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.
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
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)
The root corresponding to k = 1 is
The n n-th roots of 1 are given by
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.