Why Are Congruence Relations Written as $”a\equiv b \pmod m”$

Question

Why when writing a congruence relation, do we have to add $\pmod{m}$ at the end. I understand it's notation, but $b$ does not have to be equal to $a\bmod b$ if I understand correctly. The definition states that "$a$ and $b$ are said to be congruent modulo $m$, if their difference $a − b$ is an integer multiple of $m$." If that is true, then one can write $a\equiv b \pmod{m}$

So, for example, if we let $a=10$ and $m = 2$, then the set $b_n$, where $b_n$ contains all the integers that satisfy $a\equiv b\pmod{m}$ contains $2,4,6,8\dots$ But $10\bmod 2 = 0$. So why do we write $a\equiv b\pmod{m}$ to indicate a congruence relation? Thanks, any help is greatly appreciated.

https://hsm.stackexchange.com/questions/3340/is-gauss-the-first-who-introduced-congruences might be worth a look. — Barry Cipra, Apr 22 '20 at 18:37
If there is only one modulus in sight then often the modulus is omitted. However, it seems you may not realize the difference between congruence (an equivalence relation) vs. the $!\bmod$ (remainder) operation. See here for much more on that. It would help to elaborate to make it more precise what you are asking about. — Bill Dubuque, Apr 22 '20 at 18:37

score 2 · Accepted Answer · answered Apr 22 '20 at 18:28

This stems from seeing $\mathrm{mod}$ as an operation, which is true in computing, whereas in number theory it is not viewed that way most of the time. The way we describe the operation $\mathrm{mod}$ that you are speaking of, which we'll denote by $\%$, is that $a\% m$ is the smallest nonnegative integer $b$ such that $a\equiv b\pmod m$ (as long as $a$ is a nonnegative integer).

In mathematics, except perhaps for areas strongly tied to computer science, this operation is not used, and we consider the definition as an equivalence relation instead. If we need to consider $a\% m$ specifically in this sense, which is rarely, we would simply state it's the smallest such nonnegative integer.

score 0 · Answer 2 · answered Apr 22 '20 at 18:41

The equality relation is reflexive,symmetric and transitive.Notice that the congruence relation symbol ($\equiv $) is same with equality relation symbol.The main reason for similarity of the symbol (from various source) is that congruence symbol and equality relation symbol, share many of the same properties in each.In particular,congruence relation is also equivalence relation. Throughout this section m is a natural number and a,b are integers.

Why Are Congruence Relations Written as $”a\equiv b \pmod m”$

2 Answers2