Arithmetic Course/Number Operation/Addition

Addition

Addition is a mathematical operation of adding two quantities which can be represented by a mathematical expression

A + B

Rules of arithmetic and algebra

The following laws are true for all values of a, b, and c, whether a, b, and c are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.

Addition

• Commutative Law: $a+b=b+a\,$.
• Associative Law: $(a+b)+c=a+(b+c)\,$.
• Additive Identity: $a+0=a\,$.
• Additive Inverse: $a+(-a)=0\,$.

Subtraction

• Definition: $a-b=a+(-b)\,$.

Multiplication

• Commutative law: $a\times b=b\times a\,$.
• Associative law: $(a\times b)\times c=a\times (b\times c)\,$.
• Multiplicative identity: $a\times 1=a\,$.
• Multiplicative inverse: $a\times {\frac {1}{a}}=1$, whenever $a\neq 0\,$
• Distributive law: $a\times (b+c)=(a\times b)+(a\times c)\,$.

Division

• Definition: ${\frac {a}{b}}=a\times {\frac {1}{b}}$, whenever $b\neq 0\,$.

Let's look at an example to see how these rules are used in practice.

${\frac {(x+2)(x+3)}{x+3}}$	= $\left[(x+2)\times (x+3)\right]\times \left({\frac {1}{x+3}}\right)$ (from the definition of division)
	= $(x+2)\times \left[(x+3)\times \left({\frac {1}{x+3}}\right)\right]$ (from the associative law of addition)
	= $((x+2)\times (1)),\qquad x\neq -3\,$ (from multiplicative inverse)
	= $x+2,\qquad x\neq -3.$ (from multiplicative identity)

Of course, the above is much longer than simply cancelling $x+3$ out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:

${\frac {2\times (x+2)}{2}}={\frac {2}{2}}\times {\frac {x+2}{2}}={\frac {x+2}{2}}$.

The correct simplification is

${\frac {2\times (x+2)}{2}}={\frac {2}{2}}\times {\frac {x+2}{1}}=1\times {\frac {x+2}{1}}=x+2$,

where the number $2$ cancels out in both the numerator and the denominator.

Rules

Any number add zero is equal to the number

a + 0 = a

Any number add to itself is doubled

a + a = 2a

Any number add to its negative is zero

a + (-a) = 0