2.4 Complex Numbers x²+1=0 The Imaginary Unit i A complex conjugate is simply the original complex number (a+bi) in the form (a-bi). Complex Conjugates Addition and Subtraction Operations with Complex Numbers Complex Solutions of Quadratic Equations pg 133 (7-10, 15-90 [mod 5]) Suggested Homework Example 1 A square root of a negative creates an imaginary number

So i=√-1 i²=-1 √-9 -5=0

3√-1-5=0

3i-5=0 You cannot combine imaginary and real numbers. Multiply the complex number by its complex conjugate.

6-4i Division Using Complex Conjugates To divide two complex numbers, multiply both complex numbers by the bottom number's complex conjugate. 8-7i (1+2i)

1-2i (1+2i) 1) 2) Use the distributive property on both top and bottom. 8+16i-7i-14i²

1-2i+2i-4i² Remember that i²=-1, so you substitute in -1 for every i². 3) 8+9i-14(-1)

1-4(-1) 4) This becomes +14. This becomes +4. 22+9i

5 This is your final answer. 5i

(2+3i)² Example: Combine like terms. The normal rules of addition and subtraction apply. When subtracting, remember to distribute the negative. Example: (7+6i)+(3+12i) 1) 1) 2) (6-4i)(6+4i) Use the Distributive Property. 3) 36-24i+24i-16i² Remember i²=-1 4) 36-16(-1) 5) 52 2) Combine like terms. (7+3)+(6i+12i) Simplify. 3) 10+18i This is your final answer. 1) (13i)-(14-7i) Distribute the negative. 2) 13i-14+7i Combine like terms. 3) -14+20i Your final answer is in standard form. Example: 3

4 7

5 i + ( ) - 5

6 1

6 - ( ) i Sometimes when you use the quadratic formula, you end up with a negative under the radical. If you factor out the i, you can give your answer in standard form. Example: x²-2x+2=0 Set up the problem using the quadratic formula. x=2±√(-2)²-(4(1)(2))

2(1) Simplify whatever possible. x=2±√-4

2 Think of √-4 as two terms. x=2±(√4)(√-1)

2 Remember that √-1=i. x=2±2i

2 = x=1±i This is your final answer. 1) 2) 3) 4) 5) Example: 9x²-6x+37=0 Using your calculator, you can enter the number i. Shift and will get you i. . You can also change your calculator's mode from real to imaginary. If you want an answer in decimal form, use parentheses around the expression. Khan Academy has a series of videos with step-by-step explanations of the number i and operations using complex numbers. You can access Khan Academy without an account at www.khanacademy.org Multiplication Example: (6-2i)(2-3i) Remember to use the distributive property. When am I ever going to use this? According to Technology Integration Partnerships, engineers and physicists both use imaginary numbers to measure electrical currents and the flow of liquids. Z=V+Ii Impedance Voltage Current

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# Complex Numbers

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