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When talking about modulos why is that we use congruency and not equality?

From the accepted answer in the post What is the difference between congruency and equality? It states that:

... two figures are equal if they have the same points.

Congruent figures have the same shape and size (informally) but possibly different points.

Which confuses me since I think 7 mod 4 results in the same value or the same point as 3.

Mandelbrotter
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    $7\mod4=3\mod4;$ but $7\neq3;$ – drhab Mar 24 '21 at 15:51
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    Using $\mathtt{mod}$ as a binary operator is standard in programming languages, but not in mathematics: in maths you write $ 7 \equiv 3 (\mathrm{mod},4)$ (or minor variants of that). And that's a congruence, not an equality (as drhab points out above). You can use equality, but to do that you would say that you were working tin he group or ring $\Bbb{Z}_4$ of integers modulo 4. – Rob Arthan Mar 24 '21 at 16:03
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    @drhab Isn't it better to write: $\ 7\equiv 3\quad (\mod 4)\ ?$ – Adam Rubinson Mar 24 '21 at 16:05
  • @AdamRubinson Yes, I think so. I took over what I encountered in the question without further thinking about it. – drhab Mar 24 '21 at 16:46
  • I think OP's question is a very good one, but it has been asked before. See: https://math.stackexchange.com/questions/1397067/why-do-we-use-congruent-to-instead-of-equal-to – Adam Rubinson Mar 24 '21 at 17:48

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This is mostly a question of taste; but there's very, very good reason for that taste. Indeed, it's important to disambiguate things thoroughly to make sure we're correct when it concerns technical details.

Generally, the core factor where you can say "7 is equal to 3" without the fear mistaken is being in the right context, namely, being in the space $\mathbb{Z}/4\mathbb{Z}$. In this "four hour clock", $7$ and $3$ are indeed equal, but it's not the $7$ and $3$ that you know from $\mathbb{Z}$, it's one element, generally written with a hat, like so: $\hat{-1} = \hat3 = \hat 7$. The way we write this element refers to $\mathbb{Z}$, since we construct $\mathbb{Z}/4\mathbb{Z}$ from $\mathbb{Z}$; so it helps our intuition. But I could just as well forbid you from writing in a way that refers to $\mathbb{Z}$, and decide to write $\mathbb{Z}/4\mathbb{Z} = \{ 0, \alpha, \beta, \gamma \}$.

In the context of $\mathbb{Z}$ itself, $3$ and $7$ are NEVER equal. They are only EQUIVALENT, relative to the equivalence relation of congruence modulo 4. From any equivalence relation, you can make an algebraic quotient, and in the resulting space, the elements now become equal: $7$ and $3$ are both mapped to a single element $\gamma$, and their results are then equal. But you must never lose track of the space/context in which things are happening, since this is ripe for misuse and mistake.

Tristan Duquesne
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