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Addition

From AoPSWiki

Addition is the mathematical operation which combines two quantities. The result of addition is called a sum.

Notation

The sum of two numbers $a$ and $b$ is denoted $a+b$ , which is read "a plus b." 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 $a+b+c$ .
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)

See also

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