1. Section 7.7 Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. That’s still true. However, we will now introduce a new set of numbers. Imaginary numbers which includes the imaginary unit i.

2. Real numbers and imaginary numbers are both subsets of a new set of numbers. Complex numbers Every complex number can be written as a sum of a real number and an imaginary number.

3. The imaginary unit i is the number whose square is –1. We can write the square root of a negative number in terms of i. i i

4. Write the following with the i notation. Example i i ii

5. Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers. a is a real number and bi would be an imaginary number. If b = 0, a + bi is a real number. If a = 0, a + bi is an imaginary number.

6. Example Write each of the following in the form of a complex number in standard form a + bi. 6 =6 + 0i 8i =0 + 8i i 6 + 5i i

7. Adding and subtracting complex numbers is accomplished by combining their corresponding components. (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) – (c + di) = (a – c) + (b – d)i

8. Add or subtract the following complex numbers. Write the answer in standard form a + bi. Example (4 + 3) + (6 – 2)i =7 + 4i (8 + 2i) – (4i) = (8 – 0) + (2 – 4)i = 8 – 2i (4 + 6i) + (3 – 2i) =

9. The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms.

10. Note that the product rule for radicals does NOT apply for imaginary numbers.

11. Multiply the following complex numbers. 8i 7i 56i 2 56(-1) -56 Example

12. Multiply the following complex numbers. Write the answer in standard form a + bi. 5i(4 – 7i) 20i – 35i 2 20i – 35(-1) 20i + 35 35 + 20i Example

13. Multiply the following complex numbers. Write the answer in standard form a + bi. (6 – 3i)(7 + 4i) 42 + 24i – 21i – 12i 2 42 + 3i – 12(-1) 42 + 3i + 12 54 + 3i Example

14. In the previous chapter, when trying to rationalize the denominator of a rational expression containing radicals, we used the conjugate of the denominator. Similarly, to divide complex numbers, we need to use the complex conjugate of the number we are dividing by.

15. The conjugate of a + bi is a – bi. The conjugate of a – bi is a + bi. The product of (a + bi) and (a – bi) is (a + bi)(a – bi) a 2 – abi + abi – b 2 i 2 a 2 – b 2 (-1) a 2 + b 2, which is a real number.

16. Divide the following complex numbers. Write the answer in standard form. Example

17. Divide the following complex numbers. Example

18. So far, we have looked at only two powers of i, i and i 2 There is an interesting pattern within the powers of i.

19. The powers recycle through each multiple of 4.

20. Simplify each of the following powers. Example