Complex Numbers

1
Complex Numbers

• Understand complex numbers
• Simplify complex number expressions

2
Imaginary Numbers

• What is the square root of 9?

• What is the square root of -9?

3
Imaginary Numbers

• There is no real number that when multiplied by
  itself gives a negative number.
• A new type of number was defined for this
  purpose. It is called an Imaginary Number.
  Imaginary numbers are NOT in the Real Set.

4
Imaginary Numbers

• The constant, i, is defined as the square root of
  negative 1

• Multiples of i are called Imaginary Numbers

5
Imaginary Numbers

• The square root of -9 is an imaginary number...

• When we simplify a radical with a negative
  coefficient inside the radical, we write it as an
  imaginary number.

6
Imaginary Numbers

• Simplify these radicals

7
Multiples of i

• Consider multiplying two imaginary numbers

• So...

8
Multiples of i

• Powers of i

9
Multiples of i

• This pattern repeats

10
Multiples of i

• We can find higher powers of i using this
  repeating pattern i, -1, -i, 1

What is the highest number less than or equal to
85 that is divisible by 4?
84 ? 85
1
So the answer is
11
Powers of i - Practice

• i28
• i75
• i113
• i86
• i1089

• 1
• -i
• i
• -1
• i

12
Solutions Involving i

• Solve

13
Solutions Involving i

• Solve

14
Solutions Involving i

• Solutions

15
Complex Numbers

• When we add a real number and an imaginary number
  we get a Complex Number .
• Since the real and imaginary numbers are not like
  terms, we write complex numbers in the form a
  bi
• Examples 3 - 7i, -2 8i, -4i, 5 2i

16
Complex Numbers A/S

• To add or subtract two complex numbers, combine
  like terms (the real imaginary parts).
• Example (3 4i) (-5 - 2i) -2 2i

17
Practice

• Add these Complex Numbers

• (4 7i) - (2 - 3i)
• (3 - i) (7i)
• (-3 2i) - (-3 i)

2 10i 3 6i i
18
Complex Numbers M

• To multiply two complex numbers, FOIL them

• Replace i2 with -1

19
Practice

• Multiply
• 5i(3 - 4i)
• (1 - 3i)(2 - i)
• (7 - 4i)(7 4i)

20 15i -1 - 7i 65
20
Complex Numbers D

• We leave complex quotients in fraction (rational)
  form

• But since i represents a square root, we cannot
  leave an i term in the denominator...

21
Complex Numbers D

• We must rationalize any fraction with i in the
  denominator.

Binomial Denominator
Monomial Denominator
22
Complex Rationalize

• If the denominator is a monomial, multiply the
  top bottom by i.

23
Complex Rationalize

• If the denominator is a binomial multiply the
  numerator and denominator by the conjugate of the
  denominator ...

24
Complex Rationalize

• When you multiply conjugate complex numbers, the
  imaginary part drops out

25
Practice

• Simplify