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Commutative

Addition

Addition

For all real a, b

a + b = b + a

a + b = b + a

Associative

Addition

Addition

For all real a, b, c

a + (b + c) = (a + b) + c

a + (b + c) = (a + b) + c

Identity Addition

the sum of any number and zero is the original number

a+0=a

a+0=a

Additive Inverse

(Opposite)

(Opposite)

For every real number a there exist a real number, denoted (-a), such that

a + (-a) = 0

a + (-a) = 0

Distributive Law

For all real a, b, c

a(b + c) = ab + ac, and (a + b)c = ac + bc

a(b + c) = ab + ac, and (a + b)c = ac + bc

Commutative

Multiplication

Multiplication

Changing the order of the factors does not change the product; for example 10 x 9 = 9 x 10; a ** b = b ** a

Associative Multiplication

changing the grouping of factors will not change the product, (ab)c = a(bc)

Reflexive

a=a

Inverse Multiplication

states that a number multiplied by its reciprocal is equal to one

Identity Multiplication

the product of any number times 1 is that number

Ax1=A

Ax1=A

Symmetric

if a=b then b=a

Transitive

If a=b and b=c then a=c

Multiplication by Zero

a x 0 = 0; also, if a x b = 0, then at least one of a or b must equal 0