1. 5.7 Complex Numbers 12/4/2013

2. Quick Review If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 8 1, x = x 1, -5 = -5 1 Also, any number raised to the Zero power is equal to 1 Ex: 3 0 = 1 -4 0 = 1 Exponent Rule: When multiplying powers with the same base, you add the exponent. x 2 x 3 = x 5 y y 7 = y 8

3. The square of any real number x is never negative, so the equation x 2 = -1 has no real number solution. To solve this x 2 = -1, mathematicians created an expanded system of numbers using the IMAGINARY UNIT, i.

4. Simplifying i given any powers Do you see the pattern yet? The pattern repeats after every 4. So you can find i raised to any power by dividing the exponent by 4 and see what the remainder is. Based on that remainder, you can determine it’s value. Step 1. 22÷ 4 has a remainder of 2 Step 2. i 22 = i 2 Step 1. 51 ÷ 4 has a remainder of 3 Step 2. i 51 = i 3

5. Checkpoint Find the value of 1. i 15 2. i 20 3. i 61 4. i 122

6. Complex Number Is a number written in the standard form a + b i where a is the real part and b i is the imaginary part. Ex: 3 + 2 i Add/Subtract the real parts, then add/subtract the imaginary parts Adding and Subtracting Complex Numbers

7. Example 2 Add Complex Numbers Write as a complex number in standard form. 2i2i3 ( ( +i1 ( ( – + SOLUTION Group real and imaginary terms. 2i2i3 ( ( +i1 ( ( – + = 13+1 2i2i i + – Write in standard form. = 4+i

8. Example 3 Subtract Complex Numbers Write as a complex number in standard form. = 5+0i0i 2i2i6 ( ( –– 2i2i1 ( ( – SOLUTION 2i2i6 ( ( = –– 2i2i1 ( ( – Write in standard form. = 5 -1 + 2 i 6 – 1 – 2i + 2i

9. Checkpoint Write the expression as a complex number in standard form. Add and Subtract Complex Numbers 6. 2i2i4 ( ( – +3i3i1 ( ( + ANSWER i5+ 7. i3 ( ( – +4i4i2 ( ( + ANSWER 3i3i5+ 3i3i2+ 8. 6i6i4 ( ( +3i3i2 ( ( + – 9. 4i4i2 ( ( +7i7i2 ( ( + –– ANSWER 3i3i4 ––

10. Checkpoint Write the expression as a complex number in standard form. Add and Subtract Complex Numbers 11. 2i2i1 ( ( – +5i5i4 ( ( + ANSWER 3i3i5+ 3i3i3+ 12. i2 ( ( – ( ( – 4i4i1 ––

11. Example 4 Multiply Complex Numbers Write the expression as a complex number in standard form. a. b. 1 ( 3i3i ( +–2i2i3i3i6 ( ( +3i3i4 ( ( – SOLUTION Multiply using distributive property. 1 ( 3i3i ( + – 2i2i = 2i2i6i 26i 2 – + a. 1 ( ( – 2i2i6 – = + Use i 2 1. = – 62i2i –– = Write in standard form.

12. Example 4 Multiply Complex Numbers Multiply using FOIL. b. 3i3i6 ( ( +3i3i4 ( ( – 2418i – 12i+9i 29i 2 – = 246i6i –– 9i 29i 2 = Simplify. 246i6i –– 1 ( ( – 9 = Use i 2 1. = – 6i6i33 – = Write in standard form.

13. Homework WS 5.7 Do problems 13-38 only

14. Complex Conjugates Two complex numbers of the form a + b i and a - b i Their product is a real number because (3 + 2 i )(3 – 2 i ) using FOIL 9 – 6 i + 6i -4 i 2 9 – 4i 2 i 2 = -1 9 – 4(-1) = 9 + 4 = 13 Is used to write quotient of 2 complex numbers in standard form (a + bi)

15. SOLUTION 2i2i3+ 2i2i1 – 2i2i3+ 2i2i1 – 2i2i1+ 2i2i1 + = Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i. + – Example 5 Divide Complex Numbers Write as a complex number in standard form. 2i2i3+ 2i2i1– Multiply using FOIL. 1 2i2i 36i6i+ + 4i 24i 2 + 2i2i 2i2i+ – 4i 24i 2 – = 38i8i+1 ( ( – 4+ 1 – 1 ( ( – 4 = Simplify and use i 2 1. = – 8i8i+ – 1 5 = Simplify. 5 1 – 5 8 i + = Write in standard form. a + b i

16. Checkpoint Write the expression as a complex number in standard form. Multiply and Divide Complex Numbers i2+ i1 – ANSWER 2 1 + 2 3 i

17. Properties of Square Root of Negative Number

18. Example 1 Solve a Quadratic Equation Solve the equation. = 7x 27x 2 49 – a.b. = 3x 23x 2 5 – 29 – SOLUTION = 7x 27x 2 49 – a. Divide each side by 7. = x 2x 2 7 – Take the square root of each side. = x + – 7 – Write in terms of i. = x + – 7 i

19. Example 1 Solve a Quadratic Equation Write original equation. b. = 3x 23x 2 29 – 5 – Add 5 to each side. = 3x 23x 2 24 – Divide each side by 3. = x 2x 2 8 – Write in terms of i. = x + – 8 i Take the square root of each side. = x + – 8 – Simplify the radical. = x + – 2 i2

20. Checkpoint Solve the equation. Solve a Quadratic Equation 1. = x 2x 2 3 – ANSWER 3,3, i 3i – 2. = x 2x 2 20 – 3. = x 2x 2 32+ – ANSWER 5,5, 2 52 – i i 5,5, i 5i –

21. Graphing Complex Number Real axis Imaginary axis

22. Ex: Graph 3 – 2 i 3 2 To plot, start at the origin, move 3 units to the right and 2 units down 3 – 2 i

23. Ex: Name the complex number represented by the points. A D C B Answers: A is 1 + i B is 0 + 2 i = 2 i C is -2 – i D is -2 + 3 i