Download Section 7-7 De Moivre`s Theorem

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 (1667–1754), 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:
1. (x ⫹ iy)2 ⫽ (re␪i)2 ⫽ r2e2␪i
2. (x ⫹ iy)3 ⫽ (re␪i)3 ⫽ r3e3␪i
3. (x ⫹ iy)4 ⫽ (re␪i)4 ⫽ r4e4␪i
Based on forms 1–3, 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
1
zn ⫽ (x ⫹ iy)n ⫽ (rei␪)n ⫽ rnen␪i
The Natural Number Power of a Complex Number
Use De Moivre’s theorem to find (1 ⫹ i)10. Write the answer in exact rectangular form.
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7 ADDITIONAL TOPICS IN TRIGONOMETRY
Solution
(1 ⫹ i)10 ⫽ (兹2e45°i)10
Convert 1 ⫹ i to polar form.
⫽ (兹2)10e(10 ⴢ 45°)i
Use De Moivre’s theorem.
⫽ 32e
Change to rectangular form.
450°i
⫽ 32(cos 450° ⫹ i sin 450°)
⫽ 32(0 ⫹ i)
⫽ 32i
MATCHED PROBLEM
1
EXAMPLE
2
Solution
Rectangular form
Use De Moivre’s theorem to find (1 ⫹ i兹3)5. Write the answer in exact polar
and rectangular forms.
The Natural Number Power of a Complex Number
Use De Moivre’s theorem to find (⫺兹3 ⫹ i)6. Write the answer in exact rectangular form.
(⫺ 兹3 ⫹ i)6 ⫽ (2e150°i)6
Convert ⫺兹3 ⫹ i to polar
form.
⫽ 26e(6 ⴢ 150°)i
Use De Moivre’s theorem.
⫽ 64e900°i
Change to rectangular form.
⫽ 64 (cos 900° ⫹ i sin 900°)
⫽ 64 (⫺1 ⫹ i0)
⫽ ⫺64
Rectangular form
[Note: ⫺兹3 ⫹ i must be a sixth root of ⫺64, since (⫺兹3 ⫹ i)6 ⫽ ⫺64.]
MATCHED PROBLEM
2
Use De Moivre’s theorem to find (1 ⫺ i兹3)4. Write the answer in exact polar
and rectangular forms.
nth Roots of z
We now consider roots of complex numbers. We say w is an nth root of z, n a
natural number, if wn ⫽ z. For example, if w2 ⫽ z, then w is a square root of z.
If w3 ⫽ z, then w is a cube root of z. And so on.
If z ⫽ rei␪, then use De Moivre’s theorem to show that r1/2e(␪/2)i is a
square root of z and r1/3e(␪/3)i is a cube root of z.
2
We can proceed in the same way as in Explore/Discuss 2 to show that r1/ne(␪/n)i
is an nth root of rei␪, n a natural number:
7-7 De Moivre’s Theorem
563
[r1/ne(␪/n)i]n ⫽ (r1/n)nen(␪/n)i
⫽ re␪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
3
Solution
Finding All Sixth Roots of a Complex Number
Find six distinct sixth roots of ⫺1 ⫹ i兹3, and plot them in a complex plane.
First write ⫺1 ⫹ i兹3 in polar form:
⫺1 ⫹ i兹3 ⫽ 2e120°i
Using the nth-root theorem, all six roots are given by
21/6e(120°/6⫹k 360°/6)i ⫽ 21/6e(20°⫹k60°)i
k ⫽ 0, 1, 2, 3, 4, 5
Thus,
w1 ⫽ 21/6e(20°⫹0ⴢ60°)i ⫽ 21/6e20°i
FIGURE 1
w2 ⫽ 21/6e(20°⫹1ⴢ60°)i ⫽ 21/6e80°i
w3 ⫽ 21/6e(20°⫹2ⴢ60°)i ⫽ 21/6e140°i
w4 ⫽ 21/6e(20°⫹3ⴢ60°)i ⫽ 21/6e200°i
w5 ⫽ 21/6e(20°⫹4ⴢ60°)i ⫽ 21/6e260°i
w6 ⫽ 21/6e(20°⫹5ⴢ60°)i ⫽ 21/6e320°i
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 ⫹ i. Leave the answers in polar form and plot
them in a complex plane.
