I know the percent sign as the modulo operator in a few programming languages but in maths everyone uses "mod". Why isn't there a symbol like + for addition or minus for subtraction?
I suspect it is because "modulo" describes an equivalence class, rather than a particular number: 4 (mod 7) is technically the set of all integers n such that 7|(n-4); this set is precisely the same set as -3 (mod 7), so what should "25%7" equate to? 4? -3? 25? I don't believe it is common enough as an operation in mathematics to warrant identifying a canonical result: e.g. as the unique representative between 0 and n-1; it is more commonly used to set up the structure being worked on/in (e.g. Z_p, integers modulo p).
Many programming languages use %. http://en.wikipedia.org/wiki/Modulo_operation
But there is ambiguity on the output, and you have to be careful about what you really want.
elementary number theory - Notation for modulo - Mathematics Stack Exchange
Understanding The Modulus Operator % - Stack Overflow
Why isn't there a symbol for the modulo operator?
I suspect it is because "modulo" describes an equivalence class, rather than a particular number: 4 (mod 7) is technically the set of all integers n such that 7|(n-4); this set is precisely the same set as -3 (mod 7), so what should "25%7" equate to? 4? -3? 25? I don't believe it is common enough as an operation in mathematics to warrant identifying a canonical result: e.g. as the unique representative between 0 and n-1; it is more commonly used to set up the structure being worked on/in (e.g. Z_p, integers modulo p).
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I think the three most common notations are:
The first is an abuse of notation and somewhat misleading to people learning the subject, but it is common. The others, IMO, would be readily understood if you state once at the beginning of whatever you're writing what the notation means.
For your specific application you could also use the iverson bracket:
The Iverson bracket is basically the mathematical incarnation of the usual ways to coerce a boolean value to an integer: it gives 
when false and 
when true.
In the specific formula you write, a possibly more useful alternative is
since this form is more amenable to doing series manipulations. In fact, the context of series manipulations is the first time I saw extensive use of the Iverson bracket.
Given the integer 
and the modulus 
, it would be standard to write
To be a little more precise, if 
is the remainder of 
after division by 
in accordance with the Division Algorithm, we have
(This explanation is only for positive numbers since it depends on the language otherwise)
Definition
The modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation. (source: wikipedia)
For instance, 9 divided by 4 equals 2 but it remains 1. Here, 9 / 4 = 2 and 9 % 4 = 1.
Image source: Wikimedia
In your example: 5 divided by 7 gives 0 but it remains 5 (5 % 7 == 5).
Calculation
The modulo operation can be calculated using this equation:
a % b = a - floor(a / b) * b
floor(a / b)represents the number of times you can divideabybfloor(a / b) * bis the amount that was successfully shared entirely- The total (
a) minus what was shared equals the remainder of the division
Applied to the last example, this gives:
5 % 7 = 5 - floor(5 / 7) * 7 = 5
Modular Arithmetic
That said, your intuition was that it could be -2 and not 5. Actually, in modular arithmetic, -2 = 5 (mod 7) because it exists k in Z such that 7k - 2 = 5.
You may not have learned modular arithmetic, but you have probably used angles and know that -90° is the same as 270° because it is modulo 360. It's similar, it wraps! So take a circle, and say that its perimeter is 7. Then you read where is 5. And if you try with 10, it should be at 3 because 10 % 7 is 3.
Two Steps Solution.
Some of the answers here are complicated for me to understand. I will try to add one more answer in an attempt to simplify the way how to look at this.
Short Answer:
Example 1:
7 % 5 = 2Each person should get one pizza slice.
Divide 7 slices on 5 people and every one of the 5 people will get one pizza slice and we will end up with 2 slices (remaining). 7 % 5 equals 2 is because 7 is larger than 5.
Example 2:
5 % 7 = 5Each person should get one pizza slice
It gives 5 because 5 is less than 7. So by definition, you cannot divide whole 5items on 7 people. So the division doesn't take place at all and you end up with the same amount you started with which is 5.
Programmatic Answer:
The process is basically to ask two questions:
Example A: (7 % 5)
(Q.1) What number to multiply 5 in order to get 7?
Two Conditions: Multiplier starts from `0`. Output result should not exceed `7`.
Let's try:
Multiplier is zero 0 so, 0 x 5 = 0
Still, we are short so we add one (+1) to multiplier.
1 so, 1 x 5 = 5
We did not get 7 yet, so we add one (+1).
2 so, 2 x 5 = 10
Now we exceeded 7. So 2 is not the correct multiplier.
Let's go back one step (where we used 1) and hold in mind the result which is5. Number 5 is the key here.
(Q.2) How much do we need to add to the 5 (the number we just got from step 1) to get 7?
We deduct the two numbers: 7-5 = 2.
So the answer for: 7 % 5 is 2;
Example B: (5 % 7)
1- What number we use to multiply 7 in order to get 5?
Two Conditions: Multiplier starts from `0`. Output result and should not exceed `5`.
Let's try:
0 so, 0 x 7 = 0
We did not get 5 yet, let's try a higher number.
1 so, 1 x 7 = 7
Oh no, we exceeded 5, let's get back to the previous step where we used 0 and got the result 0.
2- How much we need to add to 0 (the number we just got from step 1) in order to reach the value of the number on the left 5?
It's clear that the number is 5. 5-0 = 5
5 % 7 = 5
Hope that helps.