Powers of Complex Numbers in Polar Form (2+I)2 =

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Powers of Complex Numbers in Polar Form (2+i)2 = (2+i)12 = Powers of Complex Numbers in Polar Form Let z = r(cosθ +isinθ) z2 = r(cosθ +isinθ)r(cosθ +isinθ) = r2(cos2θ +isin2θ) z3 = r2(cos2θ +isin2θ)r(cosθ +isinθ) = r3(cos3θ +isin3θ) z4 = r3(cos3θ +isin3θ)r(cosθ +isinθ) = r4(cos4θ +isin4θ) The formula for the nth power of a complex number in polar form is known as DeMoivre's Theorem (in honor of the French mathematician Abraham DeMoivre (1667‐1754). DeMoivre's Theorem Let z = r(cosθ+isinθ) be a complex number in polar form. If n is a positive integer, z to the nth power, zn, is zn = [r(cosθ+isinθ)]n = rn[cos(nθ)+isin(nθ)] = rnCiS(nθ) Ex1. Try. Ex2. Try. Roots of a Complex Number in Polar Form Yesterday we saw that [2(cos20o +isin20o)]6 = 64(cos120o + isin120o) Thinking backwards we could note that: A sixth root of 64(cos120o + isin120o) is 2(cos20o +isin20o) It is one of SIX distinct 6th roots of 64cis120o. Nth Root of a Complex Number In general, if a complex number, z, satisfies the equation zn = w, then we can say that z is a complex nth root of w. Also, z is one of n complex nth roots of w. Given the fact that roots are actually powers (remember x1/2 = √x), DeMoivre's Theorem can also be applied to roots... DeMoivre's Theorem for Finding Complex Roots Let w = rcisθ be a complex number in polar form. If w ≠ 0, then w has n distinct complex nth roots given by the formula: (radians) or (degrees) where k = 0, 1, 2, ..., n‐1. Important Notes about roots: 1. There are two distinct square roots There are three distinct cube roots There are four distinct fourth roots, and so on. 2. Each one of the distinct nth roots has the same modulus, . 3. Successive roots have arguments (angles) that differ by the same amount. 4. Successive roots are equally spaced around a circle centered about the pole. Ex1. Find all complex 4th roots of 16cis120o. TRY. Find all complex 5th roots of 243cis(80o). Ex2. Use DeMoivre's Theorem to find all cube roots of 8. TRY. Use DeMoivre's Theorem to find all cube roots of 27. ASSIGNMENT Blitzer P. 696‐697 #53‐76, 87‐90.

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The Nth Root of a Number Page 1
Lecture Notes The nth Root of a Number page 1 Caution! Currently, square roots are dened differently in different textbooks. In conversations about mathematics, approach the concept with caution and rst make sure that everyone in the conversation uses the same denition of square roots. Denition: Let A be a non-negative number. Then the square root of A (notation: pA) is the non-negative number that, if squared, the result is A. If A is negative, then pA is undened. For example, consider p25. The square-root of 25 is a number, that, when we square, the result is 25. There are two such numbers, 5 and 5. The square root is dened to be the non-negative such number, and so p25 is 5. On the other hand, p 25 is undened, because there is no real number whose square, is negative. Example 1. Evaluate each of the given numerical expressions. a) p49 b) p49 c) p 49 d) p 49 Solution: a) p49 = 7 c) p 49 = unde ned b) p49 = 1 p49 = 1 7 = 7 d) p 49 = 1 p 49 = unde ned Square roots, when stretched over entire expressions, also function as grouping symbols. Example 2. Evaluate each of the following expressions. a) p25 p16 b) p25 16 c) p144 + 25 d) p144 + p25 Solution: a) p25 p16 = 5 4 = 1 c) p144 + 25 = p169 = 13 b) p25 16 = p9 = 3 d) p144 + p25 = 12 + 5 = 17 . Denition: Let A be any real number. Then the third root of A (notation: p3 A) is the number that, if raised to the third power, the result is A.
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