Complex numbers have a real component and an imaginary component. They have specific operations of addition, division, multiplication and subtraction that expand on the corresponding operations for real numbers. The rules for the operations of complex numbers were first developed in the sixteenth century in an attempt to solve cubic equations that involved the square root of negative numbers. Complex numbers are currently used in a variety of fields such as chaos theory, electromagnetism and quantum physics.

Define a complex number. A complex number uses the notation a + bi, where "a" and "b" are real numbers and "i" is the square root of 1; "a" is the real component of the complex number and "b" is the imaginary component of the complex number.

Define the rules for the addition and subtraction of complex numbers. The addition of complex numbers uses the identity (a + bi) + (c +di) = (a + c) + (b + d)i. The subtraction of complex numbers uses the identity (a + bi) i (c +di) = (a  c) + (b  d)i.

Use the rules in Step 2 to add specific complex numbers. For example, (3 + 4i) + (2 + 3i) = (3 + 2) + (4  3)i = 5 + i.

Define the rules for the multiplication of complex numbers. These multiplication of complex numbers uses the identity (a + bi) (c + di) = (ac  bd) + (bc + ad)i. The division of complex numbers uses the identity (a + bi) / (c + di) = (ac + bd)/(c^2 +d^2) + (bc  ad)/(c^2 +d^2)i.

Use the rules in Step 4 to multiply specific complex numbers. For example, (3 + 4i)(2  3i) = (3 x 2  4 x (3)) + (4 x 2 + 3 x (3))i = 6  (12) + (8 + 9)i = 18  i.