5-6: Complex Numbers Day 1 (Essential Skill Review)

1. 5-6: Complex NumbersDay 1 (Essential Skill Review) Essential Question: How do we simplify square roots of negative numbers?

2. 5-6: Complex Numbers • Operations with Radicals • Simplifying a radical: Option #1 • Break a number into prime factors. • Pull any pairs out as one number outside the radical • Multiply any numbers remaining inside & outside the radical • Example #1 • 75 = 5 • 5• 3 = • Example #2 • 96 = 2•2•2•2•2•3 = 2•2

3. 5-6: Complex Numbers • Operations with Radicals • Simplifying a radical: Option #2 • Divide by perfect squares. • A perfect square is any number times itself: • 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, etc. • Simplify all portions of the radical • Example #1 • Example #2

4. 5-6: Complex Numbers • Some rules with radicals • When numbers INSIDE a radical match, the numbers OUSTIDE can be added/subtracted (rule of like terms) • Sometimes, radicals must be simplified before they can be combined • Example:

5. 5-6: Complex Numbers • Some rules with radicals (continued) • You shouldn’t leave a radical in the denominator of a fraction • To remove it, we rationalize the denominator. Multiply the top and bottom of the fraction by the radical in the denominator. • Examples:

6. 5-6: Complex Numbers • Assignment • Page 883 • 2 – 30, evens

7. 5-6: Complex NumbersDay 2 Essential Question: How do we simplify square roots of negative numbers?

8. 5-6: Complex Numbers • The imaginary number i is defined as the number whose square is -1. • So i2 = -1, and • To simplify square roots of negative numbers • Take the negative sign outside the square root, replace it with i. • Simplify the number underneath the square root as normal. Numbers outside the square root come before the i. • Example:

9. 5-6: Complex Numbers • Solve

10. 5-6: Complex Numbers • Imaginary numbers and real numbers make up the set of complex numbers. • Complex numbers are written in the form a + bi • That means the real number gets written first, followed by the imaginary number. • Example: • Write the complex number in the form a + bi

11. 5-6: Complex Numbers • Write the complex number in a + bi form.

12. 5-6: Complex Numbers • You can apply real number concepts to complex numbers. • Complex numbers have additive inverses (or “opposites”) • It’s simply the opposite of the real number added to the opposite of the imaginary number • Example: Find the additive inverse of -2 + 5i. • The opposite of -2 is 2 • The opposite of 5i is -5i. • So the additive inverse of -2 + 5i is 2 – 5i.

13. 5-6: Complex Numbers • Find the additive inverse of each number

14. 5-6: Complex Numbers • Assignment • Page 274 • Problems 1 – 18 and 24 – 28 • All problems

15. 5-6: Complex NumbersDay 3 Essential Question: How do we simplify square roots of negative numbers?

16. 5-6: Complex Numbers • Adding & Subtracting Complex Numbers • Simply combine the real parts with the imaginary parts • Example • (5+ 7i) + (-2+ 6i) • 5 + -2+ 7i + 6i • 3 + 13i

17. 5-6: Complex Numbers • Simplify each expression

18. 5-6: Complex Numbers • Multiplying Complex Numbers • If i = , then i2 = -1 • Example • (5i)(-4i) • -20i2 • Replace i2 with -1 • -20(-1) • 20

19. 5-6: Complex Numbers • Simplify the expression

20. 5-6: Complex Numbers • Multiplying Complex Numbers • FOIL Example • (2 + 3i)(-3 + 5i) • -6 + 10i – 9i + 15i2 Combine like terms • -6 + i + 15(-1) Replace i2 with -1 • -6 + i – 15 Combine like terms again • -21 + i

21. 5-6: Complex Numbers • Simplify the expression

22. 5-6: Complex Numbers • Simplify the expression

23. 5-6: Complex Numbers • Assignment • Page 274 • Problems 29-40 (all) and 58-66 (even)