Problems involving complex numbers, geometry in the plane, and related algebraic techniques.
These practice problems treat the complex numbers both as an algebraic system and as points in the plane. You will compute with the imaginary unit, the conjugate, and the modulus, pass between rectangular and polar form, take powers and roots, and read simple loci. The set fits a first course that stops short of complex analysis. Each solution is written step by step, including the geometric reading of an algebraic step.
Algebra of Complex Numbers covers arithmetic, conjugates, modulus, and equations that stay inside that arithmetic. Geometry in the Plane reads addition and multiplication as transformations and asks for loci defined by a modulus condition. Powers and Roots uses De Moivre's formula for integer powers and for nth roots. Limits and Functions is a short bridge onward: limits and the first elementary functions of a complex variable.
No earlier topic is required. Geometry helps if you want vector arguments in the plane before this arithmetic, and it is the heavier set. Real limits in much more detail are in Limits. Characteristic equations with complex roots, which reuse only the algebra from this topic, are solved in Second-Order Differential Equations.
Problem 1
MediumAlgebra of Complex Numbers
Find the complex numbers corresponding to the points:
$$
1^{\circ}\,(0,1); \quad 2^{\circ}\,(1,0); \quad 3^{\circ}\,(0,-1); \quad 4^{\circ}\,(-1,0); \quad 5^{\circ}\,(-2,3).
$$
Find the points corresponding to the complex numbers and determine their position in the complex plane:
$$
1^{\circ}\; i; \quad 2^{\circ}\; -2i; \quad 3^{\circ}\; -1+i; \quad 4^{\circ}\; 2-3i.
$$
Determine the real and imaginary parts of the complex numbers:
$$
1^{\circ}\; -7i; \quad 2^{\circ}\; 2-4i; \quad 3^{\circ}\; \cos\alpha + i\sin\alpha - 1; \quad 4^{\circ}\; a+ib+c+i.
$$
Part A: Given complex numbers $z_1 = \cos\dfrac{t}{2} + i\sin\dfrac{t}{2}$ and $z_2 = \dfrac{\sqrt{2}}{2}\sin t + i$, determine the real parameter $t$ from the condition $z_1 = z_2$.
Part B: Given complex numbers $z_1 = 1 + 2ni$ and $z_2 = 3n - 4i$, determine the real parameter $n$ such that $\text{Im}(z_1+z_2) = 0$.
Where do the points lie in the complex plane for which the following conditions are satisfied:
$$
\begin{aligned}
& 1^{\circ}\ \ 2<|z|<3,\ \ \frac{\pi}{4}<\varphi<\frac{\pi}{3}; \quad 2^{\circ}\ \ |z|>1,\ \ \frac{\pi}{3}<\varphi<\frac{\pi}{2}; \\
& 3^{\circ}\ \ |z-z_0|<1; \quad 4^{\circ}\ \ |z-i|<1\ \text{and}\ |z-1|<1; \quad 5^{\circ}\ \ |z|+\text{Re}(z)<2.
\end{aligned}
$$
Show that the point
$$ z = 1+i-\alpha-i\beta, \qquad \text{for} \qquad 0<\alpha<1,\ \ 0<\beta<1,\ \ 0<\alpha+\beta<1 $$
is located in the interior of the triangle with vertices $1,\ 1+i,\ i$.
If the points $z_1, z_2, z_3, z_4$ lie on the same circle, show that the following conditions are satisfied:
$$
\begin{aligned}
\text{Re}(z_1)\text{Im}(z_1) &= \text{Re}(z_2)\text{Im}(z_2)=\text{Re}(z_3)\text{Im}(z_3)=\text{Re}(z_4)\text{Im}(z_4), \\
\text{Re}(z_4) &= \text{Re}(z_1)\text{Re}(z_2)\text{Re}(z_3).
\end{aligned}
$$
If the point $z'$ is located in the semicircle $|z'|<1$ and $|\arg(z')|<\dfrac{\pi}{2}$, show that the point $z=z'^2$ is located in the entire circle $|z|<1$, except for the negative part of the real axis.
The numbers $z_1, z_2, z_3$ are such that:
$$ \arg(z_1)=\arg(z_3)=\frac\pi6; \quad \arg(z_2)=-\frac\pi3; \quad |z_1|=1; \quad |z_3|=\frac{\sqrt2}{2}. $$
Determine $z_2$ such that $\text{Im}(z_1+z_2+z_3)=0$.
