Chapter 1

Show that \(2-\sqrt{-1}=\sqrt[3]{2-11 \sqrt{-1}}\).

Evaluate the following quantities. Be sure to show your work. (a) \(|(1+i)(2+i)|\). (b) \(\left|\frac{4-3 i}{2-i}\right|\). (c) \(\left|(1+i)^{50}\right|\). (d) \(|z \bar{z}|\), where \(z=x+i y\). (e) \(|z-1|^{2}\), where \(z=x+i y\).

Find \(\mathrm{Arg} z\) for the following values of \(z\). (a) \(1-i\). (b) \(-\sqrt{3}+i\). (c) \((-1-i \sqrt{3})^{2}\) (d) \((1-i)^{3}\). (e) \(\frac{2}{1+i \sqrt{3}}\). (f) \(\frac{2}{i-1}\). (g) \(\frac{1+i \sqrt{5}}{(1+i)^{2}}\). (h) \((1+i \sqrt{3})(1+i)\)

Calculate the following. (a) \((1-i \sqrt{3})^{3}(\sqrt{3}+i)^{2}\) (b) \(\frac{(1+6)^{3}}{(1-i)^{3}}\). (c) \((\sqrt{3}+i)^{6}\).

Find a parametrization of the line that (a) joins the origin to the point \(1+i\). (b) joins the point. 1 to the point \(1+i\). (c) joins the point \(i\) to the point \(1+i\). (d) joins the point 2 to the point \(1+i\).

Explain why cubic equations, rather than quadratic equations, played a pivotal role in helping to obtain the acceptance of complex numbers.

Locate \(z_{1}\) and \(z_{2}\) vectorially and use vectors to find \(z_{1}+z_{2}\) and \(z_{1}-z_{2}\) when (a) \(z_{1}=2+3 i\) and \(z_{2}=4+i\) (b) \(z_{1}=-1+2 i\) and \(z_{2}=-2+3 i\) (c) \(z_{1}=1+i \sqrt{3}\) and \(z_{2}=-1+i \sqrt{3}\)

Sketch the curve \(z(t)=t^{2}+2 t+i(t+1)\) (a) for \(-1 \leq t \leq 0\). (b) for \(1 \leq t \leq 2\).

Find all solutions to the following depressed cubics. (a) \(27 x^{3}-9 x-2=0 .\) Hint: Get an equivalent monic polynomial. (b) \(x^{3}-27 x+54=0\)

Show that \(z \bar{z}\) is always a real number.

Which of the following points lie inside the circle \(|z-i|=2\) ? Explain your answers. (a) \(\frac{1}{2}+i\). (b) \(\sqrt{2}+i(\sqrt{2}+1)\) (c) \(2+3 i\). (d) \(\frac{-1}{2}+i \sqrt{3}\)

Represent the following complex numbers in polar form. (a) \(-4\). (b) \(6-6 i\). (c) \(-7 i\), (d) \(-2 \sqrt{3}-2 i\) (e) \(\frac{1}{(1-i)^{2}}\). (f) \(\frac{6}{i+\sqrt{3}}\). (g) \(3+4\) i. (h) \((5+5 i)^{3}\)

Let \(P(z)=a_{n} z^{n}+a_{n-1} z^{n-1}+\cdots+a_{1} z+a_{0}\) be a polynomial of degree \(n .\) (a) Suppose that \(a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}\) are all real. Show that if \(z_{1}\) is a root of \(P\), then \(\overline{z_{1}}\) is also a root. In other words, the roots must be complex conjugates, something you likely learned without proof in high school. (b) Suppose not all of \(a_{\mathrm{m}}, a_{n-1}, \ldots, a_{1}, a_{0}\) are real. Show that \(P\) has at least one root whose complex conjugate is not a root. Hint: Prove the contrapositive. (c) Find an example of a polynomial that has some roots occurring as complex conjugates, and some not.

Express the following in \(a+i b\) form. (a) \(e^{\frac{1 \pi}{2}}\). (b) \(4 e^{-i \frac{\pi}{2}}\) (c) \(8 e^{i \frac{1 \pi}{3}}\), (d) \(-2 e^{i \frac{2 x}{8}}\). (e) \(2 i e^{-i \frac{2 \pi}{4}}\), (f) \(6 e^{i \frac{\pi}{3}} e^{i \pi}\). (g) \(e^{2} e^{i \pi}\) (h) \(e^{i \frac{\pi}{6}} e^{-i n}\).

