1. Complex Numbers n Understand complex numbers n Simplify complex number expressions

2. Imaginary Numbers n What is the square root of 9? n What is the square root of -9?

3. Imaginary Numbers n There is no real number that when multiplied by itself gives a negative number. n A new type of number was defined for this purpose. It is called an Imaginary Number. Imaginary numbers are NOT in the Real Set.

4. Imaginary Numbers n The constant, i, is defined as the square root of negative 1: n Multiples of i are called Imaginary Numbers

5. Imaginary Numbers n The square root of -9 is an imaginary number... n When we simplify a radical with a negative coefficient inside the radical, we write it as an imaginary number.

6. Imaginary Numbers n Simplify these radicals:

7. Multiples of i n Consider multiplying two imaginary numbers: n So...

8. Multiples of i n Powers of i:

9. Multiples of i n This pattern repeats:

10. Multiples of i n We can find higher powers of i using this repeating pattern: i, -1, -i, 1 What is the highest number less than or equal to 85 that is divisible by 4? 84 + ? = 85 1 84 So the answer is:

11. Powers of i - Practice n i 28 n i 75 n i 113 n i 86 n i 1089 41414141 4-i4-i4-i4-i 4i4i4i4i 4 -1 4i4i4i4i

12. Solutions Involving i Solve: Solve:

13. Solutions Involving i Solve: Solve:

14. Solutions Involving i Solutions: Solutions:

15. Complex Numbers n When we add a real number and an imaginary number we get a Complex Number. n Since the real and imaginary numbers are not like terms, we write complex numbers in the form a + bi n Examples: 3 - 7i, -2 + 8i, -4i, 5 + 2i

16. Complex Numbers: A/S n To add or subtract two complex numbers, combine like terms (the real & imaginary parts). n Example: (3 + 4i) + (-5 - 2i) = -2 + 2i

17. Practice Add these Complex Numbers: n (4 + 7i) - (2 - 3i) n (3 - i) + (7i) n (-3 + 2i) - (-3 + i) = 2 +10i = 3 + 6i = i

18. Complex Numbers: M n To multiply two complex numbers, FOIL them: n Replace i 2 with -1:

19. Practice Multiply: n 5i(3 - 4i) n (1 - 3i)(2 - i) n (7 - 4i)(7 + 4i) = 20 + 15i = -1 - 7i = 65

20. Complex Numbers: D n We leave complex quotients in fraction (rational) form: n But since i represents a square root, we cannot leave an i term in the denominator...

21. Complex Numbers: D n We must rationalize any fraction with i in the denominator. Binomial Denominator: Monomial Denominator:

22. Complex #: Rationalize n If the denominator is a monomial, multiply the top & bottom by i.

23. Complex #: Rationalize n If the denominator is a binomial multiply the numerator and denominator by the conjugate of the denominator...

24. Complex #: Rationalize n When you multiply conjugate complex numbers, the imaginary part drops out:

25. Practice n Simplify: