Complex Numbers

The set of complex numbers is a superset of the set of real numbers. The standard form of a complex number is a + bi, where a and b are real numbers and i₂ = -1. The real number a is called the real part, and the real number b is called the imaginary part, of the complex number a + bi. Because no square of a real number is negative, i cannot represent a real number.

If a = 0, the complex number a + bi becomes bi ; such a complex number, with a zero real part, is known as a pure imaginary number.

If b = 0, then the complex number a + bi is a, a real number; thus, a real number is a complex number with its imaginary part equal to zero.

The conjugate of a complex number is the complex number having the same real part, but whose imaginary part is the additive inverse of the former. That is, the conjugate of the complex number a + bi is a - bi. Since the imaginary part of a real number is zero, the conjugate of a real number is simply the real number itself.

EX. Given the following complex numbers:

(1) 2 + 3i : the real part is 2, the imaginary part is 3
(2) 3 - 5i : the real part is 3, the imaginary part is -5
(3) 2 : the real part is 2, the imaginary part is 0
(4) 5i : the real part is 0, the imaginary part is 5
(5) 6 + 3i : its conjugate is 6 - 3i
(6) 5 - 4i : its conjugate is 5 + 4i

The addition of complex numbers simply involves the addition of the real parts and the addition of the imaginary parts. That is,

( a + bi ) + ( c + di ) = ( a + c) + ( b + d ) i

EX. (2 + 5i) + (3 + 2i) = (2 + 3) + (5 + 2) i = 5 + 7i
(5 - 2i) + (-2 + 7i) = (5 + (-2)) + ((-2) + 7) i = 3 + 5i

The subtraction of complex numbers simply involves the subtraction (in the appropriate order) of the real parts and the subtraction of the imaginary parts. That is,

( a + bi ) - ( c + di ) = ( a - c) + ( b - d ) i

EX. (6 + 3i) - (2 + i) = (6 - 2) + (3 - 1) i = 4 + 2i
(8 - 2i) - (-2 - 3i ) = (8 - (-2)) + ((-2) - (-3)) i = 10 + i

In the multiplication of complex numbers, the distributive property is applied to the real and imaginary parts of the complex numbers. The identity i ₂ = -1 is used to simplify the resulting expression of the product.

EX.
( 2 + 5i )( 3 + 2i )
= 2 ( 3 + 2i ) + 5i ( 3 + 2i )
= 6 + 4i + 15i + 10i ₂
= 6 + 19i + 10(-1)
= 6 + 19i - 10
= -4 + 19i

In the division of complex numbers, the division operation is expressed as a fraction. If the denominator of the fraction is a real number, the real and imaginary parts of the numerator are simply divided by the real denominator. Otherwise, both the numerator and denominator are multiplied by the conjugate of the denominator. That is,

(a + bi)/c = a/c + b/ci

(a + bi)/(c + di) = [(a + bi)/(c + di)] · [(c-di)/(c - di)]
= [a(c - di) + bi(c - di)] / [c(c - di) + di(c - di)]
= [ac + (-ad + bc)i - bdi²] / [c² + (-cd + cd)i - d²i²]

where i ₂ = -1

EX.