Exercises – Long Day

Exercises – Long Day#

Exercise 1: The number \(i\)#

In this exercise you will acquire some initial insight into the fundamentals of the complex numbers.

Question a#

What are \(i^2\), \(i^3\), \(i^4\), \(i^5\), \((-i)^2\), \((-i)^3\), \((-i)^4\), and \((-i)^{-5}\,\)? (Note the negative exponent in the last number.)

Hint

If you are in doubt about how to simplify \(i^2\), then see Definition 3.1.1 in the textbook.

Answer

\[-1,-i,1,i,-1,i,1,i\]

Question b#

What is the real part and the imaginary part of the three complex numbers \(10+i\), \(3\), and \(i\)? What about the real part and imaginary part of the number \(-5-i7\,\)?

Hint

If you are in doubt about what the real part and the imaginary part of a complex number are, then read the bit in the textbook from Definition 3.1.1 to Example 3.1.1.

Remember that the imaginary unit \(i\) is not itself included in the imaginary part (in fact, the imaginary part is real).

Answer

\(10+i\): The real part is \(10\) and the imaginary part \(1\).

\(3\): The real part is \(3\) and the imaginary part \(0\).

\(i\): The real part is \(0\) and the imaginary part \(1\).

\(-5-i7\): The real part is \(-5\) and the imaginary part \(-7\).

Question c#

What are \(\mathrm{Re}(-5-7i)\) and \(\mathrm{Im}(-5-7i)\)?

Answer

\(\mathrm{Re}(-5-7i)=-5\) and \(\mathrm{Im}(-5-7i)=-7.\)

Question d#

Write the complex numbers \(\,7i-5\,\), \(\,i(7i-5)\,\), and \(\,i(7i-5)i\,\) in rectangular form.

Hint

Definition 3.2.2 explains how two complex numbers are multiplied.

Answer

\[-5+7i \, \, , \, \, -7-5i \, \,,\,\, \text{and} \, \, 5-7i.\]

Exercise 2: The Complex Plane#

Question a#

Consider the following ten numbers: \(-2,\,0,\,i,\,2-i,\,1+2i,\,1,\,-2+3i,\,-5i,\,3,\,\) and \(\,-1-2i\,.\)

Which of them are complex, which are real, and which are purely imaginary?

Draw the ten numbers in the complex plane.

Hint

If you need to brush up on how to draw numbers in the complex plane or on what exactly “purely imaginary” means then have a look in the textbook’s Section 3.1, in particular the text after Definition 3.1.1.

Answer

All ten numbers are complex numbers. The numbers \(-2,0,1,3\) are real numbers. The numbers \(0,i,-5i\) are purely imaginary numbers.

Note that \(0\) can be considered as both a real number and a purely imaginary number since it is located both on the real axis and on the imaginary axis.

Question b#

We are given the number \(z=4+i\,\).

  1. Draw the four numbers \(\,z\,,\,iz\,,\,i^2z\,\), and \(\,i^3z\,\) in the complex plane.

  2. What happens geometrically in the complex plane when a complex number is multiplied by \(i\,\)?

  3. And divided by \(i\,\)?

Exercise 3: Fundamental Arithmetics#

Question a#

Determine using elementary calculations the rectangular form of the following complex numbers.

  1. \((5+i)(1+9i).\)

  2. \(i+i^2+i^3+i^4.\)

  3. \(\displaystyle{\frac{1}{1+3i}+\frac{1}{(1+3i)^2}}.\)

  4. \(\displaystyle{\frac{1}{(1+i)^4}}.\)

  5. \(\displaystyle{\frac{5+i}{2-2i}}.\)

  6. \(\displaystyle{\frac{3i}{4}}\,\) and \(\displaystyle{\frac{i2}{4}}.\)

Hint

To simplify a fraction where both numerator and denominator are complex numbers in rectangular form, Equation (3.2) in the textbook will show helpful. See also Example 3.2.3.

Answer

  1. \(-4+46i\)

  2. \(0\)

  3. \(\frac{1}{50} -\frac{9}{25} i\)

  4. \(- \frac{1}{4}\)

  5. \(1+ \frac{3}{2}i\)

  6. \(\frac{3}{4} i\) and \(\frac{1}{2} i\).

