1. Imaginary and Complex Numbers

2. Complex Number System The basic algebraic property of i is the following: i² = −1 Let us begin with i 0, which is 1. Each power of i can be obtained from the previous power by multiplying it by i. We have: i 0 = 1 i² = −1 i 3 = i² * i = -1 * i = - i i 4 = i² * i 2 = -1 * -1= 1

3. Complex Number System

4. Complex Number Systems

5. Practice Problems i 35 = i 40 = Tip: Even powers of i will be either 1 or −1, since the exponent is a multiple of 4 or 2 more than a multiple of 4. Odd powers will be either i or −i. Example: 3i· 4i = 12i² = 12(−1) = −12. Example: −5i· 6i = −30i² = 30. −i−i 1

6. Complex Numbers

7. Complex Number Systems The operation of division reviews all of the previous operations We are going to use the operations we have practiced and complex conjugates to solve division problems using imaginary numbers.

8. Complex Conjugates

9. Therefore (a + bi)(a − bi) = a² + b² The product of a conjugate pair is equal to the sum of the squares of the components. Notice that the middle terms cancel. This fact helps us rationalize denominators, which is a form of division.

10. Division Example