1. Section 8.7 Complex Numbers

2. Overview In previous sections, it was not possible to find the square root of a negative number using real numbers: is not a real number, or “n.a.r.n.” In this section we will discover a way to find such square roots. We will also learn about complex numbers and how to add, subtract, and multiply them.

3. The imaginary number i First, a definition: the imaginary number i is defined to be the square root of –1: It follows that

4. Definition In general, if a is a real number and a > 0, then

5. Examples Write each number as a product of a real number and i:

6. “Take the i out” When multiplying and dividing square roots with a negative radicand, first use the definition to change the radical.

7. Examples: Multiply and Divide

8. Complex numbers A number in the form a + bi, where a and b are real numbers and is called a complex number. Complex numbers can be: 1.added (add the real parts together and the imaginary parts together) 2.subtracted (subtract the real parts and the imaginary parts) 3.multiplied (use the FOIL method and the fact that )

9. Examples: Add and Subtract (-3 + 2i) + (4 + 7i) (7 + 10i) – (3 + 5i) (5 – i) + (-3 + 3i) + (6 – 4i) (-1 + 12i) – (-1 – i)

10. Examples: Multiply 6i(4 + 3i) (6 – 4i)(2 + 4i) (3 + 2i)(3 + 4i)

11. Remember that “conjugate” thing? Dividing complex numbers is very much like dividing radicals. To divide complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator.

12. Examples: Divide