Question b#

We are given two real numbers \(a\) and \(b\,\) that are not both equal to \(0\).

  1. Why is the number \(\,\,\displaystyle{\frac{1}{a+ib}}\,\,\) not in rectangular form?

  2. Compute \(\mathrm{Re}\displaystyle{\left(\frac{1}{a+ib}\right)}\,\) and \(\mathrm{Im}\displaystyle{\left(\frac{1}{a+ib}\right)}\,\,\).

Answer

\(\mathrm{Re}\displaystyle{\left(\frac{1}{a+ib}\right)}\, = \frac{a}{a^2+b^2}\) and \(\mathrm{Im}\displaystyle{\left(\frac{1}{a+ib}\right)} = -\frac{b}{a^2+b^2}\).

Exercise 4: Conjugation#

In Definition 3.2.3 the complex conjugate \(\overline{z}\) of a complex number \(z\) is defined. You might want to read this definition again before continuing so that you have the precise meaning of the notation \(\overline{z}\) fresh in mind.

Question a#

Show that \(\overline{\overline{2+3i}}=2+3i\) and that \(\overline{(2+3i)\cdot (2+3i)}=\overline{2+3i}\cdot \overline{2+3i}\). Now show that in general we have \(\,\overline{\overline{z}}=z\,\) and \(\,\overline{z_1\cdot z_2}=\overline{z_1}\cdot\overline{z_2}\,\) for all complex numbers \(z\), \(z_1\), and \(z_2\).

Hint

If \(z=a+bi\) is a complex number in rectangular form, then what will happen if you perform complex conjugation twice (two times) in a row? In a similar manner, try computing both \(\,\overline{z_1\cdot z_2}\) and \(\overline{z_1}\cdot\overline{z_2}\,\) where \(z_1=a+bi\) and \(z_2=c+di\) are the numbers \(z_1\) and \(z_2\) written in rectangular form.

Question b#

Let \(z=a+ib\neq 0\) be a given complex number. Which complex number corresponds to the mirror image of \(z\) about

  1. the origin in the complex plane,

  2. the real axis,

  3. the imaginary axis, and

  4. a line passing through the origin with a slope of \(1\)?

Provide the answers both in rectangular form and as expressions written in terms of \(z\), \(\overline{z}\), and \(i\).

Hint

Start with a clear overview by drawing everything mentioned in the complex plane.

Answer

  1. Rectangular form: \(-a-bi\). Expression: \(-z\).

  2. Rectangular form: \(a-bi\). Expression: \(\overline{z}\).

  3. Rectangular form: \(-a+bi\). Expression: \(-\overline{z}\).

  4. Rectangular form: \(b+ai\). Expression: \(i\cdot \overline{z}\).

Exercise 5: Absolute Value#

The absolute value \(\,\left|z\right|\,\) of a complex number \(z\) can be interpretted as the distance between \(0\) (the origin) and \(z\) in the complex plane. See Figure 3.4 in the textbook for an illustration. The absolute value is also called the modulus in the context of complex numbers.

Question a#

We are given real numbers \(a,b\) and a complex number in rectangular form \(\,z=a+ib\,.\) Compute \(\,\left|z\right|\,.\) What is \(|2+3i|\)?

Answer

\(|z|=\sqrt{a^2+b^2}\) and \(|2+3i|=\sqrt{2^2+3^2}=\sqrt{13}\).

Question b#

Investigate which geometric meaning the absolute value \(\,\left|z_1-z_2\right|\,\) has for two arbitrary complex numbers \(\,z_1\,\) and \(\,z_2\,\). You may illustrate with examples.

Answer

\(\,\left|z_1-z_2\right|\,\) is the distance between \(z_1\) and \(z_2\) in the complex plane.

Question c#

A set in the complex plane is given by

\[\big\{z \in {\mathbb C} \, \mid \, |z-1|\, = \, 3\big\}\,.\]

Provide a geometric description of the set.

Answer

The set describes a circle in the complex plane centred at \(1\) with a radius of \(3\).

