set of integers $\equiv 0.\,$ But this is a set of the form $\,m\Bbb Z,\,$ which is invariant under negation $\,-m\Bbb Z\, =\, m\Bbb Z$

When you study rings you can view this as a special case of the fact that ideals are preserved under $\rm\color{#c00}{unimodular}$ basis transformations, e.g. $\,aR + bR\, =\, cR + dR \ $ if $\, (a,b)M = (c,d)\,$ for some matrix $\,M\,$ having $\ \color{#c00}{\det M = \pm1 = \rm unit}$ (invertible) element, e.g. $\ a\Bbb Z + b \Bbb Z\, =\, a\Bbb Z + (b\!-\!a)\,\Bbb Z,\,$ which is the inductive step in the Euclidean algorithm for the gcd (it computes the modulus $\,m=\gcd(a,b)\,$ corresponding to the congruence generated by $\,a\equiv 0\equiv b,\,$ i.e. $\,a\Bbb Z + b\Bbb Z = m\Bbb Z).\,$ When the ideal $= a\Bbb Z\,$ then this amounts to multiplying $\,(a)\,$ by a $\,1\times 1\,$ matrix $\,[u]\,$ with $\det = \pm1,\,$ i.e. $\, u = \pm1,\,$ yielding $\,a\Bbb Z = -a\Bbb Z,\,$ precisely the above equality.

Just as for gcd unit normalization it often proves convenient to normalize the modulus (generator) to be positive (or 0), and similarly for the lead coefficient of a polynomial modulus.

As for the choice of canonical reps for the congruence classes, it is your freedom to choose which system you find most convenient. For example, in manual computations it often proves most convenient to choose reps of least magnitude, e.g. $\, 0,\pm1,\pm2\,$ mod $\,5,\,$ e.g. the capability to use $\,-1$ simplifies many things, e.g. compare $(-1)^n$ vs. $\,4^n.$

Answer 2

According to definition, $a\equiv b\ (\mod c)$ iff there exists $n\in \mathbb Z$ so that $a = b + nc$. In particular, all of the following statements are true:

• $-13\equiv 2\ (\mod 5)$
• $-13\equiv -3\ (\mod 5)$
• $-13\equiv -13\ (\mod 5)$
• $-13 \equiv 222\ (\mod 5)$
• $13 \equiv -2\ (\mod -5)$
• $13 \equiv 3\ (\mod -5)$
• $13 \equiv 13\ (\mod -5)$

Answer 3

The multiples of 5 are the exact same as the multiples of $-5$, it's just they're indexed differently. So $13 \equiv -2 \bmod -5$ is just as valid as $13 \equiv 3 \bmod -5$, since $13 = -3 \times -5 - 2 = -2 \times -5 + 3$.

It is true that $-3 \not\in \{0, 1, 2, 3, 4\}$. But $-3$ is in $\{0, -4, -3, -2, -1\}$. We could say that 2 and $-3$ are "representatives" of the same "coset" (I hope this doesn't sound like I'm inventing the terminology off the cuff). This coset consists of all numbers that are 2 more than a multiple of 5 or 3 less than a multiple of 5: $5n + 2$ or $5n - 3$, where $n$ is any integer.

Obviously there are infinitely many other numbers to represent a particular coset. But something like $13 \equiv 13 \bmod -5$ looks like an obfuscation. In general, for $a \equiv b \bmod c$, you want $|b| < |c|$. But the preference is usually for $b$ positive, unless $b = c - 1$, in which case mathematicians much prefer to write $a \equiv -1 \bmod c$. Look up Fermat's little theorem or Wilson's theorem for examples of this.