1. Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities
Chapter 5 – Quadratic Functions and Inequalities
5.4 – Complex Numbers

2. 5.4 – Complex Numbers In this section we will learn how to:
Find square roots and perform operations with pure imaginary numbers
Perform operations with complex numbers

3. 5.4 – Complex Numbers
Square root – the square root of a number n is a number with a square of n.
Ex. 7 is a square root of 49 because 72 = 49
Since (-7)2 = 49, -7 is also a square root

4. 5.4 – Complex Numbers Product and Quotient Properties of Square Roots
For nonnegative real numbers a and b,
√ab = √a  √b
Ex. √3  2 = √3  √2
√a/b = √a / √b
Ex. √1/4 = √1 / √4

5. 5.4 – Complex Numbers
Simplified square root expressions DO NOT have radicals in the denominator.
Any number remaining under the square root has no perfect square factor other than 1.

6. 5.4 – Complex Numbers
Example 1
Simplify
√18
√10/81

7. 5.4 – Complex Numbers
Example 2
Simplify √-9
IMAGINARY NUMBER!!

8. 5.4 – Complex Numbers
Imaginary number – created so that square roots of negative numbers can be found
Imaginary unit – i
i = √ i2 = – i3 = – I i4 = 1
Pure imaginary number – square roots of negative real numbers
Ex. 3i, -5i, and i√2
For any positive real number b,
√-b2 = √b2  √-1 or bi

9. 5.4 – Complex Numbers
Example 3
Simplify
√-28
√-32y4

10. 5.4 – Complex Numbers
Example 4
Simplify
-3i  2i
√-12  √-2
i35

11. 5.4 – Complex Numbers
You can solve some quadratic equations by using the square root property
Square Root Property
For any real number n, if x2 = n, then x = ±√n

12. 5.4 – Complex Numbers
Example 5
Solve 5y = 0.

13. 5.4 – Complex Numbers
HOMEWORK Page 264 #22 – 29

14. 5.4 – Complex Numbers
Complex number – any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. a is called the real part, and b is called the imaginary part
Ex i and 2 – 6i = 2 + (-6)I
If b = 0, the complex number is a real number
If b ≠ 0, the complex number is imaginary
If a = 0, the complex number is a pure imaginary number

15. 5.4 – Complex Numbers
Two complex numbers are equal if and only if (IFF) their real parts are equal AND their imaginary parts are equal.
a + bi = c + di IFF a = c and b = d

16. 5.4 – Complex Numbers Example 6
Find the values of x and y that make that equation 2x + yi = -14 – 3i true.

17. 5.4 – Complex Numbers
To add or subtract complex numbers, combine like terms.
Combine the real parts
Combine the imaginary parts

18. 5.4 – Complex Numbers Example 7 Simplify (3 + 5i) + (2 – 4i)

19. 5.4 – Complex Numbers
You can also multiply 2 complex numbers using the
FOIL method

20. 5.4 – Complex Numbers Example 8
In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I  Z. Find the voltage in a circuit with current 1 + 3j amps and impedance 7 – 5j ohms.

21. 5.4 – Complex Numbers
HOMEWORK Page 264 #30 – 39

22. 5.4 – Complex Numbers
Complex conjugates – two complex numbers of the form a + bi and a – bi.
The product of complex conjugates is always a real number.
You can use this to simplify the quotient of two complex numbers.

23. 5.4 – Complex Numbers
Example 9
Simplify

24. 5.4 – Complex Numbers
HOMEWORK Page 264 #40 – 49