Exercise 6: Sets in the Complex Plane#

In the complex plane we consider the set of numbers \(\,M=\left\{z\,|\,\,|z-1+2i|\leq 3\,\right\}\,.\)

Question a#

Describe \(M\) and sketch \(M\) in the complex plane.

Question b#

Determine \(M \cap \mathbb{R}\), meaning the subset of \(\,M\,\) that contains all real numbers within \(M\).

Hint

First sketch \(M \cap \mathbb{R}\) on your drawing from Question a.

Answer

\[M \cap \mathbb{R}=\left[\,1-\sqrt 5\,,\,1+\sqrt 5\,\right]\,.\]

Exercise 7: Brackets and the Hierarchy of the Arithmetic Operations#

Question a#

Given the number \(\,\,z=3(i-10)-5(7-2i)-i(3i-5)+3i(i-5)\,,\) determine the rectangular form of \(z\,.\)

Hint

In expressions where both multiplication and addition/subtraction are involved, multiplicaton is to be performed first.

Answer

\[z=-65 +3i.\]

Question b#

We are given the numbers

\[a=5-i(3-i)+6i\,\,\, \mathrm{and} \,\,\,b=-5-4(-2i+1)\,.\]

Write the number \(\,z=a+ib\,\) in rectangular form.

Answer

\[z=-4-6i.\]

Exercise 8: Fractions#

Question a#

Determine the real part and the imaginary part of \((-2+3i)/i\) and bring the number to its rectangular form.

Hint

You can find som examples of how to rewrite a fraction of complex numbers to its rectangular form in Example 3.2.3 in the textbook.

Answer

\[\frac{(-2+3i)}i=3+2i.\]

Question b#

Reduce the following expression and bring it to its rectangular form:

\[\frac{3}{5}- \frac{3-2i}{2+i}\,.\]

Hint

The easiest might be to start with the second fraction and rewrite it to rectangular form. For this you can use an approach similar to what took place in Question a.

Answer

\[\frac{3}{5}- \frac{3-2i}{2+i}=-\frac{1}{5} + \frac{7}{5} i.\]

Question c#

Let \(b,c\), and \(d\) be the following real numbers:

\[b=5 \,,\, \, c=\frac{6}{7} \,,\, \, d=\frac{2}{3} \,.\]

Compute the following numbers:

\[c+d\,,\, \, d \cdot b\,,\, \,\frac{b}{d}, \, \, \, \frac{d}{c}.\]

Question d#

Let \(k,n,m\), and \(s\) be the following complex numbers:

\[k=1+i \cdot \sqrt{3} \,,\, \, n=5 \cdot i \,,\, \, m=1+i \,,\, \, s=i \cdot 4 +3.\]

Write the following complex numbers in rectangular form:

\[\frac{m}{n} \,,\, \, \frac{k}{s} \,,\, \, \frac{1}{m} + s.\]

Exercise 9: Ordering of Complex Numbers#

Within the real numbers we have the well-known less than order relation \(\,<\,\) that for all \(\,a,b,\) and \(c\) in \(\mathbb R\) fulfill:

  1. Only one of the statements \(\,a<b,\) \(\,b<a,\) or \(\,a=b\,\) is true.

  2. If \(\,a<b\,\) and \(\,b<c\,\), then \(\,a<c\,.\)

  3. If \(\,a<b\,\), then \(\,a+c<b+c\,.\)

  4. If \(\,a<b\,\) and \(\,0<c\,\), then \(\,ac<bc\,.\)

Question a#

Test the four requirements with a few examples.

Question b#

Show that it is not possible to extend the order relation \(\,<\,\) from the real numbers to also apply to all complex numbers. More precisely, show that no order relation \(\,<\,\) exists for \(\mathbb C\) which can extend the order relation \(\,<\,\) from the real numbers and which fulfills the four requirements as mentioned above for all \(\,a,b,\) and \(\,c\,\) in \(\mathbb C\).

Hint

Try to find a counter example. Meaning, assume that such order relation does exist and then try to reach a contradiction. A single counter example is enough.

Hint

If such order relation does exist, then either \(0<i\) or \(i<0\) applies due to the first of the above requirements. Try to reach a contradiction in every case.