2. Imaginary and Complex Numbers Negative numbers do not have square roots in the real-number system. However, a larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex- number system and it will allow us to solve equations like x 2 + 1 = 0. The complex-number system makes use of i, a number that is, by definition, a square root of –1.

3. Example Solution Express in terms of i:

7. Addition and Subtraction The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials. Example Solution

8. Multiplication To multiply square roots of negative real numbers, we first express them in terms of i. For example,

9. Multiply and simplify. When possible, write answers in the form a + bi. Example Solution

10. Solution continued

11. Conjugates and Division Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators. Solution Example Find the conjugate of each number. The conjugate is 4 – 3i. The conjugate is –6 + 9i. The conjugate is –i.

12. Solution ExampleDivide and simplify to the form a + bi.

13. Powers of i

14. Example Simplify: