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7-7 De Moivre's Theorem

561

Section 7-7 De Moivre's Theorem

De Moivre's Theorem, n a Natural Number nth Roots of z

We now come to one of the great theorems in mathematics, De Moivre's theorem. Abraham De Moivre (16671754), of French birth, spent most of his life in London doing private tutoring, writing, and publishing mathematics. He belonged to many prestigious professional societies in England, Germany, and France and was a close friend of Isaac Newton. Using the polar form for a complex number, De Moivre established a theorem that still bears his name for raising complex numbers to natural number powers. More importantly, the theorem is the basis for the nth root theorem, which enables us to find all n nth roots of any complex number, real or imaginary.

De Moivre's Theorem, n a Natural Number

We start with Explore/Discuss 1 and generalize from this exploration.

1

By repeated use of the product formula for the exponential polar form rei , discussed in the last section, establish the following: (re i)2 r2e2 i 1. (x iy)2 3 i 3 2. (x iy) (re ) r3e3 i 3. (x iy)4 (re i)4 r4e4 i Based on forms 13, and for n a natural number, what do you think the polar form of (x iy)n would be?

If you guessed that the polar form of (x iy)n is rnen i, you have arrived at De Moivre's Theorem, which we now state without proof. A full proof of the theorem for all natural numbers n requires a method of proof, called mathematical induction, which is discussed in Section 9-2.

DE MOIVRE'S THEOREM If z x iy rei , and n is a natural number, then

1

EXAMPLE

zn

(x

iy)n

(rei )n

rnen

i

The Natural Number Power of a Complex Number

Use De Moivre's theorem to find (1 gular form. i)10. Write the answer in exact rectan-

1

562

7 ADDITIONAL TOPICS IN TRIGONOMETRY

Solution

(1

i)10

( 2e45°i)10 ( 2)10e(10 32e

450°i 45°)i

Convert 1

i to polar form.

Use De Moivre's theorem. Change to rectangular form.

32(cos 450° 32(0 32i i)

i sin 450°)

Rectangular form

MATCHED PROBLEM

1

EXAMPLE

Use De Moivre's theorem to find (1 and rectangular forms.

i 3)5. Write the answer in exact polar

The Natural Number Power of a Complex Number

Use De Moivre's theorem to find ( tangular form. ( 3 i)6 (2e150°i)6 26e(6

150°)i

2

Solution

3

i)6. Write the answer in exact recConvert form. 3 i to polar

Use De Moivre's theorem. Change to rectangular form.

64e900°i 64 (cos 900° 64 ( 1 64 [Note: 3 i must be a sixth root of i0) i sin 900°)

Rectangular form

64, since (

3

i)6

64.]

MATCHED PROBLEM

2

Use De Moivre's theorem to find (1 and rectangular forms.

i 3)4. Write the answer in exact polar

nth Roots of z

We now consider roots of complex numbers. We say w is an nth root of z, n a z. For example, if w2 z, then w is a square root of z. natural number, if wn 3 z, then w is a cube root of z. And so on. If w

If z rei , then use De Moivre's theorem to show that r1/2e( square root of z and r1/3e( /3)i is a cube root of z.

/2)i

is a

2

We can proceed in the same way as in Explore/Discuss 2 to show that r1/ne( is an nth root of rei , n a natural number:

/n)i

7-7 De Moivre's Theorem

563

[r1/ne(

/n)i n

]

(r1/n)nen( re

i

/n)i

But we can do even better than this. The nth-root theorem (Theorem 2) shows us how to find all the nth roots of a complex number.

nTH-ROOT THEOREM For n a positive integer greater than 1,

2

r1/ne(

/n k360°/n)i

k

0, 1, . . . , n

1

are the n distinct nth roots of rei , and there are no others.

The proof of Theorem 2 is left to Problems 31 and 32 in Exercise 7-7.

EXAMPLE

Finding All Sixth Roots of a Complex Number

Find six distinct sixth roots of First write 1 1 i 3 1 i 3, and plot them in a complex plane.

3

Solution

i 3 in polar form: 2e120°i

Using the nth-root theorem, all six roots are given by 21/6e(120°/6 Thus, w1

FIGURE 1

k 360°/6)i

21/6e(20°

k60°)i

k

0, 1, 2, 3, 4, 5

21/6e(20° 21/6e(20° 21/6e(20° 21/6e(20° 21/6e(20° 21/6e(20°

0 60°)i 1 60°)i 2 60°)i 3 60°)i 4 60°)i 5 60°)i

21/6e20°i 21/6e80°i 21/6e140°i 21/6e200°i 21/6e260°i 21/6e320°i

w2 w3 w4 w5 w6

All roots are easily graphed in the complex plane after the first root is located. The root points are equally spaced around a circle of radius 21/6 at an angular increment of 60° from one root to the next (Fig. 1).

MATCHED PROBLEM

3

Find five distinct fifth roots of 1 them in a complex plane.

i. Leave the answers in polar form and plot

564

7 ADDITIONAL TOPICS IN TRIGONOMETRY

EXAMPLE

Solving a Cubic Equation

Solve x3 1 0. Write final answers in rectangular form, and plot them in a complex plane. x3 1 x

3

4

Solution

0 1

We see that x is a cube root of 1, and there are a total of three roots. To find the three roots, we first write 1 in polar form: 1 1e180°i 1 are given by

Using the nth-root theorem, all three cube roots of 11/3e(180°/3 Thus,

FIGURE 2

k360°/3)i

1e(60°

k120°)i

k

0, 1, 2

w1 w2 w3

1e60°i le180°i le300°i

cos 60° cos 180° cos 300°

i sin 60°

1 2

i

3 2 1

i sin 180° i sin 300° 1 2

i

3 2

[Note: This problem can also be solved using factoring and the quadratic formula--try it.] The three roots are graphed in Figure 2. Solve x3 1 0. Write final answers in rectangular form, and plot them in a complex plane.

MATCHED PROBLEM

4

1. 32e300°i 3. w1 16 i16 3

Answers to Matched Problems

2. 16e(

240°)i

8

i8

3 21/10e297°i 4. 1, 1 2 i 3 2 , 1 2 i 3 2

21/10e9°i, w2

21/10e81°i, w3

21/10e153°i, w4

21/10e225°i, w5

7-7 De Moivre's Theorem

565

EXERCISE 7-7 A

In Problems 16, use De Moivre's theorem to evaluate each. Leave answers in polar form. 1. (2e30°i)3 4. ( 2e15°i)8 2. (5e15°i)3 5. (1 i 3)3 3. ( 2e10°i)6 6. ( 3 i)8

26. (A) Show that 2 is a root of x3 8 0. How many other roots does the equation have? (B) The root 2 is located on a circle of radius 2 in the complex plane as indicated in the figure. Locate the other two roots of x3 8 0 on the figure and explain geometrically how you found their location. (C) Verify that each complex number found in part B is a root of x3 8 0.

B

In Problems 712, find the value of each expression and write the final answer in exact rectangular form. (Verify the results in Problems 712 by evaluating each directly on a calculator.) 7. ( 10. ( 3 3 i)4 i)5 8. ( 1 11. 1 2 i)4 3 2

3

9. (1 i 12. 1 2

i)8 3 2

3

i

For n and z as indicated in Problems 1318, find all nth roots of z. Leave answers in polar form. 13. z 15. z 17. z 8e30°i, n 81e 1

60°i

In Problems 2730, solve each equation for all roots. Write final answers in polar and exact rectangular form 27. x3 64 27 0 0 28. x3 30. x3 64 27 0 0

3 4 5

14. z 16. z 18. z

8e45°i, n 16e 1

90°i

3 4 3

,n

,n i, n

29. x3

i, n

For n and z as indicated in Problems 1924, find all nth roots of z. Write answers in polar form and plot in a complex plane. 19. z 21. z 23. z 8, n 3 4 20. z 22. z 24. z 1, n 8, n i, n 4 3 5

C

31. Show that [r1/ne(

/n k360°/n)i n

16, n i, n 6

]

rei

for any natural number n and any integer k. 32. Show that r1/ne(

/n k360°/n)i

25. (A) Show that 1 i is a root of x4 4 0. How many other roots does the equation have? (B) The root 1 i is located on a circle of radius 2 in the complex plane as indicated in the figure. Locate the other three roots of x4 4 0 on the figure and explain geometrically how you found their location. (C) Verify that each complex number found in part B is a root of x4 4 0.

is the same number for k

0 and k

n.

In Problems 3336, write answers in polar form. 33. Find all complex zeros for P(x) 34. Find all complex zeros for P(x) 35. Solve x

5 3

x5 x

6

32. 1.

1 i

0 in the set of complex numbers. 0 in the set of complex numbers.

36. Solve x

In Problems 37 and 38, write answers using exact rectangular forms. 37. Write P(x) 38. Write P(x) x6 x

6

64 as a product of linear factors. 1 as a product of linear factors.

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