Complex Numbers

1
Complex Numbers
2
A First Observation

• There is no real number solution to the equation
  x2 10.
• The square root of 1 is called the imaginary
  unit and is denoted by i.
• This provides a solution to x210, but it is not
  a real number.

3
Definition of Complex Number

• A complex number z is a number of the form a
  bi, where i ?-1.
• a is called the real part of z, and b is called
  the imaginary part of z.
• The standard form of a complex number is a bi,
  where a and b are real numbers.

4
More on Complex Numbers

• If b ? 0, z a bi is also called an imaginary
  number.
• If b 0, z is a real number.
• If a 0 and b ? 0, z is called a pure imaginary
  number.

5
Equality of Complex Numbers

• Two complex numbers abi, and cdi are equal if
  and only if a c and b d

6
Addition and Subtraction of Complex Numbers

• (abi) (cdi) (ac)(bd)i
• (abi) (cdi) (a-c)(b-d)i

7
Multiplication of Complex Numbers

• (abi)(cdi)
  (ac-bd)(adbc)i
• Note that this is simply the
  first-outer-inner-last rule, and we replace i2 by
  1.

8
Complex Conjugates

• If z abi, the conjugate of z is a bi.
• Note that z and its conjugate are equal if and
  only if the imaginary part of z 0, i.e. z is a
  real number.
• (abi)(a-bi) a2 b2

9
Division of Complex Numbers
10
Real and Complex Numbers

• The real numbers are a subset of the complex
  numbers. I.e. every real number is a complex
  number.
• The properties of addition, subtraction,
  multiplication, and division that hold for real
  numbers also hold for complex numbers.