The Cube Roots of Unity

We will discuss here about the cube roots of unity and their properties.

Suppose let us assume that the cube root of 1 is z i.e., 1 = z.

Then, cubing both sides we get, z\(^{3}\) = 1

or, z\(^{3}\) - 1 = 0

or, (z - 1)(z\(^{2}\) + z + 1) = 0

Therefore, either z - 1 = 0 i.e., z = 1 or, z\(^{2}\) + z + 1 = 0

Therefore, z = \(\frac{-1\pm \sqrt{1^{2} - 4\cdot 1\cdot 1}}{2\cdot 1}\) = \(\frac{-1\pm \sqrt{- 3}}{2}\) = -\(\frac{1}{2}\) ± i\(\frac{√3}{2}\)

Therefore, the three cube roots of unity are

1, -\(\frac{1}{2}\) + i\(\frac{√3}{2}\) and -\(\frac{1}{2}\) - i\(\frac{√3}{2}\)

among them 1 is real number and the other two are conjugate complex numbers and they are also known as imaginary cube roots of unity.

Properties of the cube roots of unity:

Property I: Among the three cube roots of unity one of the cube roots is real and the other two are conjugate complex numbers.

The three cube roots of unity are 1, -\(\frac{1}{2}\) + i\(\frac{√3}{2}\) and -\(\frac{1}{2}\) - i\(\frac{√3}{2}\).

Hence, we conclude that from the cube roots of unity we get 1 is real and the other two i.e., \(\frac{1}{2}\) + i\(\frac{√3}{2}\) and -\(\frac{1}{2}\) - i\(\frac{√3}{2}\) are conjugate complex numbers.

 

Property II: Square of any one imaginary cube root of unity is equal to the other imaginary cube root of unity.

\((\frac{-1 + \sqrt{3}i}{2})^{2}\) = \(\frac{1}{4}\)[(- 1)^2 - 2 1 √3i + (√3i)\(^{2}\)]

               = \(\frac{1}{4}\)[1 - 2√3i - 3]

               = \(\frac{-1 - \sqrt{3}i}{2}\),

And \((\frac{-1 - \sqrt{3}i}{2})^{2}\) = \(\frac{1}{4}\)[(1^2 + 2 1 √3i + (√3i)\(^{2}\)]

                    = \(\frac{1}{4}\)[1 + 2√3 i - 3]

                    = \(\frac{-1 + \sqrt{3}i}{2}\),

Hence, we conclude that square of any cube root of unity is equal to the other.

Therefore, suppose ω\(^{2}\) is one imaginary cube root of unity then the other would be ω.

 

Property III: The product of the two imaginary cube roots is 1 or, the product of three cube roots of unity is 1.

Let us assume that, ω = \(\frac{-1 - \sqrt{3}i}{2}\); then, ω\(^{2}\) = \(\frac{-1 + \sqrt{3}i}{2}\)

Therefore, the product of the two imaginary or complex cube roots = ω ω\(^{2}\) = \(\frac{-1 - \sqrt{3}i}{2}\) × \(\frac{-1 + \sqrt{3}i}{2}\)

Or, ω\(^{3}\) = \(\frac{1}{4}\)[(-1)\(^{2}\) - (√3i)\(^{2}\)] = \(\frac{1}{4}\)[1 - 3i\(^{2}\)] = \(\frac{1}{4}\)[1 + 3] = \(\frac{1}{4}\) × 4 = 1.

Again, the cube roots of unity are 1, ω, ω\(^{2}\). So, product of cube roots of unity = 1 ω ω\(^{2}\) = ω\(^{3}\) = 1.

Therefore, product of the three cube roots of unity is 1.

 

Property IV: ω\(^{3}\) = 1

We know that ω is a root of the equation z\(^{3}\) - 1 = 0. Therefore, ω satisfies the equation z\(^{3}\) - 1 = 0. 

Consequently, ω\(^{3}\) - 1 = 0

or, ω = 1.

Note: Since ω\(^{3}\) = 1, hence, ω\(^{n}\) = ω\(^{m}\), where m is the least non-negative remainder obtained by dividing n by 3.


