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Chapter 1: Problem 3

Investigate the multiplication and division of two complex numbers. Show that in general the product (quotient) of their real parts does not equal the real part of the product (quotient) of the two numbers.

Short Answer

Expert verified
Answer: No, the product (quotient) of the real parts of two complex numbers does not equal the real part of the product (quotient) of the two complex numbers in general. This is because the imaginary parts have an effect on both the product and the quotient.

Step by step solution

01

Complex Number Multiplication

To multiply two complex numbers, we first need to know the format of a complex number. A complex number can be written as a + bi, where a is the real part and bi is the imaginary part. To multiply two complex numbers, we use the distributive property and then simplify the result using i^2 = -1. If we have two complex numbers A = a + bi and B = c + di: Product of A and B, denoted as AB = (a + bi)(c + di).
02

Multiply the Complex Numbers

Now, let's multiply the two complex numbers using the distributive property and simplify the expression: AB = (a + bi)(c + di) = ac + adi + bci + bdi^2. Since i^2 = -1, we substitute this value into the equation to get the simplified product: AB = (ac - bd) + (ad + bc)i.
03

Complex Number Division

To divide two complex numbers, we first need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number is the complex number with the opposite sign for the imaginary part. If we have two complex numbers A = a + bi and B = c + di: Quotient of A and B, denoted as A/B = (a + bi) / (c + di).
04

Divide the Complex Numbers

Now, let's multiply both the numerator and the denominator by the conjugate of the denominator (c - di): A/B = (a + bi) / (c + di) * (c - di) / (c - di) = [(a + bi)(c - di)] / [(c + di)(c - di)]. Now let's perform the multiplications in both the numerator and the denominator and simplify: A/B = [(ac + bd) + (bc - ad)i] / (c^2 + d^2).
05

Show the Product (Quotient) of Real Parts Doesn't Equal the Real Part of the Product (Quotient)

Now, we compare the product (quotient) of the real parts with the real part of the product (quotient) of two complex numbers: The product of the real parts: (a * c) The real part of the product: (ac - bd) The quotient of the real parts: (a / c) The real part of the quotient: (ac + bd) / (c^2 + d^2) We can see that in general, the product (quotient) of the real parts does not equal the real part of the product (quotient) of the two complex numbers. This is because the imaginary parts have an effect on both the product and the quotient as shown in the previous steps.

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Most popular questions from this chapter

Prove the following trigonometric identities by expressing the left side in complex form: (a) \(\cos (x+y)=\cos x \cos y-\sin x \sin y\) (b) \(\sin (x+y)=\sin x \cos y+\cos x \sin y\) (c) \(\sum_{n=1}^{N} \cos n \theta=\frac{\cos \frac{1}{3}(N+1) \theta \sin \frac{1}{2} N \theta}{\sin \frac{1}{2} \theta}\) (d) \(\sum_{n=1}^{N} \sin n \theta=\frac{\sin \frac{1}{2}(N+1) \theta \sin \frac{1}{2} N \theta}{\sin \frac{1}{2} \theta}\) Many other trigonometric identities can be similarly proved. 1.3.3 Investigate the multiplication and division of two complex numbers. Show that in general the product (quotient) of their real parts does not equal the real part of the product (quotient) of the two numbers.

Three long identical strings of linear mass density \(\lambda_{0}\) are joinzd together at a common point forming a symmetrical Y. Thus they lie in a plane \(120^{\circ}\) apart. Each is given the same tension \(\tau_{0 .}\) A distant source of sinusoidal waves sends transverse waves, with motion perpendicular to the plane of the strings, down one of the strings. Find the reflection and transmission coefficients that characterize the junction.

A uniform string of linear mass density \(\lambda_{0}\) and under a tension \(\tau_{0}\) has a small bead of mass \(m\) attached to it at \(x=0\). Find expressions for the complex amplitude and the power reflection and transmission coefficients for sinusoidal waves brought about by the mass discontinuity at the origin. Do these coefficients hold for a wave of arbitrary shape?

The two waves \(\eta_{1}=6 \cos \left(k x-\omega t+\frac{1}{8} \pi\right)\) and \(\eta_{2}=8 \sin \left(k x-\omega t+\frac{1}{8} \pi\right)\) are traveling on a stretched string. ( \(a\) ) Find the complex representation of these waves. \((b)\) Find the complex wave equivalent to their sum \(\eta_{1}+\eta_{2}\) and the physical (real) wave that it represents. (c) Endeavor to combine the two waves by working only with trigonometric identities in the real domain.

Show how the solution \((1.5 .5)\) of the wave equation can be transformed into a general traveling-wave solution.
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