**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.