If $k$ is a real positive number, determine the complex numbers $z_1$ and $z_2$ from the conditions:
$$ \frac{z_1}{\bar z_2} = i\sqrt2, \qquad z_1\bar z_2 = k. $$
Find the equation of the locus of points satisfied by the complex conjugate roots of the equations:
$$ 1^{\circ}\ \ z^2-2pz+6p-9=0; \qquad 2^{\circ}\ \ z^2-4pz+2p^2-3p+2=0; \qquad 3^{\circ}\ \ z^2-2pz+4p=0. $$
If the circle $|z-s|=r$ passes through the points $a,b,c$, show that $s=\dfrac{\Delta_1}{\Delta}$ and $r^2-|s|^2=-\dfrac{\Delta_2}{\Delta}$, where
$$
\Delta = \begin{vmatrix} 1&a&\bar a\\ 1&b&\bar b\\ 1&c&\bar c \end{vmatrix}, \quad
\Delta_1 = \begin{vmatrix} 1&a&|a|^2\\ 1&b&|b|^2\\ 1&c&|c|^2 \end{vmatrix}, \quad
\Delta_2 = \begin{vmatrix} a&\bar a&|a|^2\\ b&\bar b&|b|^2\\ c&\bar c&|c|^2 \end{vmatrix}.
$$
Find the real and imaginary parts of the numbers:
$$ 1^{\circ}\,\, \frac{3i}{i-2}; \quad 2^{\circ}\,\, \frac{1+i}{2-i}(3+2i); \quad 3^{\circ}\,\, \frac{1-3i}{i+1}-\frac{i}{i+2}; \quad 4^{\circ}\,\, \frac{\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}}{3-4i}; \quad 5^{\circ}\,\, \frac{i}{\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}}. $$
Calculate the value of the expression:
$$ 1^{\circ}\,\, \frac{z-\bar z}{1+z\bar z}, \text{ if } z=1+i; \qquad 2^{\circ}\,\, \frac{z+\bar z}{2z+3}, \text{ if } z=\frac{i-1}{2}. $$
Show that
$$ \text{Re}\left(\frac{b}{a}\right) = \frac{\bar a b+a\bar b}{2|a|^2}; \qquad \text{Im}\left(\frac{b}{a}\right) = \frac{\bar a b-a\bar b}{2i|a|^2}. $$
Given $z=\dfrac{a+tb}{c+td}$, where $t$ is a real parameter and $a,b,c,d$ are complex numbers, show that the point $z$ describes a circle for $t\in(-\infty,\infty)$, except in a case that should be specified.
Let $0<a<2$. Chords of length $a$ are drawn through the point $1$ of the circle $|z|=1$, symmetric about the real axis. Show that
$$ \frac{|1-z|}{1-|z|} < \frac4a $$
holds for every point $z$ in the region bounded by these chords and the part of the circle $\left|z-\frac12\right|=\frac12$ that lies between them.
Show by example that it is not always true that $\arg(ab)=\arg a+\arg b$ when using the principal value determination $-\pi<\arg z\leq\pi$ $(z\in\mathbb{C}^*)$.
A complex number $a$ is unimodular if $|a|=1$. Show that $a$ is unimodular if and only if it can be written as the quotient of two complex conjugate numbers.
If two of the complex numbers $\dfrac{a-b}{c-d}$, $\dfrac{b-c}{a-d}$, $\dfrac{c-a}{b-d}$ are purely imaginary, show that the third is also purely imaginary.
Show that the equation of a line passing through points $z_1$ and $z_2$ has the form
$$ \begin{vmatrix} z & \bar z & 1 \\ z_1 & \bar z_1 & 1 \\ z_2 & \bar z_2 & 1 \end{vmatrix} = 0. $$
Prove the general formulas: $\ 1^{\circ}\,\, \sin^{2n}x=\dfrac{(-1)^n}{2^{2n-1}}\left[\displaystyle\sum_{k=0}^{n-1}\binom{2n}k(-1)^k\cos(2n-2k)x+\dfrac{(-1)^n}2\binom{2n}n\right];$
$\quad2^{\circ}\,\, \sin^{2n+1}x=\dfrac{(-1)^n}{2^{2n}}\displaystyle\sum_{k=0}^{n}\binom{2n+1}k(-1)^k\sin\big((2n+1-2k)x\big).$
Prove the general formula for powers of cosine:
$$ \cos^nx=2^{1-n}\left[\cos nx+n\cos(n-2)x+\frac{n(n-1)}{1\cdot2}\cos(n-4)x+\cdots+A\right] $$
where $A=\dfrac12\dbinom{n}{n/2}$ if $n$ is even, and $A=\dbinom{n}{(n-1)/2}$ if $n$ is odd.