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7 ADDITIONAL TOPICS IN TRIGONOMETRY
EXAMPLE
4
Solution
Solving a Cubic Equation
Solve x3 ⫹ 1 ⫽ 0. Write final answers in rectangular form, and plot them in
a complex plane.
x3 ⫹ 1 ⫽ 0
x3 ⫽ ⫺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
Using the nth-root theorem, all three cube roots of ⫺1 are given by
11/3e(180°/3⫹k360°/3)i ⫽ 1e(60°⫹k120°)i
k ⫽ 0, 1, 2
Thus,
FIGURE 2
w1 ⫽ 1e60°i ⫽ cos 60° ⫹ i sin 60° ⫽
1
兹3
⫹i
2
2
w2 ⫽ le180°i ⫽ cos 180° ⫹ i sin 180° ⫽ ⫺1
w3 ⫽ le300°i ⫽ cos 300° ⫹ i sin 300° ⫽
1
兹3
⫺i
2
2
[Note: This problem can also be solved using factoring and the quadratic formula—try it.]
The three roots are graphed in Figure 2.
MATCHED PROBLEM
4
Solve x3 ⫺ 1 ⫽ 0. Write final answers in rectangular form, and plot them in a
complex plane.
Answers to Matched Problems
1. 32e300°i ⫽ 16 ⫺ i16兹3
2. 16e(⫺240°)i ⫽ ⫺8 ⫹ i8兹3
3. w1 ⫽ 21/10e9°i, w2 ⫽ 21/10e81°i, w3 ⫽ 21/10e153°i, w4 ⫽ 21/10e225°i, w5 ⫽ 21/10e297°i
1
兹3 1
兹3
4. 1, ⫺ ⫹ i
,⫺ ⫺i
2
2
2
2
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26. (A) Show that ⫺2 is a root of x3 ⫹ 8 ⫽ 0. How many
other roots does the equation have?
EXERCISE 7-7
(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.
A
In Problems 1–6, use De Moivre’s theorem to evaluate each.
Leave answers in polar form.
1. (2e30°i)3
2. (5e15°i)3
3. (兹2e10°i)6
4. (兹2e15°i)8
5. (1 ⫹ i兹3)3
6. (兹3 ⫹ i)8
(C) Verify that each complex number found in part B is a
root of x3 ⫹ 8 ⫽ 0.
B
In Problems 7–12, find the value of each expression and write
the final answer in exact rectangular form. (Verify the results
in Problems 7–12 by evaluating each directly on a calculator.)
7. (⫺兹3 ⫺ i)4
10. (⫺兹3 ⫹ i)5
9. (1 ⫺ i)8
8. (⫺1 ⫹ i)4
11.
冢
1 兹3
⫺ ⫹
i
2
2
冣
3
12.
冢
1 兹3
⫺ ⫺
i
2
2
冣
3
For n and z as indicated in Problems 13–18, find all nth roots
of z. Leave answers in polar form.
13. z ⫽ 8e30°i, n ⫽ 3
15. z ⫽ 81e
,n⫽4
60°i
17. z ⫽ 1 ⫺ i, n ⫽ 5
14. z ⫽ 8e45°i, n ⫽ 3
16. z ⫽ 16e
,n⫽4
90°i
18. z ⫽ ⫺1 ⫹ i, n ⫽ 3
In Problems 27–30, solve each equation for all roots. Write
final answers in polar and exact rectangular form
27. x3 ⫹ 64 ⫽ 0
28. x3 ⫺ 64 ⫽ 0
29. x3 ⫺ 27 ⫽ 0
30. x3 ⫹ 27 ⫽ 0
For n and z as indicated in Problems 19–24, find all nth roots
of z. Write answers in polar form and plot in a complex plane.
C
19. z ⫽ 8, n ⫽ 3
20. z ⫽ 1, n ⫽ 4
31. Show that
21. z ⫽ ⫺16, n ⫽ 4
22. z ⫽ ⫺8, n ⫽ 3
23. z ⫽ i, n ⫽ 6
24. z ⫽ ⫺i, n ⫽ 5
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 兹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.
[r1/ne(␪/n⫹k360°/n)i]n ⫽ rei␪
for any natural number n and any integer k.
32. Show that
r1/ne(␪/n⫹k360°/n)i
is the same number for k ⫽ 0 and k ⫽ n.
In Problems 33–36, write answers in polar form.
33. Find all complex zeros for P(x) ⫽ x5 ⫺ 32.
34. Find all complex zeros for P(x) ⫽ x6 ⫹ 1.
35. Solve x5 ⫹ 1 ⫽ 0 in the set of complex numbers.
36. Solve x3 ⫺ i ⫽ 0 in the set of complex numbers.
In Problems 37 and 38, write answers using exact rectangular
forms.
37. Write P(x) ⫽ x6 ⫹ 64 as a product of linear factors.
38. Write P(x) ⫽ x6 ⫺ 1 as a product of linear factors.