I'm currently reading a book where the congruence $\equiv $ is appearing often, the author using it as Equivalent.

I understand the congruence relation through a geometric lens.

But through the abstract algebra lens, I understand the modulo concept but I'm lost on the meaning and the use of it.

Questions are the following:

  • Is congruence means equivalent, algrebrically and geometrically? I would say no but I have trouble in explaining why.
  • Why are we using modulo in abstract algebra to explain the $\equiv $?
    • Is that modulo chosen in an arbitrary way?
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    $\begingroup$ These are two different notions. $\endgroup$ – Wuestenfux Jul 13 '17 at 14:10
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    $\begingroup$ The congruence symbol, for example $a\equiv b,$ means that $a$ and $b$ are equivalent given a particular relation. Usually this is realized via "modulo" or "mod" which is to say that $a$ and $b$ represent the same equivalence class in an algebraic object. $\endgroup$ – Chickenmancer Jul 13 '17 at 14:11
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    $\begingroup$ $\equiv$ means different things in different contexts, but is generally reserved for use for equivalence relations. In the case of modular arithmetic, one says $a\equiv b\pmod{p}$ if and only if $a-b$ is an integer multiple of $p$. $\endgroup$ – JMoravitz Jul 13 '17 at 14:11
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    $\begingroup$ @AndyK To better understand algebraic congruences and ideals in algebraic structures you can start here. I had planned to say much more about how the various notions of congruence are related, but all of the destructive nonsense and censorship in the prior (deleted) comments left a very bad taste. Maybe in some future question. $\endgroup$ – Bill Dubuque Jul 13 '17 at 16:05

The word congruent has non-mathematical connotations of sameness and consistency, so we shouldn't be surprised if it is used to mean similar things in different places and with different strict definitions.

In some senses, both the geometric notion and algebraic notion are the same idea: two different objects being considered as the same. The process for how that is fleshed out will look different for each type of object that we're considering.

Informally speaking, congruence (in algebra) is used to denote when two things which may not literally be the same object, can be considered effectively the same in a certain context (i.e. in a particular relation) .

You are somewhat familiar with the modulo concept, so consider a clock: 1AM today and 1AM tomorrow are two very different times (they are separated by 24 hours!). We will usually refer to them, however, as the same time.

A more mathematical set of objects that can be seen this way are vectors in $\mathbb{R}^2$. One can draw vector $\vec{u}$ with initial point $(0,0)$ and terminal point $(1,0)$ and $\vec{v}$ with $(1,0)$ and $(2,0)$ respectively. If you graph these vectors using arrows, you will literally see two different arrows, but they have the same magnitude and direction and as vectors are considered to be the same (so much so that we would actually say that $\vec{u}=\vec{v}$).

More formally two elements in a set are congruent (or more commonly equivalent) under some equivalence relation (not sure what that is? See https://en.wikipedia.org/wiki/Equivalence_relation#Definition) if they are related to each other in that relation. The set of all elements that are congruent/equivalent with each other is called an equivalence class (or a congruence class).

For example, if we are interested in the integers modulo 3, $$0\equiv 3\equiv 6\equiv 9$$

So those four numbers, while distinct in $\mathbb{Z}$ are treated exactly the same when the relation is congruence modulo 3. The congruence class will then be $$\{3k|k\in \mathbb{Z}\}$$

You ask the question,

Why are we using modulo in abstract algebra to explain the $\equiv$?

Hopefully the examples that I've given demonstrate that modular arithmetic is but one way that equivalence/congruence is defined and that there are many other useful relations in which two things would be considered equivalent. The important thing is that if two things are called equivalent or congruent, that can mean many different things and you must know in what ways they are considered "the same" in that context.

  • $\begingroup$ hi @rnrstopstraffic , many thanks for your very clear and articulate answer. Cheers. $\endgroup$ – Andy K Jul 13 '17 at 15:45
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    $\begingroup$ I'm open to suggestions for those who are downvoting. If it's because it lacks a certain amount of rigor, that was intentional (see the level of the OP). If it's because you think that the question shouldn't be answered, then don't answer it. If for some other reason, I certainly would be interested in improving if there are thoughtful recommendations. $\endgroup$ – rnrstopstraffic Jul 13 '17 at 15:48
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    $\begingroup$ @AndyK You're welcome. $\endgroup$ – rnrstopstraffic Jul 13 '17 at 15:49

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