Exercises – Long Day#
Exercise 1: Polar Form#
This exercise builds further upon Exercise 4a from the Short Day of Week 3. The numbers \(z_1=1+i\sqrt{3}\,\), \(z_2=-1+i\sqrt{3}\,\), \(z_3=-1-i\sqrt{3}\,\), and \(\,z_4=1-i\sqrt{3}\,\) are given.
Question a#
State the four numbers in polar form.
Hint
See Definition 3.6.1 to recap on what the polar form of a complex number is all about. You can use the results from Exercise 4a from Short Day of Week 3 to avoid double work.
Answer
\(z_1=2\mathrm e^{\frac{\pi}{3} i}\), \(z_2=2\mathrm e^{\frac{2\pi}{3} i}\), \(z_3=2\mathrm e^{\frac{-2\pi}{3} i}\), \(z_4=2\mathrm e^{\frac{-\pi}{3} i}.\)
Question b#
Use the polar form to compute \(z_1^{3}\), \(z_2^{3}\), \(z_3^{3}\), and \(z_4^{3}.\)
Hint
You can use the last part of Theorem 3.6.2 from the textbook to compute modulus and argument of an integer power of a complex number.
Answer
\(-8,8,8,-8.\)
Question c#
Show that \(z_2\) and \(z_3\) are roots in the polynomial \(Z^3-8\).
Hint
See Definition 4.1.2 to read more about what precisely a root of a polynomial is.
Question d#
Determine a polynomial \(p(Z)\) in \(\mathbb{C}[Z]\) of degree three that has \(z_1\) and \(z_4\) as roots.
Answer
\(Z^3+8\) is a valid answer, and there are more options.
Exercise 2: First-Degree Polynomials#
A polynomial \(p(Z) \in \mathbb{C}[Z]\) is given by \(p(Z)=(2-i)Z+i.\)
Question a#
Find a root of the polynomial \(p(Z)\).
Answer
\(\frac15-\frac25 i\).
Question b#
Solve the polynomial equations \(p(z)=2\) and \(p(z)=-2+2i\).
Answer
The equation \(p(z)=2\) has the solution \(1\).
The equation \(p(z)=-2+2i\) has the solution \(-1\).
Exercise 3: Polynomial Arithmetics#
The following three polynomials in \(\mathbb{C}[Z]\) are given:
Question a#
Determine the degrees and leading coefficients of the three given polynomials.
Hint
Note that \(p_3(Z)\) is the same polynomial as \((1+i)Z^5-1\).
Answer
\(p_1(Z)\) has degree \(3\) and leading coefficient \(2\).
\(p_2(Z)\) has degree \(1\) and leading coefficient \(1\).
\(p_3(Z)\) has degree \(5\) and leading coefficient \(1+i\).
Question b#
Compute \(p_1(Z)+p_2(Z)+p_3(Z)\), \(ip_3(Z)\), and \(p_1(Z)p_2(Z)\).
Answer
\(p_1(Z)+p_2(Z)+p_3(Z)=(1+i)Z^5+2Z^3.\)
\(i \cdot p_3(Z)=(-1+i)Z^5-i.\)
\(p_1(Z)p_2(Z)=2Z^4+4Z^3-Z^2-3Z-2.\)
Exercise 4: Binomial Equations#
Question a#
Solve the binomial equation \(z^3=-8i\).
Hint
Binomial equations are equations of the form \(z^n=w\). These are solved using Theorem 4.4.1. In our case we have \(n=3\) and \(w=-8i\).
Answer
\(z^3=-8i\) has the solutions \(2i\), \(\sqrt{3}-i\), and \(-\sqrt{3}-i\).
Exercise 5: Binomial Second-Degree Equations with Real Right-Hand Sides#
Question a#
Let \(r\) be a positive real number. Use Theorem 4.2.1 to justify that the equation
has exactly two solutions which are given by \(i\sqrt{r}\) and \(-i\sqrt{r}\).
Question b#
Solve Question a again, but this time use Theorem 4.4.1.
Hint
Using Theorem 4.4.1 for solving equations of the form \(z^n=w\), in our case we have \(n=2\) and \(w=-r\).
Hint
What is the principal argument of a negative real number?
