Complex Numbers
C.A-1.5
Imaginary numbers
i represents the square root of – 1.
Complex Numbers
A Complex Number is a number that can be
written in the form:
Where a and b are real numbers and
The a is the Real part,
and b is called the imaginary part.
Equality of Complex Numbers
• Two complex numbers are equal if their real
parts are equal and their imaginary parts are
equal.
• If a + bi and c + di are two complex numbers,
then
a + bi = c + di if and only if a = c and b = d
Addition and Subtraction
of Complex Numbers
• Two complex numbers such as a + bi and c + di
are added and subtracted as if they were
binomials:
• (a + bi) + (c + di) = (a + c) + (b + d )i
• (a + bi) - (c + di) = (a - c) + (b - d )i
Multiplication of
Complex Numbers
• The numbers a + bi and c + di are multiplied as
if they were binomials, with i2 = -1:
• (a + bi)(c + di) = (ac – bd) + (ad + bc)i
Errors involving signs
• To avoid errors in determining the sign of the
result, always express numbers in a + bi form
before attempting any algebraic
manipulations.
Complex Conjugates
The complex numbers
and
Are complex conjugates of each other.
Powers of i
Powers of i
If n is a natural number that has a
remainder of r when divided by 4, then
When n is divisible by 4, the remainder
r is 0 and i0 = 1.
Absolute Value of a
Complex Number
If a + bi is a complex number, then