Find all the roots in both polar and Cartesian form for each expression. (a) \((-2+2 i)^{8}\). (b) \((-1)^{\frac{1}{3}}\). (c) \((-64)^{\frac{1}{4}}\) (d) \((8)^{\frac{1}{6}}\). (e) \((16 i)^{\frac{1}{2}} .\)

Find a parametrization of the curve that is a portion of the circle \(|z|=1\) that joins the point \(-i\) to \(i\) if (a) the curve is the right semicircle. (b) the curve is the left semicircle.

Let \(z_{1}=\left(x_{1}, y_{1}\right)\) and \(z_{2}=\left(x_{2}, y_{2}\right)\) be arbitrary complex numbers. Prove or disprove the following. (a) \(\operatorname{Re}\left(z_{1}+z_{2}\right)=\operatorname{Re}\left(z_{1}\right)+\operatorname{Re}\left(z_{2}\right)\) (b) \(\operatorname{Re}\left(z_{1} z_{2}\right)=\operatorname{Re}\left(z_{1}\right) \operatorname{Re}\left(z_{2}\right)\). (c) \(\operatorname{lm}\left(z_{1}+z_{2}\right)=\operatorname{lm}\left(z_{1}\right)+\operatorname{lm}\left(z_{2}\right)\). (d) \(\operatorname{Im}\left(z_{1} z_{2}\right)=\operatorname{Im}\left(z_{1}\right) \operatorname{Im}\left(z_{2}\right)\).

Sketch the sets of points determined by the following relations. (a) \(|z+1-2 i|=2\). (b) \(\operatorname{Re}(z+1)=0\) (c) \(|z+2 i| \leq 1\) (d) \(\operatorname{Im}(\varepsilon-2 i)>6\)

Show that arg \(z_{1}=\arg z_{2}\) iff \(\mathrm{z}_{2}=c z_{1}\), where \(c\) is a positive real constant.

Prove that the complex number \((1,0)\) (which we identify with the real number 1 ) is the multiplicative identity for complex numbers.

Prove that \(\sqrt{2}|z| \geq|\operatorname{Re}(z)|+|\operatorname{Lm}(z)|\).

Let \(z_{1}=-1+i \sqrt{3}\) and \(z_{2}=-\sqrt{3}+i\). Show that the equation \(\operatorname{Arg}\left(z_{1} z_{2}\right)=\operatorname{Arg} z_{1}+\operatorname{Arg} z_{2}\) does not hold for the specific choice of \(z_{1}\) and \(z_{2} .\)

Find all the roots of the equation \(z^{4}-4 z^{3}+6 z^{2}-4 z+5=0\) if \(z_{1}=i\) is a root

Find a parametrization of the curve that is a portion of the circle \(C_{1}(0)\) that joins the point 1 to \(i\) if (a) the parametrization is counterclockwise along the quarter circle. (b) the parametrization is clockwise.

Show that the equation \(\operatorname{Arg}\left(z_{1} z_{2}\right)=\) Arg \(z_{1}+\) Arg \(z_{2}\) is true if \(\frac{-\pi}{2}<\operatorname{Arg} z_{1} \leq \frac{\pi}{2}\) and \(\frac{-\pi}{2}<\mathrm{Arg} \pi_{2} \leq \frac{\pi}{2} .\) Describe the set of points that meets this criterion.

Solve the equation \((z+1)^{3}=z^{3}\).

Let's use the symbol \(*\) for a new type of multiplication of complex numbers defined by \(z_{1} * z_{2}=\left(x_{1} x_{2}, y_{1} y_{2}\right) .\) This exercise shows why this is an unfortunate definition. (a) Use the definition given in property (P7) and state what the multiplicative identity \(\zeta\) would have to be for this new multiplication. (b) Show that if you use this new multiplication, nonzero complex numbers of the form \((0, a)\) have no inverse. That is, show that if \(z=(0, a)\), there is no complex number \(w\) with the property that \(z * w=\zeta\), where C is the multiplicative identity you found in part (a).

