Problem 27 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(n\) is a positive integer not a multiple of 3 , then \(1+\omega^{n}+\omega^{2 n}=\) (a) 3 (b) 1 (c) 0 (d) none of these

Step-by-Step Solution

Answer

The expression evaluates to \(0\) when \(n\) is not a multiple of 3.

Step 1: Understanding the Roots of Unity

The cubic roots of unity are 1, \( \omega \), and \( \omega^2 \), where \( \omega = e^{\frac{2\pi i}{3}} \) and \( \omega^2 = e^{\frac{4\pi i}{3}} \). These satisfy the equation \( \omega^3 = 1 \), and \( 1 + \omega + \omega^2 = 0 \). Using these properties, we'll find an expression for \( 1 + \omega^n + \omega^{2n} \).

Step 2: Replace and Simplify Using Roots of Unity

Since \( n \) is not a multiple of 3, consider modulo operations on \( n \):- If \( n \equiv 1 \pmod{3} \), then \( \omega^n = \omega \) and \( \omega^{2n} = \omega^2 \).- If \( n \equiv 2 \pmod{3} \), then \( \omega^n = \omega^2 \) and \( \omega^{2n} = \omega \).In both cases,\[1 + \omega^n + \omega^{2n} = 1 + \omega + \omega^2 = 0.\]

Step 3: Verify With Known Identity

We know from the identity of the cube roots of unity that:\[1 + \omega + \omega^2 = 0.\]Given that for \(n\) not a multiple of 3, the exponents \(n\) and \(2n\) cycle through \(\omega\), \(\omega^2\), and 1, ensuring the equation always sums to zero.

Key Concepts

Roots of Unity

Roots of unity are special solutions to the equation \(x^n = 1\), where \(n\) is a positive integer. They are complex numbers that, when raised to the \(n\)-th power, result in one. For cubic roots of unity, \(n = 3\), so we have the roots: 1, \(\omega = e^{\frac{2\pi i}{3}}\), and \(\omega^2 = e^{\frac{4\pi i}{3}}\). These roots lie on the complex plane, forming the vertices of an equilateral triangle with its centroid at the origin.

• \(\omega\) is a primitive root, meaning it cycles through the other roots with successive powers.
• The main property of cubic roots is \(\omega^3 = 1\) and the sum of the roots, \(1 + \omega + \omega^2 = 0\).

This sum of zero is used to evaluate expressions like \(1 + \omega^n + \omega^{2n}\) under different conditions. Understanding these properties helps solve equations involving roots of unity.

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to the complex plane using a two-dimensional layout. A complex number has both a real part and an imaginary part, represented as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property \(i^2 = -1\).Complex numbers are essential when dealing with roots of unity. Key characteristics of complex numbers include:

• They can be represented visually on the complex plane, aiding in intuitive understanding of operations like addition and multiplication.
• The modulus (or magnitude) of a complex number \(z = a + bi\) is \(|z| = \sqrt{a^2 + b^2}\), representing the distance from the origin in the complex plane.
• The argument of a complex number is the angle it makes with the positive x-axis, which is especially useful for polar representation.

Roots of unity demonstrate how complex numbers elegantly solve polynomial equations that lack real solutions, shedding light on their significance in mathematical analysis.

Modular Arithmetic

Modular arithmetic focuses on the properties and relationships of numbers within a specific modulus. It involves finding the remainder of division, often expressed as \(a \equiv b \pmod{n}\).Within our problem, modular arithmetic is crucial for simplifying exponents in expressions involving roots of unity. Here's why it matters:

• When \(n\) is not a multiple of 3, it can be \(n \equiv 1 \pmod{3}\) or \(n \equiv 2 \pmod{3}\). This simplifies \(\omega^n\) and \(\omega^{2n}\).
• Modulo 3 ensures that the exponents of \(\omega\) cycle predictably, which helps verify identities like \(1 + \omega^n + \omega^{2n} = 0\).
• It simplifies computation by reducing potentially large numbers into smaller, more manageable expressions.

Engaging with modular arithmetic offers a powerful tool for tackling cyclic or periodic phenomena in number theory and beyond.