Addition

Addition is the binary mathematical operation of putting numbers or quantities together (adding). The plus sign "+" means that objects are added up. The numbers that being added are called "addends" and the result is called "sum".

Commutativity
Addition is commutative, meaning a + b = b + a.

Associativity
Addition is associative, meaning a + (b + c) = (a + b) + c

Zero and one
Adding 0 to a number is not changing the number: a + 0 = 0 + a = a. For any integer a, (a + 1) means least integer greater than a, also known as successor of a.

Sum
If S is a finite set of numbers, then all its elements can be added together using sigma notation. $$\sum_{x \in S} x$$

Rational numbers (fractions)
Two fractions can be added up using this formula: $$\frac ab + \frac cd = \frac{ad+bc}{bd}$$. However it is easier to use least common denominator.