5.7 Complex Numbers 12/17/2012

Quick Review Exponent Rule: If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 81 , x = x1 , -5 = -51 Also, any number raised to the Zero power is equal to 1 Ex: 30 = 1 -40 = 1 Exponent Rule: When multiplying powers with the same base, you add the exponent. x2 • x3 = x5 y • y7 = y8

The square of any real number x is never negative, so the equation x2 = -1 has no real number solution. To solve this x2 = -1 , mathematicians created an expanded system of numbers using the IMAGINARY UNIT, i.

Simplifying i given any powers 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. i22 = i2 Step 1. 51 ÷ 4 has a remainder of 3 Step 2. i51 = i3 Do you see the pattern yet?

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

Properties of Square Root of Negative Number

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

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

Checkpoint Solve the equation. 1. x 2 = – 3 ANSWER 3, i 3 – 2. = x 2 7 Solve a Quadratic Equation Solve the equation. 1. x 2 = – 3 ANSWER 3, i 3 – 2. = x 2 7 – ANSWER 7, i 7 – 3. = x 2 20 – ANSWER 5, 2 5 – i 4. = x 2 3 2 + – ANSWER 5, i 5 – 5. = y 2 4 – 12 ANSWER 2, 2 – i

Adding and Subtracting Complex Numbers Is a number written in the standard form a + bi where a is the real part and bi is the imaginary part. Add/Subtract the real parts, then add/subtract the imaginary parts Complex Number Adding and Subtracting Complex Numbers

Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i ( Example 2 Add Complex Numbers Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i ( SOLUTION Group real and imaginary terms. 2i 3 ( + i 1 – = 2 i Write in standard form. = 4 + i 11

Write as a complex number in standard form. 2i 6 ( – 1 Example 3 Subtract Complex Numbers Write as a complex number in standard form. 2i 6 ( – 1 SOLUTION Group real and imaginary terms. 2i 6 ( = 1 2 i + – -1 + 2i Simplify. = 5 + 0i Write in standard form. = 5 12

Write the expression as a complex number in standard form. Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 6. ( 4 – ( 2i ( + 1 + 3i ( ANSWER i 5 + 7. i 3 ( – + 4i 2 ANSWER 3i 5 + 8. 6i 4 ( + 3i 2 – ANSWER 3i 2 + 9. 4i 2 ( + 7i – ANSWER 3i 4 –

Write the expression as a complex number in standard form. Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 11. 2i 1 ( – + 5i 4 ANSWER 3i 5 + 12. i 2 ( – 4i 1 ANSWER 3i 3 +

Write the expression as a complex number in standard form. Example 4 Multiply Complex Numbers Write the expression as a complex number in standard form. a. 1 ( 3i + – 2i b. 3i 6 ( + 3i 4 ( – SOLUTION Multiply using distributive property. 1 ( 3i + – 2i = 6i 2 a. 1 ( – 2i 6 = + Use i 2 1. 6 2i – = Write in standard form.

Example 4 b. 3i 6 ( + 4 – 24 18i 12i 9i 2 = 24 6i – 9i 2 = 24 6i – 1 ( Multiply Complex Numbers b. 3i 6 ( + 4 – 24 18i 12i 9i 2 = Multiply using FOIL. 24 6i – 9i 2 = Simplify. 24 6i – 1 ( 9 = Use i 2 1. 6i 33 – = Write in standard form. 16

Complex Conjugates Two complex numbers of the form a + bi and a - bi Their product is a real number because (3 + 2i)(3 – 2i) using FOIL 9 – 6i + 6i -4i2 9 – 4i2 i2 = -1 9 – 4(-1) = 9 + 4 = 13 Is used to write quotient of 2 complex numbers in standard form (a + bi)

Write as a complex number in standard form. 2i 3 + 1 – a + bi SOLUTION Example 5 Divide Complex Numbers Write as a complex number in standard form. 2i 3 + 1 – a + bi SOLUTION 2i 3 + 1 – = • Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i. Multiply using FOIL. 1 2i 3 6i + – 4i 2 = 3 8i + 1 ( – 4 = Simplify and use i 2 1. 8i + – 1 5 = Simplify. 5 1 – 8 i + = Write in standard form. 18

Write the expression as a complex number in standard form. Checkpoint Multiply and Divide Complex Numbers Write the expression as a complex number in standard form. 13. i 2 ( – 3i ANSWER 6i 3 + 14. ( 2i 1 + i 2 – ANSWER 3i 4 + 15. i 2 + 1 – ANSWER 2 1 + 3 i

Graphing Complex Number Imaginary axis Real axis

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

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

Homework 5.7 p.264 #17-20, 27/29, 33-35, 40, 43, 45, 46, 52-54, 64-71 5.6

Checkpoint Solve the equation. 1. x 2 = – 3 2. = x 2 7 – 3. = x 2 20 – Solve a Quadratic Equation Solve the equation. 1. x 2 = – 3 2. = x 2 7 – 3. = x 2 20 – 4. = x 2 3 2 + – 5. = y 2 4 – 12

Write the expression as a complex number in standard form. Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 6. ( 4 – ( 2i ( + 1 + 3i ( 7. i 3 ( – + 4i 2 8. 6i 4 ( + 3i 2 – 9. 4i 2 ( + 7i –

Write the expression as a complex number in standard form. Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 10. 2i 1 ( – + 5i 4 11. i 2 ( – 4i 1 Write the expression as a complex number in standard form. 12. i 2 ( – 3i 14. i 2 + 1 – 13. ( 2i 1 + i 2 –