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5-6 Complex Numbers

5-6 Complex Numbers. Objectives. Identifying Complex Numbers Operations with Complex Numbers. Vocabulary. The imaginary number i is defined as a number whose square is -1. So i ² = -1 and i = . Square Root of a Negative Real Number For any positive real number a, .

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5-6 Complex Numbers

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  1. 5-6 Complex Numbers

  2. Objectives Identifying Complex Numbers Operations with Complex Numbers

  3. Vocabulary The imaginary number iis defined as a number whose square is -1. So i² = -1 and i = . Square Root of a Negative Real Number For any positive real number a,

  4. –54 = –1 • 54 = –1 • 54 = i • 54 = i • 3 6 = 3i 6 Simplifying Numbers Using i Simplify –54 by using the imaginary number i.

  5. Vocabulary A complex number can be written in the form a + bi, where a and b are real numbers, including 0. a + bi Real Part Imaginary Part

  6. –121 – 7 = 11i – 7 Simplify the radical expression. Simplifying Imaginary Numbers Write –121 – 7 in a + bi form. = –7 + 11iWrite in the form a + bi.

  7. Vocabulary The absolute value of a complex number is its distance from the origin on the complex number plane. 4i 2i You can plot the points on the graph use the Pythagorean Theorem to find the distance. -4 -2 2 4 -2i -4i

  8. |10 + 24i| = 102 + 242 = 100 + 576 = 26 Finding Absolute Value Find each absolute value. a. |–7i| –7i is seven units from the origin on the imaginary axis. So |–7i| = 7 b. |10 + 24i|

  9. Additive Inverse of a Complex Number Find the additive inverse of –7 – 9i. –7 – 9i –(–7 – 9i) Find the opposite. 7 + 9iSimplify.

  10. Adding Complex Numbers Simplify the expression (3 + 6i) – (4 – 8i). (3 + 6i) – (4 – 8i) = 3 + (–4) + 6i + 8i Use commutative and associative properties. = –1 + 14i Simplify.

  11. Multiplying Complex Numbers Find each product. a. (3i)(8i) (3i)(8i) = 24i2Multiply the real numbers. = 24(–1)Substitute –1 for i2. = –24 Multiply. b. (3 – 7i)(2 – 4i) (3 – 7i)(2 – 4i) = 6 – 14i – 12i + 28i2 Multiply the binomials. = 6 – 26i + 28(–1) Substitute –1 for i2. = –22 – 26iSimplify.

  12. x= ±i 6 Find the square root of each side. Check: 9x2 + 54 = 0 9x2 + 54 = 0 9(i 6)2 + 54 0 9(i(– 6))2 + 54 0 9(6)i2 + 54 0 9(6i2) + 54 0 54(–1) –54 54(–1) –54 54 = 54 –54 = –54 Finding Complex Solutions Solve 9x2 + 54 = 0. 9x2 + 54 = 0 9x2 = –54 Isolate x2. x2 = –6

  13. Use z = 0 as the first input value. f(0) = 02 – 4i = –4i f(–4i)= (–4i)2 – 4i First output becomes second input. Evaluate for z = –4i. = –16 – 4i = [(–16)2 + (–16)(–4i) + (–16)(–4i) + (–4i)2] – 4i = (256 + 128i – 16) – 4i = 240 + 124i Real-World Connection Find the first three output values for f(z) = z2 – 4i. Use z = 0 as the first input value. f(–16 – 4i)= (–16 – 4i)2 – 4i Second output becomes third input. Evaluate for z = –16 – 4i. The first three output values are –4i, –16 – 4i, 240 + 124i.

  14. Homework 5-6 Pg 278 #1, 2, 11, 12, 19, 20, 24, 25, 29, 30, 41, 42

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