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For each of the following two pairs of numbers compute \(z+w, z-w, z w,\) and \(z / w\) (a) \(z=6+8 i\) and \(w=3-4 i \quad\) (b) \(z=8 e^{i \pi / 3}\) and \(w=4 e^{i \pi / 6}\) Notice that for adding and subtracting complex numbers, the form \(x+i y\) is more convenient, but for multiplying and especially dividing, the form \(r e^{i \theta}\) is more convenient. In part (a), a clever trick for finding \(z / w\) without converting to the form \(r e^{i \theta}\) is to multiply top and bottom by \(w^{*}\); try this one both ways.

Short Answer

Expert verified
Calculations: (a) 1. \(9+4i\), 2. \(3+12i\), 3. \(50\), 4. \(-\frac{14}{25}+\frac{56}{25}i\); (b) 1. \(6+6\sqrt{3}i\), 2. \(2-2\sqrt{3}+2i\), 3. \(32i\), 4. \(\sqrt{3}+i\).

Step by step solution

01

Addition of Complex Numbers (Part a)

For the given complex numbers \(z = 6 + 8i\) and \(w = 3 - 4i\), we add the real and imaginary parts separately. \[ z + w = (6 + 3) + (8i - 4i) = 9 + 4i \]
02

Subtraction of Complex Numbers (Part a)

Subtract the real and imaginary parts of the complex numbers: \[ z - w = (6 - 3) + (8i + 4i) = 3 + 12i \]
03

Multiplication of Complex Numbers (Part a)

Use the distributive property to multiply the complex numbers: \[ z \times w = (6 + 8i)(3 - 4i) \] Expand this using the FOIL method: \[ 6 \cdot 3 + 6 \cdot (-4i) + 8i \cdot 3 + 8i \cdot (-4i) \]This simplifies to: \[ 18 - 24i + 24i - 32i^2 \] Since \(i^2 = -1\), we have:\[ 18 - 32(-1) = 18 + 32 = 50 \] So, \(z \times w = 50 \).
04

Division of Complex Numbers Using Conjugate (Part a)

Multiply the numerator and the denominator by the conjugate of the denominator (\(w^* = 3 + 4i\)):\[ \frac{z}{w} = \frac{6 + 8i}{3 - 4i} \times \frac{3 + 4i}{3 + 4i} \] Perform the multiplication:Numerator: \[ (6 + 8i)(3 + 4i) = 18 + 24i + 32i - 32 = -14 + 56i \]Denominator: \[ (3 - 4i)(3 + 4i) = 9 + 12i - 12i - 16i^2 = 9 + 16 = 25 \] So, \[ \frac{z}{w} = \frac{-14 + 56i}{25} = -\frac{14}{25} + \frac{56}{25}i \]
05

Convert Complex Numbers to Polar Form for Multiplication (Part b)

The numbers are given as \( z = 8 e^{i \pi / 3} \) and \( w = 4 e^{i \pi / 6} \). Multiply their magnitudes and add their angles: \[ z \cdot w = 8 \times 4 e^{i (\pi/3 + \pi/6)} = 32 e^{i (\pi/2)} \]Simplifying, this is \( 32i \).
06

Division of Complex Numbers in Polar Form (Part b)

For division, divide the magnitudes and subtract the angles:\[ \frac{z}{w} = \frac{8}{4} e^{i (\pi/3 - \pi/6)} = 2 e^{i \pi/6} \]Convert back to rectangular form:\[ 2 \left( \cos(\pi/6) + i \sin(\pi/6) \right) = 2 \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = \sqrt{3} + i \]
07

Addition of Complex Numbers (Part b)

Convert both numbers to rectangular form:For \( z = 8 e^{i \pi / 3} \):\[ 8 (\cos(\pi/3) + i\sin(\pi/3)) = 8 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = 4 + 4 \sqrt{3}i \] For \( w = 4 e^{i \pi /6} \):\[ 4 \left( \cos(\pi/6) + i \sin(\pi/6) \right) = 4 \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = 2 \sqrt{3} + 2i \] Add:\[ (4 + 4 \sqrt{3}i) + (2 \sqrt{3} + 2i) = 4 + 2 \sqrt{3} + (4 \sqrt{3} + 2)i = 4 + 2 \sqrt{3} + (6 \sqrt{3}i + 2i) \]
08

Subtraction of Complex Numbers (Part b)

Using the rectangular form:Subtract \( w \) from \( z \):\[ (4 + 4 \sqrt{3}i) - (2 \sqrt{3} + 2i) = (4 - 2 \sqrt{3}) + (4 \sqrt{3}i - 2i) = 4 - 2 \sqrt{3} + (4 \sqrt{3}i - 2i) \]

