1.2.1. Addition and multiplication

Addition uses the binary operator '+' and is written as:

sum = augend + addend

product = multiplicand × multiplier
product = multiplicand · multiplier
product = (multiplicand)(multiplier)
product = multiplicand multiplier

x = a + ci
y = b + di 

Addition and Subtraction

Add the real parts together and the imaginary parts together.

x + y = (a + b) + (c + d)i

When the imaginary parts are zero for both x and y, the above reduces to:

x + y = a + b

Multiplication

Multiply out the terms, replace i² with -1, and group real and imaginary parts.

x·y = (a + bi)(c + di)
    = a·c + b·d·i2 + a·di + b·ci
    = a·c - b·d + (a·d + b·c)i

When both imaginary parts are zero this reduces to:

x·y = a·c

The axioms for addition and multiplication are the following:

Closure law

x + y and x·y are unique numbers.

Commutative law The order of operations can be changed:

x + y = y + x 
x·y = y·x 

Associative law

Grouping of operands can be changed:

x + y + z = x + (y + z) = (x + y) + z
x·y·z = x·(y·z) = (x·y)·z

Distributive law

Multiplication is distributive over addition:

x·(y + z) = x·y + x·z
(x + y)·z = x·z + y·z

The numbers zero and one are identical for real and complex numbers:

0 + 0i = 0
1 + 0i = 1

Identity laws

There is a unique number 0 such that: x + 0 = 0 + x = x

There is a unique number 1 such that: x·1 = 1·x = x

Inverse laws

For any number x, there is a number -x such that:

x + (-x) = (-x) + x = 0

The number -x is the negative (additive inverse) of x.

For any non-zero number x, there is a number x^-1 such that:

x·x^-1 = x^-1·x = 1

The number x^-1 is the reciprocal (multiplicative inverse) of x.

Zero factor laws

For every number x, x·0 = 0

If x·y = 0, then either x = 0 or y = 0

Laws for negatives

-(-x) = x
(-x)(-y) = x·y
-x·y = (-x)y = x(-y) = -(-x)(-y)
(-1)x = -x