Study Tips: Finding the Roots of Unity

This tip on roots of an equation, and complex numbers was contributed by one of our top tutors, Sudheer Naidu.

    What you will learn from this tip:
  • An understanding of the relation between complex numbers and the roots of unity.
  • The trick to calculate the nth roots of unity.

This is a simple trick. All you need is a rough sheet of paper along with a very basic understanding of complex numbers and trigonometry.


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Start off by drawing a circle of unit radius on the complex plane. No need to be precise. Just a rough drawing is fine.

The total angle covered by this circle is 360o. The basic idea is that to find the nth roots of unity, the circle must be divided into n equal parts.
The angle subtended by each part is noted down and we find out the coordinates of the points subtending these angles on the unit circle. The coordinates, on the complex plane, give us the roots of unity. Here are some examples to help you understand better.

Square Roots of Unity

To find the square roots of unity, i.e, roots of the equation
x2 = 1
divide the circle into two equal parts, like so

The angles subtended by the two parts are 0o and 180o.

    Thus, the square roots of unity are :
  1. ei(0o) = cos(0o) + isin(0o) = 1
  2. ei(180o) = cos(180o) + isin(180o) = -1

Cube Roots of Unity

To find the cube roots of unity, i.e, roots of the equation
x3 = 1
divide the circle into three equal parts, like so

The angles subtended by the three parts are 0, 120o and 240o.

    Thus, the cube roots of unity are :
  1. ei(0o) = cos(0o) + isin(0o) = 1
  2. ei(120o) = cos(120o) + isin(120o) = -0.5 + ?3/2
  3. ei(240o) = cos(240o) + isin(240o) = -0.5 - ?3/2

Fourth Roots of Unity

To find the fourth roots of unity, i.e, roots of the equation
x4 = 1
divide the circle into four equal parts, like so

The angles subtended by the four parts are 0, 90o, 180o and 270o.

    Thus, the fourth roots of unity are :
  1. ei(0o) = cos(0o) + isin(0o) = 1
  2. ei(90o) = cos(90o) + isin(90o) = i
  3. ei(180o) = cos(180o) + isin(180o) = -1
  4. ei(270o) = cos(270o) + isin(270o) = -i

Similarly, the nth roots of unity can be calculated. Simple, isn't it?


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