Why the modular arithmetic requires modulus to be an integer greater than one?

Asked Aug 07 '22 at 08:59

Active Aug 07 '22 at 08:59

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The modular arithmetic article in Wikipedia tells that the modulus in modular arithmetic should be an integer greater than $1$.

But I can easily imagine a similar algebra where equivalence classes are defined as $a\equiv a+n$ or $a\equiv a+n/2$, for $n\in \mathbb{Z}$.

As such, the requirement that modulus should be integer greater than $1$ seems artificial to me.

asked Aug 07 '22 at 08:59

Anixx

2

Positive normalization of a modulus is merely for convenience - just like for denominators, and similarly for unit normalization of gcds. Further, as you mention, we can also consider fractional ideals and modules, but they are not usually introduced in a first course on number theory. – Bill Dubuque Aug 07 '22 at 11:39
The question is about the modulus, not the remainder. You can have modulus of 1 or even complex ones like in $$\exp(x) = \exp(y) \iff x\equiv y \mod 2\pi i$$ – emacs drives me nuts Aug 07 '22 at 11:55
@emacs Who mentioned remainders? At Anixx: are you familiar with ideals or modules? – Bill Dubuque Aug 07 '22 at 17:28
@BillDubuque no – Anixx Aug 07 '22 at 17:29

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