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Roots of Complex Numbers

Roots of Complex Numbers. Section 9.3. n th root. If z and w are complex numbers and if n ≥ 2 is an integer , then z is an nth root of w iff z n = w. Example 1. Write the 3 cube roots of 1331 in trigonometric form. First, z 3 = 1331 z = [r, θ ] by De Moivre’s

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Roots of Complex Numbers

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  1. Roots of ComplexNumbers Section 9.3

  2. nthroot • If z and w are complexnumbers and if n ≥ 2 is an integer, then z is an nthroot of w iffzn= w

  3. Example 1 • Write the 3 cube roots of 1331 in trigonometric form. First, z3 = 1331 z = [r, θ] by De Moivre’s z3 = [r3, 3θ] = [1331, 0] So r3 = 1331 so, r = 11 3θ = 0 + 2π or θ = 0 + 2/3π

  4. Example 1 3θ = 0 + 2π or θ = 0 + 2/3π So, we have [11, 0]  11 (cos (0) + isin (o))  11 [ 11, 2/3 π]  11 (cos (2π/3)+ isin (2π/3)) [11, 4/3 π]  11 (cos (4π/3)+ isin (4π/3)

  5. nthroot of complexnumberstheorem • The n nthroots of [r, θ] are where k = 0, 1, 2, .., n-1 The n nth roots of r(cosθ + isinθ) Where k = 0, 1, 2, …, n-1

  6. Example 2 Express the sixth roots of [729, 60˚] Nth roots of complex numbers theorem [3, 10˚] [3, 70˚] [3, 130˚] [3, 190˚] [3, 250˚] [3, 310˚]

  7. Example 3 Express the square roots of i in polar, trigonometric, and binomial. Rectangular: (0,1) Polar: [1, π/2] Square roots in polar:

  8. Example 3 to trigonometric: to binomial: to trigonometric: to binomial:

  9. Example 4 There are exactly two real nth roots of a positive number if n is even and exactly one real nth root if n is odd. The other nth roots are vertices of a regular n-gon centered at the origin. Example: z5 = 51 What figure does this make? How many real and nonreal solutions does this make? Regular pentagon Nonreal: 4 Real: 1

  10. Homework Pages 539 – 540 2, 4 – 10, 13

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