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IMAGINARY AND COMPLEX NUMBERS
Higher Concepts of Mathematics
Complex Numbers
Complex numbers are numbers which consist of a real part and an imaginary part. The solution
of some quadratic and higher degree equations results in complex numbers. For example, the
roots of the quadratic equation, x2 - 4x + 13 = 0, are complex numbers. Using the quadratic
formula yields two complex numbers as roots.
x
b ±
b2
4ac
2a
x
4 ±
16 52
2
x
4 ±
36
2
x
4 ± 6i
2
x 2 ± 3i
The two roots are 2 + 3i and 2 - 3i; they are both complex numbers. 2 is the real part; +3i and -
3i are the imaginary parts. The general form of a complex number is a + bi, in which "a"
represents the real part and "bi" represents the imaginary part.
Complex numbers are added, subtracted, multiplied, and divided like algebraic binomials. Thus,
the sum of the two complex numbers, 7 + 5i and 2 + 3i is 9 + 8i, and 7 + 5i minus 2 + 3i, is
5 + 2i. Similarly, the product of 7 + 5i and 2 + 3i is 14 + 31i +15i2. But i2 equals -1. Thus,
the product is 14 + 31i + 15(-1) which equals -1 + 31i.
Example 1:
Combine the following complex numbers:
(4 + 3i) + (8 - 2i) - (7 + 3i) =
Solution:
(4 + 3i) + (8 - 2i) - (7 + 3i) = (4 + 8 - 7) + (3 - 2 - 3)i
= 5 - 2i
MA-05
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