Complex Numbers

A complex number is a number of the form a + bi, where i =

and a and b are real numbers. For example, 5 + 3i, -

+ 4i, 4.2 - 12i, and -

i are all complex numbers. a is called the real part of the complex number and bi is called the imaginary part of the complex number. In the complex number 6 - 4i, for example, the real part is 6 and the imaginary part is -4i.

Adding and Subtracting Complex Numbers

To add two complex numbers, add their real parts and add their imaginary parts: (a₁ + b₁i) + (a₂ + b₂i) = (a₁ + a₂) + (b₁ + b₂)i.

Examples:

(12 + 6i) + (11 + 5i) = (12 + 11) + (6 + 5)i = 23 + 11i
(5 - 7i) + (4 + i) = (5 + 4) + (- 7 + 1)i = 9 - 6i.
(2 - 4i) + (- 6 - 5i) = (2 - 6) + (- 4 - 5)i = - 4 - 9i.

To subtract two complex numbers, subtract their real parts and subtract their imaginary parts: (a₁ + b₁i) - (a₂ + b₂i) = (a₁ - a₂) + (b₁ - b₂)i.

Examples:

(4 + 5i) - (2 + 3i) = (4 - 2) + (5 - 3)i = 2 + 2i.
(3 - 7i) - (4 + 6i) = (3 - 4) + (- 7 - 6)i = - 1 - 13i
(- 4 + 2i) - (3 - 11i) = (- 4 - 3) + (2 - (- 11))i = - 7 + 13i
(6 - 9i) - (- 3 - 4i) = (6 - (- 3)) + (- 9 - (- 4))i = 9 - 5i

Multiplying a Complex Number by a Scalar

To multiply a complex number by a scalar, multiply the real part by the scalar and multiply the imaginary part by the scalar: c(a + bi) = ca + cbi.

Examples:

4(2 + 5i) = 4(2) + 4(5i) = 8 + 20i

(6 - 9i) =

(6) +

(- 9i) = 2 - 3i.
-2(11 - 2i) = - 2(11) + (- 2)(- 2i) = - 22 + 4i.
2i(5 + 7i) = 2i(5) + 2i(7i) = 10i + 14i² = 10i + 14(- 1) = - 14 + 10i.

Multiplying Complex Numbers

To multiply two complex numbers, use the FOIL method and treat each complex number as an ordinary binomial. Then simplify the i² term (i² = - 1) and combine like terms.

(a₁ + b₁i)(a₂ + b₂i)	=	a₁a₂ + a₁b₂i + a₂b₁i + b₁b₂i²
	=	a₁a₂ + (a₁b₂ + a₂b₁)i + b₁b₂(- 1)
	=	(a₁a₂ - b₁b₂) + (a₁b₂ + a₂b₁)i.