Complex Root Calculator

This complex root calculator helps you tackle the task of finding the roots of complex numbers to any degree, in particular complex square roots and complex cube roots.

If you are not quite sure what a complex root is or how to find complex roots, keep reading! Besides providing basic definitions, we will also teach you how to find the roots of complex numbers by hand and explain what the roots of unity are.

If, first, you need a refresher on complex numbers, try our complex number calculator.

Let n be a fixed positive integer. We say that a complex number w is an n-th root of another complex number z if:

wⁿ = z

Remember that every complex number has exactly n distinct n-th complex roots. So, every complex number has:

• Two complex square roots;
• Three complex cube roots;
• Four complex fourth roots;
• Ten complex tenth roots, and so on.

Geometrically, the n-th roots of a complex number z constitute the vertices of a regular n-gon (regular polygon with n sides). These points lie on a circle, whose radius is given by the (this time real) n-th root of the magnitude of z. The points are equally spaced at every 2π/n radians:

• The complex square roots form a line;
• Cube roots of a complex number form an equilateral triangle;
• 4-th roots form a square;
• 5-th roots form a pentagon; and
• 10-th roots form a decagon, and so on.

Now that we know what a complex root is, let's discuss how to find them.

To find the n-th roots of a complex number z (let's denote those roots by w₁ ... w_n), use the following formula:

ⁿ√r × exp(i × (θ + 2kπ) / n)

or, equivalently,

ⁿ√r × (cos(θ/n + 2kπ/n) + i × sin(θ/n + 2kπ/n))

where k = 0, ..., n - 1.

Note that these roots are given in their trigonometric form — we use the sine and cosine functions. You can compute them with the help of our sin calculator and cosine calculator.

To use the complex root calculator:

1. Enter the form of the complex number of which you want to determine the complex roots. You can choose between inputting it in its Cartesian form or its polar form.

2. Tell us the degree n of the root you are interested in.

3. Our complex root calculator returns all n-th roots of the number you entered.

4. You may choose the form in which the calculator displays the results (the Cartesian form or the polar form).

5. If you need it, you can increase the precision (number of decimal places) with which the complex root calculator displays its results. Simply click on the "I want to set output precision." checkbox. By default, we use 4 decimal places.

As for finding roots of complex numbers by hand, you can choose either an algebraic or geometric way. You can find both approaches below.

Algebraic approach to finding roots of complex numbers

To algebraically find the n-th complex roots of a complex number z, follow these steps:

1. If your number z is given as its Cartesian coordinates, a + bi, convert it to the polar form. In other words, find its magnitude r and argument φ.

2. Compute the n-th root of r.

3. Compute φ/n and its multiplicities: 2 × φ/n, 3 × φ/n, up to (n-1) × φ/n.

4. You can find the roots you're looking for using the following formula

   ⁿ√r × exp(i × (θ + 2kπ) / n)

   or, equivalently,

   ⁿ√r × (cos(θ/n + 2kπ/n) + i × sin(θ/n + 2kπ/n))

   where k = 0, ..., n − 1.

Geometric approach to finding roots of complex numbers

To geometrically find all n-th roots of z, follow these steps:

1. The first step is the same as in the algebraic approach: if your number is in its Cartesian form, compute its magnitude r and argument φ.

2. Plot a circle with a radius equal to the n-th root of r. All the roots we're searching for lie in this circle.

3. On this circle, mark a point with argument φ/n. This number is one of the roots and the starting point to find all the remaining ones!

4. Starting from this first root, make marks at every 2π/n radians until you make a full circle and come back to the starting point. These points are all the other complex roots you're looking for.

The roots of unity are the complex roots of the number 1. All that we have said as far applies to them.

So, roots of unity lie on the unit circle, and they are equally spaced at every 2π / n radians. The formula for n-th roots of unity reads:

exp(2kπi / n),

where k = 0, ..., n − 1. Equivalently:

cos(2kπ/n) + i × sin(2kπ/n)

In particular:

• The square roots of unity are 1 and -1;

• The cubic (third) roots of unity are:

  1;

  exp(2π/3) = -1/2 + i√3/2;

  exp(-2π/3) = -1/2 − i√3/2; and

• The fourth roots of unity are 1, i, -1, -i.

Obviously, 1 is always one of its own n-th roots for any n! This is because 1 raised to any power n is 1.