1. 2.5 Introduction to Complex Numbers 11/7/2012

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 raised to any power 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. Raise i to the remainder and determine its 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 i 3 = - i i 0 = 1 i 1 = i i 2 = -1

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.

7. State the real and imaginary part of each complex number.

8. Add/Subtract the real parts, then add/subtract the imaginary parts Adding and Subtracting Complex Numbers

9. 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

10. Subtract Complex Numbers Write as a complex number in standard form. Simplify. = 5+0i0i 2i2i6 ( ( –– 2i2i1 ( ( – SOLUTION Group real and imaginary terms. 2i2i6 ( ( = 16 2i2i 2i2i + –– 2i2i1 ( ( –– – Write in standard form. = 5 -1 + 2 i

11. Checkpoint Write the expression as a complex number in a + b i form. Add and Subtract Complex Numbers 1. 2i2i4 ( ( – +3i3i1 ( ( + ANSWER i5+ 2. i3 ( ( – +4i4i2 ( ( + ANSWER 3i3i5+ 3i3i2+ 3. 6i6i4 ( ( +3i3i2 ( ( + – 4. 4i4i2 ( ( +7i7i2 ( ( + –– ANSWER 3i3i4 ––

12. 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.

13. 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.

14. Divide Complex Numbers

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

16. SOLUTION 2i2i3+ 2i2i1 – 2i2i3+ 2i2i1 – 2i2i1+ 2i2i1 + = Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i. + – Divide Complex Numbers Write as a complex number in a + b i 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.

17. Checkpoint Write the expression as a complex number in standard form. Multiply and Divide Complex Numbers 1. i2 ( ( – 3i3i ANSWER 6i6i3+ 2. ( 2i2i1 ( +i2 ( ( – ANSWER 3i3i4+ 3. i2+ i1 – ANSWER 2 1 + 2 3 i

18. Graphing Complex Number Real axis Imaginary axis

19. 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

20. 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

21. Homework WS 2.5 #1-12all, 13-27odd, 31-34all