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Addition is the mathematical operation (it is represented by the $+$ sign) which combines two quantities. The result of addition is called a sum. For example, the sum of 3 and 2 is 5 because $3+2=5$.

## Notation

The sum of two numbers $a$ and $b$ is denoted $a+b$, which is read "a plus b." The two numbers being added together, or $a$ and $b$, are called addends. The sum of $f(a), f(a+1), f(a+2), f(a+3), \cdots, f(b)$, where $f$ is a function, is denoted $\sum_{i=a}^bf(i)$. (See also Sigma notation)

## Properties

• Commutativity: The sum $a+b$ is equivalent to $b+a$.
• Associativity: The sum $(a+b)+c$ is equivalent to $a+(b+c)$. This sum is usually denoted $a+b+c$.
• Distributivity: $a(b+c)=ab+ac$
• Closure: If $a$ and $b$ are both elements of $\mathbb{R}$, then $a+b$ is an element of $\mathbb{R}$. This is also the case with $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{C}$.
• Identity: $a+0=a$ for any complex number $a$.
• Inverse: The sum of a number and its additive inverse, $a+(-a)$, is equal to zero.
• Equality: If $a=b$, then $a+c=b+c$.
• If $a$ is real and $b$ is positive, $a+b>a$.
• The sum of a number and its Complex conjugate is a real number.
• $a+(-b)=a-b$ (See also Subtraction)