Property V: The sum of the three cube roots of unity is zero i.e., 1 + ω + ω\(^{2}\) = 0.

We know that, the sum of the three cube roots of unity = 1 + \(\frac{-1 - \sqrt{3}i}{2}\) + \(\frac{-1 + \sqrt{3}i}{2}\)

Or, 1 + ω + ω\(^{2}\) = 1 - \(\frac{1}{2}\) + \(\frac{√3}{2}\)i - \(\frac{1}{2}\) - \(\frac{√3}{2}\)i = 0.

Notes:

(i) The cube roots of 1 are 1, ω, ω\(^{2}\) where, ω = \(\frac{-1 - \sqrt{3}i}{2}\) or, \(\frac{-1 + \sqrt{3}i}{2}\)

(ii) 1 + ω + ω\(^{2}\) = 0 ⇒ 1 + ω = - ω\(^{2}\), 1 + ω\(^{2}\) = - ω and ω + ω\(^{2}\) = -1

(iii) ω\(^{4}\) = ω\(^{3}\) ω = 1 ω = ω;

ω\(^{5}\) = ω\(^{3}\) ω\(^{2}\) = 1 ω\(^{2}\) = ω\(^{2}\);

ω\(^{6}\) = (ω\(^{3}\))\(^{2}\) = (1)\(^{2}\) = 1.

In general, if n be a positive integer then,

ω\(^{3n}\) = (ω\(^{3}\))\(^{n}\) = 1\(^{n}\) = 1;

ω\(^{3n + 1}\) = ω\(^{3n}\) ω = 1 ω = ω;

ω\(^{3n + 2}\) = ω\(^{3n}\) ω\(^{2}\) = 1 ω\(^{2}\) = ω\(^{2}\).

 

Property VI: The reciprocal of each imaginary cube roots of unity is the other.

The imaginary cube roots of unity are ω and ω\(^{2}\), where ω = \(\frac{-1 + \sqrt{3}i}{2}\).

Therefore, ω ω\(^{2}\) = ω\(^{3}\) = 1

⇒ ω = \(\frac{1}{ω^{2}}\) and ω\(^{2}\) = \(\frac{1}{ω}\)

Hence, we conclude that the reciprocal of each imaginary cube roots of unity is the other.

 

Property VII: If ω and ω\(^{2}\) are the roots of the equation z\(^{2}\) + z + 1 = 0 then - ω and - ω\(^{2}\) are the roots of the equation  z\(^{2}\) - z + 1 = 0.

Property VIII: Cube roots of -1 are -1, - ω and - ω\(^{2}\).






11 and 12 Grade Math 

From The Cube Roots of Unity to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

    Mar 02, 24 05:31 PM

    Fractions
    The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerato…

    Read More

  2. Subtraction of Fractions having the Same Denominator | Like Fractions

    Mar 02, 24 04:36 PM

    Subtraction of Fractions having the Same Denominator
    To find the difference between like fractions we subtract the smaller numerator from the greater numerator. In subtraction of fractions having the same denominator, we just need to subtract the numera…

    Read More

  3. Addition of Like Fractions | Examples | Worksheet | Answer | Fractions

    Mar 02, 24 03:32 PM

    Adding Like Fractions
    To add two or more like fractions we simplify add their numerators. The denominator remains same. Thus, to add the fractions with the same denominator, we simply add their numerators and write the com…

    Read More

  4. Comparison of Unlike Fractions | Compare Unlike Fractions | Examples

    Mar 01, 24 01:42 PM

    Comparison of Unlike Fractions
    In comparison of unlike fractions, we change the unlike fractions to like fractions and then compare. To compare two fractions with different numerators and different denominators, we multiply by a nu…

    Read More

  5. Equivalent Fractions | Fractions |Reduced to the Lowest Term |Examples

    Feb 29, 24 05:12 PM

    Equivalent Fractions
    The fractions having the same value are called equivalent fractions. Their numerator and denominator can be different but, they represent the same part of a whole. We can see the shade portion with re…

    Read More