Complex Numbers
Question: Does the equation
=
have a solution?
Certainly not in the real number system, but we can create a larger number
system by adding a special number i defined as:
=
(we call i the imaginary unit)
Then
=
=
Also
=
=
=
Thus both i and -i are solutions to the equation given above.
Remember that
=
Definition: A complex number has the form
where a and b are real numbers. This is the standard form of a complex number.
Also, a complex number a+bi is called an imaginary number if b is not zero.
If a is zero, then the complex number reduces to bi and is called a pure
imaginary number.
Definition: The conjugate of
is
Definitions of equality, addition, and multiplication:
=
means
=
and
=
=
=
Note: People do not usually use this complicated definition of multiplication.
Instead, they multiply using the usual First, Outer, Inner, Last pattern and then use
=
Example:
=
=
=
Theorem 1: A useful shortcut to remember is that
=
Properties:
1) Addition and multiplication are commutative:
=
=
2) Addition and multiplication are associative:
=
=
3)
=
=
4)
=
is the multiplicative identity:
=
5) Every complex number
6) Every nonzero complex number
has a multiplicative inverse:
=
=
=
=
Definitions of negation, subtraction, and division:
=
=
=
where we assume that c+di is not zero and
can thus find the reciprocal as above
Definition: The principal square root of a negative real number is defined by
=
where a > 0
The other square root of -a is
=
Warning:
=
holds for non-negative reals a and b, but it may
fail to hold in other cases. To be precise, it fails when both a and b are
negative.
Example:
compared with
We do not get the same answer.