Problem 174
Question
If \(1, \omega, \omega^{2}, \ldots \omega^{n-1}\) are the \(n, n\)th roots of unity and \(z_{1}\) and \(z_{2}\) are any two complex numbers, then $$ \sum_{k=0}^{n-1}\left|z_{1}+\omega^{k} z_{2}\right|^{2}= $$ (A) \(n\left[\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right]\) (B) \((n-1)\left[\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right]\) (C) \((n+1)\left[\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right]\) (D) None of these
Step-by-Step Solution
Verified Answer
The answer is (A) \(n\left[|z_1|^2 + |z_2|^2\right]\).
1Step 1: Understanding Roots of Unity
The nth roots of unity are solutions to the equation \(z^n = 1\). These roots are \(1, \omega, \omega^2, \ldots, \omega^{n-1}\), where \(\omega = e^{2\pi i/n}\). They satisfy the property that \(\sum_{k=0}^{n-1} \omega^k = 0\).
2Step 2: Expanding the Expression
We need to evaluate \( \sum_{k=0}^{n-1} \left|z_1 + \omega^k z_2\right|^2 \). Start by expanding the modulus squared: \( \left|z_1 + \omega^k z_2\right|^2 = (z_1 + \omega^k z_2)(\overline{z_1 + \omega^k z_2}) = |z_1|^2 + z_1 \overline{\omega^k z_2} + \omega^k z_2 \overline{z_1} + |z_2|^2 \).
3Step 3: Simplifying the Summation
Examine the expression inside the sum: \( \sum_{k=0}^{n-1} |z_1|^2 + z_1 \overline{\omega^k z_2} + \omega^k z_2 \overline{z_1} + |z_2|^2 \). Separate terms: \( \sum_{k=0}^{n-1} |z_1|^2 + \sum_{k=0}^{n-1} |z_2|^2 + \sum_{k=0}^{n-1} z_1 \overline{\omega^k z_2} + \sum_{k=0}^{n-1} \omega^k z_2 \overline{z_1} \).
4Step 4: Applying Roots of Unity Properties
Use the property of roots of unity: \( \sum_{k=0}^{n-1} \omega^k = 0 \). Therefore, \( \sum_{k=0}^{n-1} z_1 \overline{\omega^k z_2} = z_1 \overline{z_2} \sum_{k=0}^{n-1} \overline{\omega^k} = 0 \) and \( \sum_{k=0}^{n-1} \omega^k z_2 \overline{z_1} = z_2 \overline{z_1} \sum_{k=0}^{n-1} \omega^k = 0 \).
5Step 5: Calculating the Final Sum
After applying properties, the remaining summation is \( \sum_{k=0}^{n-1} (|z_1|^2 + |z_2|^2) = n|z_1|^2 + n|z_2|^2 \). This simplifies to \( n(|z_1|^2 + |z_2|^2) \).
6Step 6: Matching with Options
Compare the calculated result with the given options. The solution matches option (A): \( n\left[|z_1|^2 + |z_2|^2\right] \).
Key Concepts
Complex NumbersModulus SquaredSummation PropertiesNth Roots of Unity
Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part. These numbers are expressed in the standard form as \( z = a + bi \) where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). This property is key to understanding how complex numbers work in different mathematical contexts.
When we combine different complex numbers, we often add, subtract, and multiply them like real numbers. However, when dividing, we often multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part. The conjugate of a complex number \(z = a + bi\) is \(\overline{z} = a - bi\).
Understanding these basic properties is essential for working with more complicated operations involving complex numbers in mathematics and engineering.
When we combine different complex numbers, we often add, subtract, and multiply them like real numbers. However, when dividing, we often multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part. The conjugate of a complex number \(z = a + bi\) is \(\overline{z} = a - bi\).
Understanding these basic properties is essential for working with more complicated operations involving complex numbers in mathematics and engineering.
Modulus Squared
The modulus of a complex number \(z = a + bi\) is denoted as \(|z|\) and is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\). It represents the length or magnitude of the vector corresponding to the complex number when represented in the complex plane.
In many problems, we encounter expressions like \(|z|^2\), the modulus squared, which simplifies to \(a^2 + b^2\). This relates to the norm of the vector and is useful in summation properties, as seen in exercises involving square roots or magnitudes of sums of complex numbers.
In many problems, we encounter expressions like \(|z|^2\), the modulus squared, which simplifies to \(a^2 + b^2\). This relates to the norm of the vector and is useful in summation properties, as seen in exercises involving square roots or magnitudes of sums of complex numbers.
- Calculates the "size" of a complex number correspondingly.
- Simplifies calculations, especially in trigonometric and summation contexts.
Summation Properties
Summation properties in the context of complex numbers often involve series of terms. One critical property of these series, especially when they relate to roots of unity, is how they can simplify expressions.
In particular, for the nth roots of unity, the sum of all roots equals zero: \(\sum_{k=0}^{n-1} \omega^k = 0\). This property is extremely useful for simplifying terms in sums involving complex numbers. For example, when expressions contain terms like \(z \overline{\omega^k}\) or \(\omega^k \overline{z}\), the summation property ensures these expression terms add up to zero.
Using these properties effectively can reduce complicated series into simpler forms that easily match solutions in mathematical exercises or problems.
In particular, for the nth roots of unity, the sum of all roots equals zero: \(\sum_{k=0}^{n-1} \omega^k = 0\). This property is extremely useful for simplifying terms in sums involving complex numbers. For example, when expressions contain terms like \(z \overline{\omega^k}\) or \(\omega^k \overline{z}\), the summation property ensures these expression terms add up to zero.
Using these properties effectively can reduce complicated series into simpler forms that easily match solutions in mathematical exercises or problems.
Nth Roots of Unity
The nth roots of unity are a fascinating concept in complex numbers. These roots are the solutions to the equation \(z^n = 1\), meaning raising these numbers to the nth power gives 1.
Defined generally as \(1, \omega, \omega^2, \ldots, \omega^{n-1}\), the fundamental root \(\omega\) is the complex number \(e^{2\pi i/n}\). It represents a rotational symmetry in the complex plane, dividing the unit circle into \(n\) equal parts. Each root has unique properties that form a symmetrical pattern around the origin.
Defined generally as \(1, \omega, \omega^2, \ldots, \omega^{n-1}\), the fundamental root \(\omega\) is the complex number \(e^{2\pi i/n}\). It represents a rotational symmetry in the complex plane, dividing the unit circle into \(n\) equal parts. Each root has unique properties that form a symmetrical pattern around the origin.
- They satisfy \(\sum_{k=0}^{n-1} \omega^k = 0\).
- Provides solutions that "revolve" around the number 1.
Other exercises in this chapter
Problem 172
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View solution Problem 173
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View solution Problem 175
If \(1, a_{1}, a_{2}, \ldots, a_{n-1}\) are the \(n, n\)th roots of unity, then \(\left(1-a_{1}\right)\left(1-a_{2}\right)\left(1-a_{3}\right) \ldots\left(1-a_{
View solution Problem 176
If \(1, \alpha, \alpha^{2}, \ldots, \alpha^{n}{\underline{\phantom{xx}}}^{1}\) are the \(n n\)th roots of unity then \(\sum_{i=1}^{n-1} \frac{1}{2-\alpha^{i}}\) is equal to (A) \(\fra
View solution