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34. Find three roots of the equation x3- 1 = 0 and represent the roots as points on the plane.

35. Find four roots of the equation x4 − 1 = 0 and represent the roots as points on the plane.

36. Find six roots of 26 10 and represent the roots as points on the plane. Show graphically that the sum of the six roots is zero.

37. Find three roots of x3 8 = O and represent the roots as points on the plane. Show graphically that the sum of the three roots is zero.

154. Polar representation. The graphical representation of complex numbers given in § 147 gives a simple graphical interpretation of the operations of addition and subtraction, but the graphical meaning of the operations of multiplication and division may be given more clearly in another manner. We have seen that we may represent x + iy by the point P (x, y) on the plane. Represent the angle between OP and the X axis by 0. This angle is called the argument of the complex number x + iy. Represent the line OP by p. This is called the modulus of x + iy. Then from the figure

YA

P

x+iy

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Hence the complex number x + iy may be written in the form

x + iy = p(cos + i sin 0),

(4)

when the relations between x, y and P, 0 are given by (1), (2), and (3). A number expressed in this way is in polar form, and may be designated by (p, 0). We observe that a complex number lies on a circle whose center is the origin and whose radius is the modulus of the number. The argument is the angle between the axis of real numbers and the line representing the modulus.

155. Multiplication in polar form. If we have two numbers p (cos + i sin ) and p'(cos ' + i sin e'), we may multiply them and obtain

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In this product pp' is the new modulus and +0' the new argument. We may now make the following statement: The product of the two numbers p(cos + i sin 0) and p'(cos 0' + i sin 0') has as its modulus pp' and as its argument 0 + 0'. Thus the product of two numbers is represented on a circle whose radius is the product of the radii of the circles on which the factors are represented. The argument of the product is the sum of the arguments of the factors.

156. Powers of numbers in polar form. When the two factors of the preceding section (p, 0) and (p', 0') are equal, that is, when p = p' and ', the expression (1) assumes the form [p (cos + i sin 0)]2 = p2 (cos 2 0 + i sin 2 0).

0

=

(1) This suggests as a form for the nth power of a complex number [p (cos + i sin 0)]" = p” (cos n0 + i sin n0).

(2)

The student should establish this expression by the method of complete induction. The theorem expressed by (2) is known as DeMoivre's theorem. Stated verbally it is as follows: The modulus of the nth power of a number is the nth power of its modulus. The argument of the nth power of a number is n times its argument.

EXERCISES

Plot, find the arguments and moduli of the following numbers and of their products.

1. 1 + i √3, √3 + i.

Solution:

YA

B

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Thus if the product has the form R(cos + i sin ), we have by § 155, R = pp' = 4, ℗ = 0 + 0' :

2. 1+ i, 2+ i.

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90°.

10. [2 (cos 15° + i sin 15°)]3. 12. [(cos 120° + i sin 120°)]2. 14. [(cos 180° + i sin 180°)]3.

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157. Division in polar form. If we have, as before, two complex numbers in polar form (p, 0) and (p', 0'), we may obtain their quotient as follows.

Rationalizing,

§ 152 and § 153,

Since sin2 + cos2 0 = 1,

p(cos + i sin ◊) p' (cos '+isin ')

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We may now make the following statement: The quotient of two complex numbers has as its modulus the quotient of the moduli of the factors, and as its argument the difference of the arguments of the factors.

158. Roots of complex numbers. We have seen that the square of a number has as its modulus the square of the original modulus, while the argument is twice the original argument.

This would suggest that the square root of any number, as (p, 0),

0

would have √p as its modulus and as its argument. Since every real number has two square roots, we should expect the same fact to hold here. Consider the two numbers

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√P (cos + ising) and VP cos (+180°) + i sin (+180°)],

2

2

where VP is the principal square root of p (§ 72). The square of the first is (p, 0), by § 155. That the square of the second is the same is evident if we keep in mind the fact that

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The graphs of these two numbers are situated at points symmetrical to each other with respect to the origin.

We may obtain as the corresponding expression for the higher roots of complex numbers the following:

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where for a given value of n, k takes on the values 0, 1, and where indicates the real positive nth root of p.

n

. . ., n − 1,

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