1.7 Roots Of Complex Numbers
A fundamental identity is the formula of De Moivre which follows.
Theorem 1.7.1 Let r > 0 be given. Then if n is a positive integer,
Proof: It is clear the formula holds if n = 1. Suppose it is true for n.
which by induction equals
by the formulas for the cosine and sine of the sum of two angles. ■
Corollary 1.7.2 Let z be a non zero complex number. Then there are always exactly k kth roots
of z in ℂ.
Proof: Let z = x + iy and let z =
be the polar form of the complex number. By De
Moivre’s theorem, a complex number
is a kth root of z if and only if
This requires rk =
and also both cos
This can only
for l an integer. Thus
and so the kth roots of z are of the form
Since the cosine and sine are periodic of period 2π, there are exactly k distinct numbers which result from
this formula. ■
Example 1.7.3 Find the three cube roots of i.
First note that i = 1
Using the formula in the proof of the above corollary, the
cube roots of i
where l = 0,1,2. Therefore, the roots are
Thus the cube roots of i are
The ability to find kth roots can also be used to factor some polynomials.
Example 1.7.4 Factor the polynomial x3 − 27.
First find the cube roots of 27. By the above procedure using De Moivre’s theorem, these cube roots are
+ 9 and so
where the quadratic polynomial x2 + 3x + 9 cannot be factored without using complex numbers.
Note that even though the polynomial x3 − 27 has all real coefficients, it has some complex zeros,
These zeros are complex conjugates of each other. It is always
this way. You should show this is the case. To see how to do this, see Problems 17
Another fact for your information is the fundamental theorem of algebra. This theorem says that any
polynomial of degree at least 1 having any complex coefficients always has a root in ℂ. This is sometimes
referred to by saying ℂ is algebraically complete. Gauss is usually credited with giving a proof of this
theorem in 1797 but many others worked on it and the first completely correct proof was due to Argand in
1806. For more on this theorem, you can google fundamental theorem of algebra and look at the
interesting Wikipedia article on it. Proofs of this theorem usually involve the use of techniques from
calculus even though it is really a result in algebra. A proof and plausibility explanation is given