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Key Concepts

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

Addition of Complex Numbers
Complex numbers are made of two parts, a real part and an imaginary part. Adding complex numbers is as simple as adding the real parts together and the imaginary parts together. For example, if we have two complex numbers \( z = a + bi \) and \( w = c + di \), their sum \( z + w \) is calculated as \((a+c) + (b+d)i\).
This means you handle the real components separately from the imaginary ones. In the provided exercise, the numbers were \( z = 6 + 8i \) and \( w = 3 - 4i \). By adding the real parts: \(6 + 3 = 9\) and the imaginary parts: \(8i - 4i = 4i\), we get the sum: \(9 + 4i\).
Remember, the addition of complex numbers is commutative, which means \(z + w = w + z\). This makes the process quite straightforward and typically easy to understand and apply.
Multiplication of Complex Numbers
Multiplying complex numbers requires a bit more consideration than addition. You can use the distributive property, often employed through the FOIL method, to multiply expressions like \((a + bi)(c + di)\). Here is how it's done:
  • Multiply the first terms: \(ac\)
  • Multiply the outside terms: \(adi\)
  • Multiply the inside terms: \(bci\)
  • Multiply the last terms: \(bdi^2\)
Remember that \(i^2 = -1\).
So, when applied, this will eventually simplify into a sum of a real part and an imaginary part. For the case of \( z = 6 + 8i \) and \( w = 3 - 4i \), multiplication gives \(18 - 24i + 24i - 32i^2\). Which simplifies to \(18 + 32 = 50\), since \(-32i^2\) becomes \(+32\). The result here was simply a real number!
Polar Form of Complex Numbers
Complex numbers can also be represented in polar form, which is very useful for multiplication and division. Instead of using \(z = a + bi\), we express the complex number in terms of a magnitude \(r\) and an angle \(\theta\), as \(z = r e^{i\theta}\).
The magnitude \(r\) is the distance from the origin to the point on the complex plane and is calculated by the formula \(r = \sqrt{a^2 + b^2}\). The angle \(\theta\) is the direction measured from the positive real axis and can be found using \(\theta = \tan^{-1}(b/a)\).
In multiplication, you simply multiply the magnitudes and add the angles. For instance, the exercise uses \( z = 8 e^{i\pi/3} \) and \( w = 4 e^{i\pi/6} \). Their product is \(32 e^{i(\pi/2)}\), simplifying directly into \(32i\). Polar form excels in simplifying such operations.
Division using Conjugates
Dividing complex numbers can be done by multiplying the denominator and numerator by the conjugate of the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\).
Why the conjugate? Multiplying by the conjugate eliminates the imaginary part in the denominator, making the division simpler.
For example, to divide \( z = 6 + 8i \) by \( w = 3 - 4i \): we multiply by \( w^* = 3 + 4i \). This means:\[\frac{z}{w} = \frac{6 + 8i}{3 - 4i} \times \frac{3 + 4i}{3 + 4i}\]You simplify the numerator and denominator separately to get the result.
In this problem, after multiplying, we find the final expression \(-\frac{14}{25} + \frac{56}{25}i\). By applying the conjugate, the division becomes manageable without converting to polar form.

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Most popular questions from this chapter

A charged particle of mass \(m\) and positive charge \(q\) moves in uniform electric and magnetic fields, \(\mathbf{E}\) and \(\mathbf{B}\), both pointing in the \(z\) direction. The net force on the particle is \(\mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})\). Write down the equation of motion for the particle and resolve it into its three components. Solve the equations and describe the particle's motion.

Problem 2.7 is about a class of one-dimensional problems that can always be reduced to doing an integral. Here is another. Show that if the net force on a one-dimensional particle depends only on position, \(F=F(x),\) then Newton's second law can be solved to find \(v\) as a function of \(x\) given by $$v^{2}=v_{\mathrm{o}}^{2}+\frac{2}{m} \int_{x_{0}}^{x} F\left(x^{\prime}\right) d x^{\prime}$$ [Hint: Use the chain rule to prove the following handy relation, which we could call the " \(v\) dv/d \(x\) rule": If you regard \(v\) as a function of \(x,\) then $$\dot{v}=v \frac{d v}{d x}=\frac{1}{2} \frac{d v^{2}}{d x}$$ Use this to rewrite Newton's second law in the separated form \(m d\left(v^{2}\right)=2 F(x) d x\) and then integrate from \(x_{\mathrm{o}}\) to \(x .\) ] Comment on your result for the case that \(F(x)\) is actually a constant. (You may recognise your solution as a statement about kinetic energy and work, both of which we shall be discussing in Chapter 4.)

Consider an object that is coasting horizontally (positive \(x\) direction) subject to a drag force \(f=-b v-c v^{2} .\) Write down Newton's second law for this object and solve for \(v\) by separating variables. Sketch the behavior of \(v\) as a function of \(t .\) Explain the time dependence for \(t\) large. (Which force term is dominant when \(t\) is large?)

The hyperbolic functions cosh \(z\) and \(\sinh z\) are defined as follows: $$\cosh z=\frac{e^{z}+e^{-z}}{2} \quad \text { and } \quad \sinh z=\frac{e^{z}-e^{-z}}{2}$$ for any \(z,\) real or complex. (a) Sketch the behavior of both functions over a suitable range of real values of \(z\). (b) Show that \(\cosh z=\cos (i z)\). What is the corresponding relation for \(\sinh z ?\) (c) What are the derivatives of cosh \(z\) and \(\sinh z ?\) What about their integrals? ( \(\mathbf{d}\) ) Show that \(\cosh ^{2} z-\sinh ^{2} z=1\) (e) Show that \(\int d x / \sqrt{1+x^{2}}=\operatorname{arcsinh} x\). [Hint: One way to do this is to make the substitution \(x=\sinh z .]\)

There are certain simple one-dimensional problems where the equation of motion (Newton's second law) can always be solved, or at least reduced to the problem of doing an integral. One of these (which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject to a force that depends only on the velocity \(v\), that is, \(F=F(v)\). Write down Newton's second law and separate the variables by rewriting it as \(m d v / F(v)=d t .\) Now integrate both sides of this equation and show that $$t=m \int_{v_{0}}^{v} \frac{d v^{\prime}}{F\left(v^{\prime}\right)}$$ Provided you can do the integral, this gives \(t\) as a function of \(v\). You can then solve to give \(v\) as a function of \(t .\) Use this method to solve the special case that \(F(v)=F_{\mathrm{o}},\) a constant, and comment on your result. This method of separation of variables is used again in Problems 2.8 and \(2.9 .\)

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