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) = is the additive identity: = 4) = is the multiplicative identity: = 5) Every complex number has additive inverse 6) Every nonzero complex number has a multiplicative inverse: = = = 7) Multiplication distributes over addition: = 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.