Cube Roots of Unity
The root of unity is a number which is complex in nature and gives 1 if raised to the power of a positive integer n. These roots are used in different branches and topics of maths like number theory. It is also called as de Moivre system. Here we will discuss the cube roots of unity in detail.
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.
How to Find Cube Root of Unity Values (Derivation)?
Suppose the cube root of 1 is “a”,
i .e ^{3}√1 = a.
According to the general cube roots definition, a^{3 } = 1 or a^{3} – 1 = 0
(a^{3 } – b^{3}) = (a – b) ( a^{2} + ab + b^{2})
Now, (a^{3 } – 1^{3}) = 0
or (a – 1)( a^{2} + a + 1) = 0
Therefore, a = 1
or
( a^{2} + a + 1) = 0
By using quadratic roots formula for the above equation, we get;
a = [(-1) ± √(1^{2}-4.1.1)]/2
= [-1 ± √-3]/2
= -1/2 ± i√(3)/2
Therefore, the three cube roots of unity are:
1, -1/2+i√(3)/2, -1/2 – i√(3)/2
Properties of Cube roots of unity
1) One imaginary cube roots of unity is the square of the other.
\(\left ( \frac{-1+\sqrt{3i}}{2} \right )^{2}\) = ¼ [(-1)^{2} – 2 × 1 × √3 i + ( √3 i)^{2}] = ¼(1 – 2√ 3i – 3) = (-1-√ 3 i) /2And \(\left ( \frac{-1-\sqrt{3i}}{2} \right )^{2}\) = ¼ [(-1)^{2} + 2 × 1 × √3 i + ( √3 i)^{2}] = ¼(1 + 2√ 3i – 3) = (-1+ √ 3 i) /2
2) If two imaginary cube roots are multiplied then the product we get is equal to 1.
Let’s take ω = (-1 + √3 i ) /2
ω^{2} = (-1 – √3 i ) /2
Now we will get the product of two imaginary cube roots as ω x ω^{2} = [(-1 + √3 i ) / 2]x [(-1 – √3 i ) /2]
Or ω^{3} = ¼[(-1)^{2} – (√3 i)^{2}]
= ¼[( 1 – 3i^{2}) = ¼ x 4 = 1.
3) As there are three cube roots of unity, their sum is zero, let’s see how
= 1 + [(-1 + √3 i ) /2] + [(-1 – √3 i ) /2]
Or
1 + ω + ω^{2 } = 1 – ½ + (√3 i ) /2 – ½ – (√3 i ) /2
= 0
Video Lesson
Cube roots of infinity
Cube Roots of unity Solved Example
Now you know that finding the cube root of unity is different from how to find the cube root of any number. Let’s practice it below.
Question: Let’s factorize the following: a^{2} + ab + b^{2}
Solution: a^{2} – (-1)ab + 1.b^{2}
Since, 1 + ω + ω^{2 }= 0 or ω + ω^{2 }= -1 and ω^{3 }= 1. Therefore,
a^{2} – (ω + ω^{2})ab + ω^{3} . b^{2 }
a^{2} – abω -ab ω^{2} + ω^{3} . b^{2}
a (a – bω) – bω^{2} (a – bω) = (a – bω^{2})(a – bω)
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Learn More | |
Algebraic Operations On Complex Numbers | Cube Roots of Numbers |
Nature of Roots of Quadratic Equation | Roots of Polynomial |
Frequently Asked Questions
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 is the Sum of Cube Root of Unity?
According to the properties of the cube root of 1, the sum of its root is zero. So, 1 + ω + ω^{2}=0.
Is Cube Root of Unity Collinear?
As 1 + ω + ω^{2}=0, it can be said that the cube root of unity is collinear.
What are the Values of Cube Roots of Unity?
The cube root of unity values are 1, −½ + i √(3/ 2), and −½ – i √(3/ 2)