Question c#
Solve the equations \(z^2=-16\).
Answer
\(z^2=-16\) har the solutions \(-4i\) and \(4i\).
Exercise 6: Polynomials with Real Coefficients#
Question a#
Check without using a solution formula that \(-1+2i\) is a root of the polynomial \(3Z^2+6Z+15.\)
Hint
Definition 4.1.2 reveals how you can check whether a specific complex number is a root.
Question b#
Find another root of the polynomial \(3Z^2+6Z+15\) without using a solution formula.
Hint
Can the theory in the last part of Section 4.3 in the textbook be of use?
Exercise 7: Integer Powers and Polar Form#
Question a#
Write \(-1+\sqrt{3}i\) in polar form and use it in a similarly manner as in Example 3.6.2 to show that
Question b#
Let \(n\) be a natural number. Show the following:
and
Hint
First show that \((-1+\sqrt{3}i)^{3}=2^3\).
Hint
If \((-1+\sqrt{3}i)^{3}=2^3\), then \(((-1+\sqrt{3}i)^{3})^n=(2^3)^n\).
Exercise 8: Equations with the Exponential Function#
Question a#
We are given the numbers \(\,w_1=1\,,\,w_2=\mathrm e\,,\) and \(\,w_3=2i\,\). Find for \(n=1,\dots,3\) the set of all solutions in \(\mathbb C\) of the equations:
Hint
Lemma 3.6.1 from the textbook describes how to find the solutions to an equation in the form \(\mathrm e^z=w\).
Hint
An arbitrary argument of \(w\) is equal to the principal argument of \(w\) plus an integer multiple of \(2\pi\).
Answer
According to Lemma 3.6.1 every solution to the equation \(\mathrm e^z=1\) has the form \(z=i \mathrm{arg}(1)\). The principal argument of \(1\) is \(0\), but all possible arguments of \(1\) are equal to the principal argument plus an integer multiple of \(2\pi\). Hence, the equation \(\mathrm e^z=1\) has the solution set \(\{ ip2\pi \, \mid \, p \in \mathbb{Z}\}.\)
\(\mathrm e^z=\mathrm e\) has the solution set \(\{ 1+ip2\pi \, \mid \, p \in \mathbb{Z}\}.\)
\(\mathrm e^z=2i\) has the solution set \(\left\{ \ln(2)+i(\frac{\pi}{2}+p2\pi) \, \mid \, p \in \mathbb{Z}\right\}.\)
Question b#
Find the set of all solutions to the equation
Answer
The solution set is the union of the solution sets to the equations corresponding to \(n=1\) and \(n=3\) above.
Question c#
Show the first claim in Theorem 3.4.2, which is that \(\,\mathrm e^z \neq 0\,\) for all \(\,z\in\mathbb C\,\).
Hint
If \(z=a+bi\) is written in rectangular form, then Definition 3.4.1 implies that \(\mathrm e^z=\mathrm e^a \cdot (\cos(b)+\sin(b) i)\). Can this expression become zero?
Exercise 9: Complex Numbers and Pythagorean Triples#
A Pythagorean Triple \((a,b,c)\) consists of three natural numbers such that \(a>b\) and \(a^2+b^2=c^2\). An example is \((4,3,5).\)
Question a#
Show that if a triple \((a,b,c)\) of natural numbers is a Pythagorean Triple, then the complex number \(z=\frac{a}{c}+\frac{b}{c}i\) fulfills \(|z|=1\) and \(\mathrm{Re}(z)>\mathrm{Im}(z)\).
Question b#
Let us now assume that a complex number \(z\) fulfills \(|z|=1\) and \(\mathrm{Re}(z)>\mathrm{Im}(z)\) and that \(z\) can be written in the form \(z=\frac{a}{c}+\frac{b}{c}i\), where \(a,b\) and \(c\) are natural numbers. Show that \((a,b,c)\) is a Pythagorean Triple.
Question c#
Use the above insights and complex numbers to construct other Pythagorean Triples based on the triple \((4,3,5).\)
Hint
If the number \(z=\frac{a}{c}+\frac{b}{c}i\) gives rise to the Pythagorean Triple \((a,b,c)\), what then about the number \(z^2\) (or maybe \(i \cdot \overline{z}^2\))?