I'm more familiar with the notation $a \equiv b \pmod c$, but I think this is equivalent to $a \bmod c = b \bmod c $, which makes it clear that we should put a $=$ instead of $\equiv$.
What's the reason for the change of sign? If it's to emphasize that modular equivalence is a congruence relation, why don't we use the $\equiv$ sign in both notations?
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- I am a human being.
- You are a human being.
Therefore, I am you: right ? Well, as it turns out, the answer is no. It simply means that we belong to the same class. Likewise, $3\neq7$, but $3\equiv7\bmod4$.
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The difference between the two is as follows:
Note that (in its usual usage) $a \mod c$ is a function that gives a particular answer, and has nothing to do with equivalence relations per se. For example, we might note that $$ \renewcommand{\mod}{\operatorname{mod}} 7 \mod 2 = 1 $$ That is, $7$ has a remainder of $1$ upon division by $2$. What we cannot say is that $$ 7 \mod 2 = 3 $$ because $1$ and $3$ are distinct values. We can say, however, that $$ 7 \equiv 3 \pmod 2 $$ Since both $7$ and $3$ have the same remainder upon division by $2$.
Why is it important? Well, we often make "substitutions" of equivalent values, and writing "mod" every time you make a substitution becomes unwieldy. For example, we can compute: $$ 5*7*9 - 2*3 + 5 + 3 \equiv 1*1*1 - 0*1 + 1 + 1 \equiv 3 \equiv 1 \pmod 2 $$ If we were to write this with mod as an operator, we'd have $$ [5*7*9 - 2*3 + 5 + 3] \mod 2 =\\ [(5 \mod 2)(7 \mod 2)(9 \mod 2) - (2 \mod 2)(3 \mod 2) + (5 \mod 2) + 3 \mod 2] \mod 2 = \\ [(1)(1)(1) - (0)(1) + (1) + 1 ]\mod 2 =\\ 3 \mod 2 = 1 $$
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We could have $-1 \bmod 2=-1$ and $1\bmod 2=1$, so $1\neq -1$, but nevertheless we have $1\equiv -1 \bmod 2$. So the equality $a\bmod c=b\bmod c$ is not clear, because it depends of the choice of a residue system.
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I prefer to use the congruence exclusively (I view the remainder operation as a source of headache even though I realize that it is a necessary evil in computing).
But I use both equal ($=$) and congruent to ($\equiv$) signs together when processing a lengthy calculation in modular arithmetic. My freshman algebra students quickly catch on with my calculations like $$ 12^{3004}\equiv5^{3004}=5^{3000}\cdot 5^4=(5^6)^{500}\cdot 5^4\equiv 1^{500}\cdot5^4=25^2\equiv4^2=16\equiv2\pmod7. $$ IOW, I use $=$ when there is an equality of integers between the steps, and $\equiv$ when I mean a congruence. The power of laws obeyed by congruences is apparent. Checking/following the progress of the calculation is easier this way. Of course, using $\equiv$ all the way is correct also. The $\equiv$ is there as a reminder that in this step we do something that only results in a congruence.
As the students become acquainted with the language of residue class rings, I gradually stop making the distinction between $=$ and $\equiv$ as well as, clarity of context permitting, the distinction between $n$ and $\overline{n}$.
Thinking about what that would look like when done by somebody who is only familiar with binary mod makes me shudder.
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The $a$ in the expression $$a \equiv b \pmod c$$ and the thing you intuitively write as $a \mod c$ in the expression $$a \bmod c = b \bmod c$$ are different kinds of things.
The symbol $a$ denotes an integer—a variable of type $\mathbb{Z}$, as a programmer might say, or an element of the ring $\mathbb{Z}$, as a mathematician might say.
The thing you denote by $a \bmod c$ is an integer modulo $c$—a variable of type $\mathbb{Z}/c\mathbb{Z}$, or an element of the ring $\mathbb{Z}/c\mathbb{Z}$. You can think of an integer modulo $c$ as an "hour" on a "$c$-hour clock," as described at the link above.
There's a natural way to turn an integer $n$ into an integer modulo $c$: you put the clock hand at noon, and then tick it forward $n$ hours. The hour you end up at is the one you call $n \mod c$. A more common notation for it is $[n]_c$, which can be abbreviated to just $[n]$ if it's clear that you're working on a $c$-hour clock.
Saying that the integers $a$ and $b$ are congruent modulo $c$, written $$a \equiv b \pmod c,$$ is the same as saying that the hours $[a]_c$ and $[b]_c$ are equal, written $$[a]_c = [b]_c.$$ In fact, this is how many people define congruence. You should be able to convince yourself, if you think about it long enough, that the hours $[a]_c$ and $[b]_c$ are equal if and only if the integers $a$ and $b$ have the same remainder when divided by $c$, which is another common definition of congruence modulo $c$.
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I don't see any problem in writing $a\bmod n = b\bmod n$ as long as one is familiar what it means.
Even "worse" is when dealing with decimal system, we write, for example, $1=0.\dot9$ even though they are not the same series - they merely converge to the same real number.
Now, to the point. We start with ring $\mathbb Z$ (i.e. a structure consisting of integers with familiar addition and multiplication) and fix some natural $n$. Then we define equivalence relation $\equiv$ such that $$ a\equiv b \iff n|b-a$$ Having the fact that one can form quotient set of equivalence classes with respect to $\equiv$, using the properties of $\mathbb Z$ (namely that multiples of $n$ form an ideal) it follows that the quotient set $\mathbb Z/\equiv$ is actually a ring itself and then we use more familiar notation $\mathbb Z/n\mathbb Z$. Very importantly, we get a surjective function $\pi:\mathbb Z\to\mathbb Z/n\mathbb Z$ that sends any $a$ to it's equivalence class $[a]$ satisfying $$\pi(a+b) = \pi(a) + \pi(b),\ \pi(ab) = \pi(a)\pi(b)\quad \text{(i.e. $\pi$ is a ring homomorphism)}$$ So, when we write $a \equiv b\ \pmod n$ we mean that $a$ and $b$ belong to the same equivalence class (which just means that they give the same residue when divided by $n$ in this context) and consequently $\pi(a) = \pi(b)$. We might as well write $\operatorname{mod} n$ instead of $\pi$ and apply convention $a\bmod n = (\operatorname{mod}\ n)(a)$.
