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Chapter 2: Problem 27

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

Short Answer

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

Step by step solution

01

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 \( 1 + \omega^n + \omega^{2n} \).
02

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,\[1 + \omega^n + \omega^{2n} = 1 + \omega + \omega^2 = 0.\]
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

If \(z_{1}\) and \(z_{2}\) are any two complex numbers then \(\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}-z_{2}\right|^{2}\) is equal to (a) \(2\left|z_{1}\right|^{2}\left|z_{2}\right|^{2}\) (b) \(2\left|z_{1}\right|^{2}+2\left|z_{2}\right|^{2}\) (c) \(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\) (d) \(2\left|z_{1}\right|\left|z_{2}\right|\)

Let \(c\) be the set of complex numbers and \(z_{1}, z_{2}\) are in \(C .\) (i). Argument \(z_{1}=\) argument \(z_{2} \Rightarrow z_{1}=z_{2}\) (ii). \(\left|z_{1}\right|=\left|z_{2}\right| \Rightarrow z_{1}=z_{2}\) Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) both 1 and 2 (d) neither 1 nor 2

If \(|z-i|<|z+i|\), then (a) \(\operatorname{Re}(z)>0\) (b) \(\operatorname{Re}(z)<0\) (c) \(\operatorname{Im}(z)>0\) (d) \(\operatorname{Im}(z)<0\)

For any complex number \(z\), which of the following is not true? (a) \(\operatorname{Re}(z)=\frac{z+\bar{z}}{2}\) (b) \(\operatorname{Im}(z) \frac{\mathrm{z}-\bar{z}}{2 i}\) (c) \(|z|^{2}=z \bar{z}\) (d) \(-|\operatorname{Re}(z)| \leq|z| \leq|\operatorname{Re}(z)|\)

If \(z^{2}=-i\), then \(z=\) (a) \(\frac{1}{\sqrt{2}}(1+i)\) (b) \(\frac{1}{\sqrt{2}}(1+i)\) (c) \(\pm \frac{1}{\sqrt{2}}(1-i)\) (d) none of these
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