1. Lesson 1.8 Complex Numbers Objective: To simplify equations that do not have real number solutions.

2. Warm up 1. (3.0 x 10 3 ) (5.0 x 10 5 ) 2. -4 3. - ( - )

3. Imaginary Number i i = i 2 = -1 i 3 = i 2 ∙ i = (-1)i = - i i 4 = i 2 ∙ i 2 = (-1)(-1) = 1 This pattern continues every 4 powers.

4. Imaginary Number i So i 13 = i 12 ∙ i = 1∙ i = i = √-1 Find : i 15 i 6 -i 7

5. Imaginary Number i = i When multiplying negative square roots, convert them to I first and then multiply. = 4i ∙ 3i = 12i 2 = -12 NOT √144

6. Complex Numbers Complex Numbers – a combination of real and imaginary numbers. a + bi a & b are real numbers Real part Imaginary part

7. Complex Numbers Writing as a complex number -1/2 = -1/2 + 0i = i = 3i = 0 + 3i -1 - = -1 – i = -1 – 2i

8. Equality of Complex Numbers a + bi = c + di if a = c and b = d So given y + 4i = 7 – xi y = 7 and 4 = -x ; -4 = x

9. Addition & Subtraction of Complex Numbers (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) – (c + di) = (a – c) + (b – d)i

10. Addition & Subtraction of Complex Numbers (7 – 2i) + (4 – 3i) 14 – (3 – 8i) (-9 + 3i) + (6 – 2i)

11. Multiplication of Complex Numbers (a + bi)(c + di) = ac+bdi 2 + (ad + bc)i Ex: (3 + 4i)(2 + 6i) (3)(2) + 18i + 8i + 24i 2 6 + 26i – 24 -18 + 26i

12. Complex Conjugate The multiplication of a complex number and its conjugate is a real number. (a + bi)(a - bi) = a 2 + b 2

13. Division The complex conjugate is used to simplify division of complex numbers. a + bi = a + bi c – di c + di c + di c – di ● Complex conjugate

14. Division Simplify to the form a + bi: 4 – 2i 5 + 2i 1 2 – 3i