Cube Roots of Unity and Their Properties
The cube roots of unity are 1, w and w2 where w = cos(2pi/3) + i sin(2pi/3), satisfying 1 + w + w2 = 0 and w3 = 1.
What is Cube Roots of Unity and Their Properties?
The cube roots of unity are 1, w and w2 where w = cos(2pi/3) + i sin(2pi/3), satisfying 1 + w + w2 = 0 and w3 = 1.
Key formula / rule: w3 = 1 and w^(3k+1) = w for every integer k.
Key points
- Derive the three cube roots of unity from z3 = 1.
- Apply the identity 1 + w + w2 = 0 to simplify symmetric expressions.
Common exam trap
Assuming w is real because it satisfies a real cubic.
Definitions
- Term
Cube Roots of Unity and Their Properties
- Meaning
The cube roots of unity are 1, w and w2 where w = cos(2pi/3) + i sin(2pi/3), satisfying 1 + w + w2 = 0 and w3 = 1.
- Term
Cube Roots of Unity and Their Properties — explanation
- Meaning
Because the three roots sum to zero, long symmetric expressions collapse in one step. Recognising a hidden w in an expression is often the whole question in an examination.
Learning objectives
Derive the three cube roots of unity from z3 = 1.
Apply the identity 1 + w + w2 = 0 to simplify symmetric expressions.
Formulae
- Key point
w3 = 1 and w^(3k+1) = w for every integer k.
- Key point
1 + w + w2 = 0 is the identity that kills symmetric sums.
- Key point
a3 + b3 + c3 - 3abc factorises using w and w2.
Prerequisites
MAT-U1-CQ-T1-S2-C1
Common mistakes
Assuming w is real because it satisfies a real cubic.
Forgetting that w2 is the conjugate of w when simplifying.
Keywords
Cube
Roots
Unity
Their
Practice preview
If x = a + b, y = aw + bw^2, z = aw^2 + bw, where w is a complex cube root of unity, then x^3 + y^3 + z^3 - 3xyz is equal to:…
hard
If w is a complex cube root of unity, then the value of the determinant | 1 w w^2 | | w w^2 1 | | w^2 1 w | is:…
hard
The value of (1 - w + w^2)(1 + w - w^2) is:…
hard