Solving complex polynomial equations

The trigonometric form of a complex number

Any complex number (where and are real numbers) can be associated to the vector on the plane . This allows to think of complex numbers as vectors in .

Now consider the polar coordinates of this point in : they are the length and oriented angle (with the horizontal axis) defining any non-zero vector in .

In one can transition from usual coordinates to polar and back using the following relations (obtained using the definitions of and together with the Pythagorean theorem):

and, similarly,

The complex number can then be rewritten in trigonometric form:

The polar coordinates and corresponding to the complex number are called the absolute value and the argument of respectively.

Let us find some easy trigonometric forms (draw this on paper to see the corresponding transition!):

1. Indeed, as a vector on the plane is degrees counterclockwise from the axis, and of length , so and .

2. The absolute value is always a positive number (remember, we are working with non-zero numbers!), and as the vector on the plane is degrees from the origin (so in radians) and of length .

3. I will leave the explanation as an exercise: draw the corresponding vector and find its length and the angle with the horizontal axis.

To go from trigonometric form to the usual form you can just expand the brackets. (Do this with the examples above to make sure you get the same answers!)

Multiplication in trigonometric form

Given two complex numbers in the standard form and , multiplication seems to be a more sophisticated operation. Moreover, we do not seem a nice interpretation of this operation in terms of the geometric representation of complex numbers as vectors on the plane. Indeed,

However, this is where the trigonometric form comes into play: let and be the trigonometric forms of and . Then

In other words, when multiplying complex numbers, their absolute values (lengths of the vectors) get multiplied, whereas the arguments add up! This provides us with a geometric description of multiplication of complex numbers. For example, multiplying by will rotate any complex number by degrees counterclockwise.

This also provides us with an easier way to think about powers of complex numbers. If , then

Finding roots of complex numbers

In real numbers, when we introduce roots, they appear as the positive solution of the equation where is a real number. For even the equation has two solutions (a positive and a negative one), whereas for odd the equation has a single solution. In particular, if we are tasked to find all that satisfy (where ) we would write that . We do not have a nicer way to write, say, the number than tautologically by its very definition: in other words, we cannot simplify roots from real numbers any further if they can’t be extracted (like or ).

While we cannot simplify any real roots any further, the situation is different in complex numbers. Let’s take an example and solve the equation .

• Let’s start off by taking the right part in trigonometric form: (why?).
• Then we are searching for the two complex numbers that satisfy:

1. The square of their absolute value is .
2. Their argument doubled is .

• Then, automatically, the absolute value of has to be (as it is a positive real number). At the same time, the argument has two possibilities: and (the second possibility arises from the fact that any angle is defined up to adding ). For any () equation all the different roots will always have the same absolute value but different arguments differing by multiples of . Therefore, the two solutions will be and .

Similarly, the same procedure works for finding roots of higher degrees: let us consider the equation . This equation has a single real root, but three complex ones: let’s find them using trigonometric forms.

• Our equation is: .
• The absolute value of has to be as well as the only positive real number that cubed gives .
• The argument has three possible values: , and as those are the three angles (up to adding ) that, when multiplied by , give you an angle of the form (so, geometrically, fall in the position of the angle ). (This is true as they are of the form for an integer .)

I am deliberately not being too careful with the argument manipulations here for the sake of brevity, and suggest you try some examples or look for this further if this is unclear.