Complex Numbers
INTRODUCTION
A complex number is a number that can be represented by an expression of the form: a + bi
where a and b are real numbers, and i is a symbol with the property
i2 = −1
We call a the real part of the complex number and we call b the imaginary part.
ADDING AND SUBTRACTING COMPLEX NUMBERS
To add or subtract complex numbers, simply add or subtract their real parts and their imaginary parts separately.
(a + bi) ± (c + di) = (a ± c) + (b ± d )i
Example: If x = 3 + 2i and y = −2 − 5i , find x + y .
Solution:
x + y = (3 + 2i) + (−2 − 5i)
= (3 − 2) + (2 − 5)i
= (1) + (−3)i
= 1 − 3i
MULTIPLYING COMPLEX NUMBERS
Multiplication is defined so that the usual laws hold: i.e., do FOIL as usual, but simplify at the end using the fact that i2 = −1 .
(a + bi) × (c + di) = ac + adi + bci + bdi2
= (ac − bd ) + (ad+ bc)i
Example: If x = 3 + 2i and y = −2 − 5i , find xy.
