1. Roots, Radicals, and Complex Numbers
Chapter 7
Roots, Radicals, and Complex Numbers

2. Chapter Sections 7.1 – Roots and Radicals 7.2 – Rational Exponents
7.3 – Simplifying Radicals
7.4 – Adding, Subtracting, and Multiplying Radicals
7.5 – Dividing Radicals
7.6 – Solving Radical Equations
7.7 – Complex Numbers
Chapter 1 Outline

3. § 7.7
Complex Numbers

4. Recognize a Complex Number
An imaginary number is a number such as It is called imaginary because when imaginary numbers were first introduced, many mathematicians refused to believe they existed!
Every imaginary number has as a factor. The , called the imaginary unit, is denoted by the letter i.

5. Recognize a Complex Number
Imaginary Unit
Square Root of a Negative Number
For any positive real number n,

6. Recognize a Complex Number
Every number of the form
Where a and b are real numbers and I is the imaginary unit, is a complex number.
Every real number and every imaginary number are also complex numbers. A complex number has two parts: a real part, a, and an imaginary part, b.

7. Recognize a Complex Number

8. Recognize a Complex Number
Complex numbers can be added, subtracted, multiplied, and divided. To perform these operations, we use the definitions that and
Definition of i2
If , then

9. Add and Subtract Complex Numbers
To Add or Subtract Complex Numbers
Change all imaginary numbers to bi form.
Add (or subtract) the real parts of the complex numbers.
Add (or subtract) the imaginary parts of the complex numbers.
Write the answer in the form a + bi.

10. Add and Subtract Complex Numbers
Example Add

11. Multiply Complex Numbers
To Multiply Complex Numbers
Change all imaginary numbers to bi form.
Multiply the complex numbers as you would multiply polynomials.
Substitute –1 for each i2.
Combine the real parts and the imaginary parts. Write the answer in a + bi form.

12. Multiply Complex Numbers
Example Multiply

13. CAUTION!

14. Divide Complex Numbers
The conjugate of a complex number a + bi is a – bi. For example,
Complex Number
Conjugate

15. Divide Complex Numbers
To Divide Complex Numbers
Change all imaginary numbers to bi form.
Multiply both the numerator and denominator by the conjugate of the denominator.
Write the answer is a + bi form.

16. Divide Complex Numbers
Example Divide

17. Find Powers of i
The successive powers of i rotate through the four values of i, -1, -i, and 1.
in = i if n = 1, 5, 9, …
in = 1 if n = 4, 8, 12, …
in = -1 if n = 2, 6, 10, …
in = -i if n = 3, 7, 11, …