Why don't we do it? Because it is tedious to write down calculations that way.
One usual way of writing down calculations is to write $\overline 1,\ \overline 2,\ldots$ for congruence classes when having fixed $n$. For example $\overline 1 = \overline 3$ in $\mathbb Z/2\mathbb Z$, instead of $1\equiv 3\pmod 2$. I wouldn't have problem even if one would write $1=3$ as long as it is clear that these merely represent same elements of the ring $\mathbb Z/2\mathbb Z$, and not equality of integers, just as $1 = 0.\dot 9$ as real numbers, but not as series.
Long story short, how you write depends on the context and most important thing is that it is not ambiguous to the reader.
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Typically $=$ is used to denote that two objects are equivalent in every single context. It is usually a primitive in logic, defined by textual replacement.
Other symbols, for example $\equiv$, are used for context sensitive equivalence. $9$ is equal to $4$ in the context of $\pmod 5$. Two binary trees may be equivalent in terms of their shape, but not where they are located in memory. A husband and wife may be equivalent for the sake of tax purposes, but not for their respective expectations of each other.
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Equal to means that they are the (exact) same thing. Congruent to means that they have some important property in common, in this case the same remainder when divided by the modulus. The relation, however, must be reflexive, symmetric, and transitive.
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In more advanced math, you will find number systems where it is not possible (or possible, but not wise) to define $a\mod c$ in an obvious way, while you will be able to write it as $a\equiv b\pmod{c}$.
There is an additional problem with $a\mod c = b\mod c$, which is that it allows for $a\mod c = b\mod d$. But this is hardly ever useful.
There are ways to write it as equality, but it is pedagogically useful to become comfortable with congruence as an instance equivalence relation.
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I (like most people, I believe) move freely between $=$ and $\equiv$ in the following sense.
Saying $a \equiv b \pmod{n}$ is the same as saying $a = b$ as elements of $ \mathbb Z / n\mathbb Z$. At least this is what I say to my students to keep in mind in case I jump around in my notation.
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"Why" questions about notation can range into non-mathematical territory: Notation is not always perfectly logical or consistent.
This isn't an explanatory answer, but an extended comment about writing $$ a \equiv b \pmod{c} \tag{1} $$ versus $$ a \bmod{c} = b \bmod{c}. \tag{2} $$
In (1), mod $c$ is grammatically an adverb, modifying the binary relation "is equivalent to". Consistency dictates writing something like $$ a \equiv_{\bmod{c}} b\qquad\text{or}\qquad a \equiv_{c} b. $$ I'm not a historian, but believe the widespread use of the notation (1) at the school level arose in the New Math era of the 1950s. (It's easy to imagine how this "afterthought" notation (mod $c$) would work psychologically for students raised on arithmetic drills involving integers. I haven't a shred of evidence, however, and will keep my speculations private.)
In (2), by contrast, mod $c$ functions grammatically as an adjective, modifying the symbols $a$ and $b$. This is a more sophisticated viewpoint: Instead of relating integers $a$ and $b$ (which are familiar to young students), (2) equates residue classes mod $c$. That's already one reason to avoid (2) in an introductory setting.
To give a completely different reason, writing "$a \bmod{c}$" to denote a residue class is clumsy. The generic equivalence class notation $[a]_{c}$ would be preferable, or even $[a]$ if the modulus $c$ were fixed during the discussion/computation. One does in fact see $[a] = [b]$ in introductory algebra at the university level.
Honestly, even square brackets are cumbersome. (They can also cause perplexing LaTeX errors when Cayley tables are typeset as arrays. Exercise: Why?) Between friends and in private, you might as well denote the residue class simply as "$a$" if the modulus is fixed, so that mod $7$ you have, e.g., $$ 5^{2015} \equiv -(2^{2015}) \equiv -(2^{3 \cdot 671 + 2}) \equiv -4 \equiv 3. $$
Ideally: Mathematical notation conveys concepts concisely, precisely, and consistently across fields and disciplines.
In practice: Changing historical perspectives and notational habits, field-specific conventions and emphases, and computer typesetting capabilities influence the ways we express mathematics in writing. Even sub-specialities of pure mathematics may use mutually-incompatible notation, to say nothing of discussions between pure mathematicians and engineers (say).
The answer to the question at hand may be as banal as "(1) is more compact than (2)" or "that's just the way everyone writes it (in 2015)". The take-away message is: "Mathematical notation is not always logical or consistent, and making notation consistent is much harder than it might first appear."
Answer to Exercise:
If a square-bracketed expression follows a newline command
\\with only white space in between, LaTeX reads the expression as an optional argument, expecting a length.
It depends on how you define your notation. The standard definition of congruence is $a \equiv b \pmod c$ if and only if $c \mid (b-a)$. The standalone expression $b \pmod c$ is undefined, hence not equal to anything.
On the other hand, some people like to write "$b \bmod c$" to stand for the least nonnegative number $d$ that satisfies $d \equiv b \pmod c$. If you adopt this notation then it is true that $a \equiv b \pmod c$ if and only if $(a \bmod c) = (b \bmod c)$.