```Website owner:  James Miller
```

[ Home ] [ Up ] [ Info ] [ Mail ]

Products, quotients and roots of complex numbers in polar form. De Moivre’s theorem. Roots of unity

Polar representation of a complex number. The polar representation of the complex number

z = x + iy

is

z = x + iy = r (cos θ + i sin θ)

where

x = r cos θ

y = r sin θ

θ = arctan (y/x)

See Figure 1. The radius vector r is called the modulus or absolute value of the complex number and the polar angle θ is called the amplitude or argument of the number. The argument θ of a complex number z is often denoted by arg z. The abbreviation cis θ is sometimes used for cos θ + i sin θ. In polar coordinates a point P is often specified by the number pair (r, θ).

Product of two complex numbers in polar form. The product z1z2 of the two complex numbers

z1 = r1 (cos θ1 + i sin θ1)

and

z2 = r2 (cos θ2 + i sin θ2)

is

1)                    z1z2 = r1r2[cos (θ1 + θ2) + i sin (θ1 + θ2)] ,

a result which can be easily obtained by utilizing the trigonometric identities

sin(A B) = sin A cos B cos A sin B

cos(A B) = cos A cos B sin A sin B .

Quotient of two complex numbers in polar form. The quotient z1/z2 of the two complex numbers

z1 = r1 (cos θ1 + i sin θ1)

and

z2 = r2 (cos θ2 + i sin θ2)

is

2)         z1/z2 = ( r1/r2) [cos (θ1 - θ2) + i sin (θ1 - θ2)]

De Moivre’s Theorem. For any complex number z = r( cos θ + i sin θ )

3)        zn = [ r(cos θ + i sin θ)]n = rn (cos nθ + i sin n)

This formula holds for every real value of the exponent n. For example, if the exponent is a fraction 1/n, we get

Rules of arguments. If

z1 = r1 (cos θ1 + i sin θ1)

z2 = r2 (cos θ2 + i sin θ2)

w = za

where a is a real number, then by 1), 2) and 3 ) above

1]        arg (z1z2) = arg z1 + arg z2

2]        arg (z1/z2) = arg z1 - arg z2

3]        arg w = a arg z

Def. N-th root of a number. Let n be a positive integer. If an = b, then a is said to be the n-th root of b.

Roots of complex numbers in polar form. The n distinct n-th roots of the complex number

z = r( cos θ + i sin θ)

can be found by substituting successively k = 0, 1, 2, ... , (n-1) in the formula

The n roots are equally spaced around the circumference of a circle in the complex plane. See Figure 2. If the complex number for which we are computing the n n-th roots is z = ρ( cos θ + i sin θ) the radius of the circle will be

and the first root w0 corresponding to k = 0 will be at an amplitude of α = θ/n. This root will be followed by the n-1 remaining roots at equal distances apart. The angular amplitude between each root is Δα = 360o/n.

Example. Suppose we wish to compute the ten 10-th roots of z = 8(cos 150o + i sin 150o) shown in Fig. 3. The ten roots w0, w1, .... , w9 would be spaced evenly around a circle as shown in Fig. 4. The first root, w0, would be at an amplitude of α = 150/10 = 15o. The rest of the roots would be spaced at Δα = 360/10 = 36o intervals.

Roots of unity. The n n-th roots of 1 are obtained from 5) above by letting r = 1 and θ = 0. They are

for k = 0, 1, 2, ... , (n-1)

Let us denote the root corresponding to k = 1 by w. This root w is then given by

The n n-th roots of 1 then correspond to powers of w:

w, w2, w3, ... ,wn

where wn = 1

The roots are equally spaced around the circumference of a unit circle in the complex plane. See Figure 5.

Primitive roots of unity. Of the n n-th roots of 1 some of the roots may be m-th roots of 1 where m is some integer less than n. For example, the 6 sixth roots of 1 are

r1 = w

r2 = w2

r3 = w3

r4 = w4

r5 = w5

r6 = w6 = 1

Of these, r3 = w3 and r6 = w6 are square roots of 1 and r2 = w2, r4 = w4 and r6 = w6 are cube roots of 1. The primitive roots of 1 are those roots which are not m-th roots of 1 for some 0 < m < n. Thus in the example just given the roots r1 = w and r5 = w5 are primitive roots of 1. In other words, of the n n-th roots of 1, a particular n-th root r is a primitive root if and only if rm 1 for any integer m less than n.

Theorem Let w, w2, w3, ... ,wn be the n n-th roots of 1. Let m be any integer 0 < m < n and let d be the greatest common divisor (m,n) of m and n. If d > 1 then , wm is an n/d-th root of 1.

Example. Let m = 3 and n = 6. Then d = (m,n) = (3,6) = 3 and n/d = 2. Thus w3 is a square root of 1.

Corollary. The primitive n-th roots of 1 are those and only those n-th roots w, w2, w3, ... ,wn of 1 whose exponents are relatively prime to n.

Roots of a complex number in terms of the roots of unity. Let

z = a + bi = r( cos θ + i sin θ )

and

Then the n n-th roots of z are

z0, wz0, w2z0, ... ,w k-1z0

where

References.

James & James. Mathematics Dictionary.

Brink. A First Year of College Mathematics.

Spiegel. College Algebra.

Hauser. Complex Variables with Physical Applications.