1st Year Math Chapter 1 Exercise 1.4 Notes: Cube and Fourth Roots of Unity (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 1.4 looks at a special and frequently tested case of complex roots: the solutions to xⁿ = 1, focusing on the cube roots (n = 3) and fourth roots (n = 4) of unity.
What Does This Exercise Cover?
This exercise finds all complex numbers that, when raised to the power 3 or 4, equal 1, and studies the useful algebraic properties these roots share.
Key Concepts
1. What Are Roots of Unity?
The nth roots of unity are all the complex numbers x that satisfy xⁿ = 1. Since this is a degree-n polynomial equation, it has exactly n roots by the Fundamental Theorem of Algebra.
2. Cube Roots of Unity
Solving x³ = 1 gives x³ − 1 = 0, which factors as (x − 1)(x² + x + 1) = 0. This gives one real root, x = 1, and two complex roots from x² + x + 1 = 0: x = (−1 + i√3)/2 and x = (−1 − i√3)/2. The complex roots are often denoted ω and ω².
- ω³ = 1 (since ω is a root of x³ = 1)
- 1 + ω + ω² = 0 (from the factor x² + x + 1 = 0)
- ω² is the complex conjugate of ω
3. Fourth Roots of Unity
Solving x⁴ = 1 gives x⁴ − 1 = 0, which factors as (x² − 1)(x² + 1) = 0, giving four roots: x = 1, x = −1, x = i, and x = −i.
Roots of Unity at a Glance
| Equation | Roots |
| x³ = 1 (Cube roots) | 1, ω = (−1+i√3)/2, ω² = (−1−i√3)/2 |
| x⁴ = 1 (Fourth roots) | 1, i, −1, −i |
Solved Examples
Example 1: Verify that 1 + ω + ω² = 0, where ω is a complex cube root of unity.
ω satisfies x² + x + 1 = 0, since it comes from factoring x³ − 1 = (x − 1)(x² + x + 1).
Substituting ω into this factor gives ω² + ω + 1 = 0 directly.
Answer: 1 + ω + ω² = 0
Example 2: Simplify ω¹⁵ + ω¹⁶ + ω¹⁷, where ω is a complex cube root of unity.
Since ω³ = 1, reduce each exponent modulo 3: 15 mod 3 = 0, 16 mod 3 = 1, 17 mod 3 = 2.
So ω¹⁵ = ω⁰ = 1, ω¹⁶ = ω¹ = ω, and ω¹⁷ = ω².
Adding these: 1 + ω + ω², which equals 0 from the identity above.
Answer: 0
Example 3: Find the fourth roots of unity and verify that their sum is zero.
Solving x⁴ = 1 gives the roots 1, i, −1, −i.
Adding them: 1 + i + (−1) + (−i) = (1 − 1) + (i − i) = 0.
Answer: Roots: 1, i, −1, −i; sum = 0
Sample MCQs
1. The complex cube roots of unity satisfy the equation:
a) x² + x + 1 = 0 b) x² − x + 1 = 0 c) x² + 1 = 0 d) x² − 1 = 0
Answer: a) x² + x + 1 = 0
2. If ω is a complex cube root of unity, ω³ equals:
a) 0 b) 1 c) ω d) −1
Answer: b) 1
3. The fourth roots of unity are:
a) 1, −1, i, −i b) 1, ω, ω² c) 1, 2, 3, 4 d) i, 2i, 3i, 4i
Answer: a) 1, −1, i, −i
4. The value of 1 + ω + ω² (ω a complex cube root of unity) is:
a) 1 b) −1 c) 0 d) 3
Answer: c) 0
5. ω¹⁰ (ω a complex cube root of unity) simplifies to:
a) 1 b) ω c) ω² d) 0
Answer: b) ω
Important Short Questions
- Define the nth roots of unity.
- List the cube roots of unity and state which ones are complex (non-real).
- State two key algebraic properties of the complex cube roots of unity.
- List all four fourth roots of unity.
- Simplify ω⁷, where ω is a complex cube root of unity.
Important Long Questions
- Solve x³ = 1 to find all cube roots of unity, and show that 1 + ω + ω² = 0.
- Solve x⁴ = 1 to find all fourth roots of unity, and verify that their sum equals zero.
- Simplify ω¹⁵ + ω¹⁶ + ω¹⁷, where ω is a complex cube root of unity, showing your method.
- Show that ω² is the complex conjugate of ω, using their standard forms (−1+i√3)/2 and (−1−i√3)/2.
How to Approach This Exercise Effectively
- Memorize the exact standard forms of ω and ω²: (−1 ± i√3)/2 — they appear repeatedly across this exercise.
- For any high power of ω, reduce the exponent modulo 3 first, since ω³ = 1.
- Keep the identity 1 + ω + ω² = 0 close at hand — it’s the fastest route to simplifying many expressions.
- For fourth roots, remember they simply cycle through 1, i, −1, −i, repeating every 4 powers, just like powers of i.
- Practice deriving the roots from x³ − 1 = 0 and x⁴ − 1 = 0 by factoring, rather than only memorizing the final answers.
FAQs
Q: What does ‘roots of unity’ mean?
A: Roots of unity are complex numbers that equal 1 when raised to a certain power n — the nth roots of unity are the n solutions of xⁿ = 1.
Q: Why is x = 1 not considered a ‘complex’ cube root of unity?
A: Because it is a real number; the term ‘complex cube roots of unity’ specifically refers to the two non-real roots, usually called ω and ω².
Q: What is the fastest way to simplify a high power of ω?
A: Divide the exponent by 3 and use only the remainder, since ω³ = 1 makes the powers of ω repeat every 3 steps.
Q: Are cube roots and fourth roots of unity related to each other?
A: They solve different equations (x³ = 1 versus x⁴ = 1) and have different values, though both sets of roots lie on the unit circle in the complex plane.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 1: Complex Numbers, Exercise 1.4.