What is complex cube root of unity - Maths - Determinants

Question

What is complex cube root of unity?

Anuradha Sharma · Accepted Answer

A cube root of unity, occasionally called a De Moivre number, is any complex number that gives 1 when raised to some integer power n.

Let the cube root of 1 be ω i.e., ³√1 = ω.

⇒ ω³ = 1

⇒ ω³ – 1 = 0 or (ω – 1) (ω² + ω + 1) = 0

So, either ω – 1 = 0 or (ω² + ω + 1) = 0

⇒ ω = 1 and (ω² + ω + 1) = 0

Now, the roots of (ω² + ω + 1) = 0 can be find out as follows:

ω = Cube Root

So, we have ω = 1 and Cube Root .

Hence, out of three cube roots of unity 1 is real number whereas other roots i.e.,

are conjugate complex numbers which are also known as imaginary cube roots of unity.

So, we have ω = 1 and

Hence, out of three cube roots of unity 1 is real number whereas other roots i e ,

are conjugate complex numbers which are also known as imaginary cube roots of unity