9 Basic Properties of Real and Complex Numbers

Distributive Law

for any real x, y, z. x(y+z) = xy +xz

Multiplicative Inverse

If x is real and not zero, x ⋅ 1/x = 1

Additive Identity (aka identity law for addition)

___ is the number (0) that when added to a number results in the original number

x+0=x

Commutativity of Multiplication

For any real x and y, x ⋅ y = y ⋅ x
(does not work for division as (for example) 8/2 ≠ 2/8)

changing order when multiplying does not affect the product

Multiplicative Identity (aka identity property of multiplication)

x ⋅ 1 = x for any real x
multiplying a real x by 1 will result in x)

Commutativity of Addition

for any real numbers x and y, x+y = y+x
(subtraction does not satisfy ___, 6-1 ≠ 1-6)

changing the order for two real numbers when adding does not affect the sum

Associativity of Addition

for any real x,y,z… (x+y) + z = x + (y+z)
(subtraction is not ___, nor are exponents)

changing order of any three real numbers when adding does not affect the sum

Additive Inverse Rule

for any real x, there’s exactly one number, its __ (written -x → not always negative, -x just stands for the __), where x+(-x) = 0

Associativity of Multiplication

for any real x,y,z … (x ⋅ y) ⋅ z = x(y ⋅ z)

changing the grouping of any three real numbers when multiplying does not affect the product