In this explainer, we will learn how to identify the cubic roots of unity using de Moivre’s theorem.

A cube (or cubic) root of unity is a complex-valued solution to the equation =1. If we only consider real-valued solutions to this equation, we can apply the cube root to both sides of the equation to obtain =√1=1, which means that there is only one real-valued solution. However, there are more solutions to this equation that are not real numbers. To solve this equation algebraically, we first rearrange this equation as −1=0. Recalling the difference of cubes formula − =( − ) + + , we can factor −1 to write ( −1) + +1=0.

The first factor −1 leads to the real root =1, while the quadratic factor + +1 will lead to complex-valued roots when we apply the quadratic roots formula. Applying the quadratic roots formula with =1, =1, and =1, we have =− ±√ −4 2 =−1±√1−42=−12±√32 .

This gives us two other complex-valued solutions to the equation =1. Hence, we have obtained three cube roots of unity.

What is the Definition of Cube Root of Unity?

The cube roots of unity can be defined as the numbers which when raised to the power of 3 gives the result as 1. In simple words, the cube root of unity is the cube root of 1 i.e.3√1.

What are Cube Roots of Unity?

There are a total of three cube roots of unity which are as follows:

Cube Root of Unity Value Nature of Cube Root
1 Real
−½ + i √(3/ 2) Complex
−½ – i √(3/ 2) Complex

Here, a = 1 is the real cube root of unity while a = – ½ + i √(3/ 2) and a = – ½ – i √(3/ 2) are the imaginary or complex cube roots of unity.

Property: Square of Complex Cubic Roots of Unity

Let be a complex cubic root of unity. Then, = .

Let us consider an example where we can apply this property to simplify an expression involving .


How do you find the roots of unity?

Sum of cube of unity

It is defined as the number that can be raised to the power of 3 and result is 1. The sum of the three cube roots of unity is zero i.e., 1++2=0.

Who discovered cube root of unity?

The Greek mathematician Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. His formula is again mentioned by Eutokios in a commentary on Archimedes.

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