Show that \(\left|z_{1}-z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\)

Describe the set of complex numbers for which \(\operatorname{Arg}\left(\frac{1}{x}\right) \neq-\operatorname{Arg}(z) .\) Prove your assertion.

Find the three solutions to \(z^{\frac{3}{2}}=4 \sqrt{2}+i 4 \sqrt{2}\).

Consider the following sets. (i) \(\\{z: \operatorname{Re}(z)>1\\}\). (ii) \(\\{z:-1<\operatorname{lm}(z) \leq 2\\}\). (iii) \(\\{z:|z-2-i| \leq 2\\}\). (iv) \(\\{z:|z+3 i|>1\\}\). (v) \(\left\\{r e^{i \theta}: 01\right.\) and \(\left.\frac{\pi}{4}<\theta<\frac{\pi}{3}\right\\}\). (vii) \(\\{z:|z|<1\) or \(|z-4|<1\\}\). (a) Sketch each set. (b) State, with reasons, which of the following terms apply to the above sets: open; connected; domain; region; closed region; bounded.

Explain why the complex number \((0,0)\) (which, you recall, we identify with the real number 0 ) has no multiplicative inverse.

Prove that \(|z|=0\) iff \(z=0\).

Let \(m\) and \(n\) be positive integers that have no common factor. Show that there are \(n\) distinct solutions to \(w^{n}=z^{m}\) and that they are given by \(w_{k}=r^{m}\left(\cos \frac{m(\theta+2 \pi k)}{n}+i \sin \frac{m(\theta+2 \pi k)}{n}\right)\) for \(k=0,1, \ldots, n-1 .\)

Show that \(D_{1}(0)\) is connected. Hint: Show that if \(z_{1}\) and \(z_{2}\) lie in \(D_{1}(0)\), then the straight-line segment joining them lies entirely in \(D_{1}(0)\).

Show that if \(z \neq 0\), the four points \(z\), \(\bar{z},-z\), and \(-\bar{z}\) are the vertices of a rectangle with its center at the otigin.

Show that \(\arg \left(\frac{1}{x}\right)=-\arg z\)

Show that if \(z \neq 0\), the four points \(z, i z,-z\), and \(-i z\) are the vertices of a square with its center at the origin.

If \(1=z_{0}, z_{1}, \ldots, z_{n-1}\) are the \(n\) th roots of unity, prove that \(\left(z-z_{1}\right)\left(z-z_{2}\right) \cdots\left(z-z_{n-1}\right)=1+z+z^{2}+\cdots+z^{n-1}\)

Prove that the boundary of the neighborhood \(D_{\varepsilon}\left(z_{0}\right)\) is the circle \(C_{\varepsilon}\left(z_{0}\right)\).

Show that the equation of the line through the points \(z_{1}\) and \(z_{2}\) can be expressed in the form \(z=z_{1}+t\left(z_{2}-z_{1}\right)\), where \(t\) is a real number.

Show that if \(z \neq 0\), then (a) \(\operatorname{Arg}(z \bar{z})=0\). (b) \(\operatorname{Arg}(z+\bar{z})=0\) when \(\operatorname{Re}(z)>0\).

Let \(z_{k} \neq 1\) be an \(n\) th root of unity. Prove that \(1+z_{k}+z_{k}^{2}+\cdots+z_{k}^{n-1}=0\).

Let \(S\) be the open set consisting of all points \(z\) such that \(|z+2|<1\) or \(|z-2|<1\). Show that \(S\) is not connected.

Show that the nonzero vectors \(z_{1}\) and \(z_{2}\) are parallel iff \(\operatorname{Im}\left(z_{1} \overline{z_{2}}\right)=0\).

Prove that the only accumulation point of \(\left\\{\frac{1}{n}: n=1,2, \ldots\right\\}\) is the point \(0 .\)

Show that \(\left|z_{1} z_{2} z_{3}\right|=\left|z_{1}\right|\left|z_{2}\right|\left|z_{3}\right|\).

Find all four roots of \(z^{4}+4=0\), and use them to demonstrate that \(z^{4}+4\) can be factored into two quadratics with real coefficients.

Regarding the relation between closed sets and accumulation points, (a) prove that if a set is closed, then it contains all its accumulations points. (b) prove that if a set contains all its accumulation points, then it is closed.

Show that \(\left|z^{n}\right|=|z|^{n}\), where \(n\) is an integer.