The Imaginary Unit i and Complex Numbers
By definition, the imaginary unit
. As a consequence, ,
.
Using the definition of i, we can represent the square root of any negative number.
Example:
Example:
A number of the form
, where a and b are real numbers and
, is called a complex number.
The number a is called the real part and the number b is called the imaginary part.
Example: In the complex number
, –3 is the real part and 5 is the imaginary part.
Example: In the complex number
,
is called the real part and
is called the imaginary part.
Example: 8i can be thought of as the complex number
in which the real part is 0 and the imaginary
part is 8. Because it has no real part, 8i is called a pure imaginary number.
Example: The real number
can be thought of as the complex number
. So real numbers are a
subset of the complex numbers.
To simplify powers of i, use the fact that i2 = –1.
Examples:
and
Example: To simplify an odd power of i first factor out one of the i's and then simplify the remaining even
power of i as in the preceding example. So
.
To add or subtract complex numbers, combine like terms.
Example:
Example:
To multiply complex numbers, use F.O.I.L., replace
Example:
with –1, and combine like terms.
Example:
Pairs of complex numbers such as 5 + 8i and 5 – 8i which have the same real part but opposite imaginary
parts are called complex conjugates. As in the preceding example, the product of complex conjugates is
always a real number.
To divide a complex number by a complex number: (1) write the division as a fraction, (2) multiply both
the numerator and denominator of that fraction by the conjugate of the denominator, (3) carry out the
multiplications and simplify to put the answer in
form.
Example: Divide
by
.
(Thomason - Fall 2015)