Why do we use logical equivalence instead of equals sign in Modular Arithmetic?

Question

When computing modular expressions, I see that $x\pmod m \equiv r$ rather than $x\pmod m = r$. Why do we use logical equivalence instead of the equals sign here?

Also, in general can we write $\equiv$ instead of $=$ between 2 numbers when they have the same numerical values?

EDIT: I am now confused on whether $\equiv$ always refers to congruence or whether it can mean numerical equality in this context, due to the following quote by my course notes:

we can define $x \pmod m$ to be the remainder $r$ when we divide $x$ by $m$. I.e., if $x \pmod m \equiv r$, then $x = mq+r$ where $0 ≤ r ≤ m−1$ and $q$ is an integer

I am confused because they are defining mod operations here and the restriction on $r$ implies they are talking about $r$ being the numerical quantity equal to the mod operation, rather than a value to which it is congruent. So does $\equiv$ at all refer to numerical equivalence rather than just congruence in modular arithmetic?

Answer 1

It's not logical equivalence, it's another use of the symbol $\equiv$ is a different context. That use is a convention.

The statement "$a=b$" means that $a$ and $b$ are different ways of talking about the same object. That's why you can say $1+1 = 2$.

If you think like a programmer you might write $$ 9 \pmod{5} = 4 \pmod{5} , $$ but you are more likely to see $$ 9 \equiv 4 \pmod{5} . $$ With an equal sign in the second expression you might be momentarily distracted into thinking $9 = 4$.

For a thorough discussion I recommend @BillDubuque 's answer at mod [= remainder] operation (and relation), name and meaning

Answer 2

We do not normally write $x\pmod m\equiv r$ but rather $x\equiv r\pmod m$. Here the $\pmod m$ does not modify $r$ (or $x$) but rather modifies $\equiv$: it gives the context under which this relationship applies. It is short for

Working modulo $m$, we have $x\equiv r$.

Now to make such a statement you don't want $=$, but something weaker, since $x$ and $r$ are not equal, just congruent/equivalent. There is (as far as I know) no good reason why we don't use $\cong$ instead of $\equiv$, other than tradition.