Using the real factorizations $\ x^{2n}-1=(x-1)(x+1)\displaystyle\prod_{k=1}^{n-1}\left(x^2-2x\cos\frac{k\pi}n+1\right)$ and $\ x^{2n+1}-1=(x-1)\displaystyle\prod_{k=1}^{n}\left(x^2-2x\cos\frac{2k\pi}{2n+1}+1\right)$, prove:
$\quad1^{\circ}\,\, \displaystyle\prod_{k=1}^{n-1}\sin\left(\frac{k\pi}{2n}\right)=\frac{\sqrt n}{2^{n-1}};\qquad 2^{\circ}\,\, \displaystyle\prod_{k=1}^{n}\cos\left(\frac{k\pi}{2n+1}\right)=\frac1{2^n}.$
Let $a$ be a fixed positive real number with $a\neq1$. Show that as $t$ ranges over $[0,2\pi)$, the curve $\ z=ae^{it}+\dfrac1{ae^{it}}\ $ traces an ellipse.
Show that the equation $\ \delta z\bar z+a\bar z+\bar az+\gamma=0\ $ (with $\delta,\gamma\in\mathbb R$, $a\in\mathbb C$) represents: $1^\circ$ a circle not through the origin if $\delta\neq0$, $\gamma\neq0$, $\gamma\delta-|a|^2<0$; $2^\circ$ a circle through the origin if $\delta\neq0$, $\gamma=0$; $3^\circ$ a line through the origin if $\delta=\gamma=0$. Conversely, show every circle or line can be written in this form.
Prove that $\ \left|\displaystyle\sum_{k=1}^nz_k\right|=\displaystyle\sum_{k=1}^n|z_k|\ $ (for nonzero $z_k$) if and only if $\arg z_k=\arg z_1$ for $k=2,\ldots,n$.
Prove that $\ \left|\displaystyle\sum_{k=1}^na_kb_k\right|^2\le\left(\displaystyle\sum_{k=1}^n|a_k|^2\right)\left(\displaystyle\sum_{k=1}^n|b_k|^2\right)\ $ for $a_k,b_k\in\mathbb C$.
On the unit sphere $\xi^2+\eta^2+\zeta^2=1$ in $O\xi\eta\zeta$-space, project each point $P=(\xi,\eta,\zeta)$ (other than the north pole $N=(0,0,1)$) from $N$ onto the plane $\zeta=-1$, identified with $\mathbb C$. Find the resulting map $z=f(\xi,\eta,\zeta)$.
Show that a circle on the sphere through the north pole $(0,0,1)$ projects (stereographically) to a line in the $z$-plane, and that every line or circle in the $z$-plane is the image of some circle on the sphere.
What is the stereographic projection of a meridian of geographic longitude $\alpha$ — the half great circle running from the south pole to the north pole at fixed longitude $\alpha$?
Let $m$ be a nonzero integer and $w\in\mathbb C$. What condition on $w$ makes every root $z$ of $\ \left(\dfrac{1+iz}{1-iz}\right)^m=w\ $ real (allowing $z=\infty$ to count as the single non-real exception)?
Suppose $z_1,\ldots,z_n$ all lie in an open half-plane bounded by a line through the origin. Show that $\dfrac1{z_1},\ldots,\dfrac1{z_n}$ also lie in such a half-plane, and that both $\ z_1+\cdots+z_n\neq0\ $ and $\ \dfrac1{z_1}+\cdots+\dfrac1{z_n}\neq0$.
If $z_1+z_2+\cdots+z_n=0$ and not all $z_k$ lie on a common line through the origin, show that every line through the origin has points of $\{z_k\}$ strictly on both sides.
Let $A_1,\ldots,A_n$ be the vertices of a regular $n$-gon inscribed in a circle of radius $R$ centered at $O$. $\ 1^\circ$ Prove that for $M$ on the circle, $\sum_k|MA_k|^2=2nR^2$, independent of $M$'s position. $\ 2^\circ$ Prove that for $M$ anywhere in the plane, $\sum_k|MA_k|^2=n(R^2+\ell^2)$ where $\ell=|OM|$. $\ 3^\circ$ Prove $2^\circ$ still holds for $M$ outside the plane.
Let $M$ lie on the arc $\overparen{A_nA_1}$ of the circle circumscribing a regular $n$-gon $A_1A_2\cdots A_n$ (the short arc between the two consecutive vertices $A_n$ and $A_1$, containing no other vertex). Prove: $\ 1^\circ$ if $n$ is even, $\displaystyle\sum_{k\text{ even}}|MA_k|^2=\sum_{k\text{ odd}}|MA_k|^2$; $\ 2^\circ$ if $n$ is odd, $\displaystyle\sum_{k\text{ even}}|MA_k|=\sum_{k\text{ odd}}|MA_k|$.
For a regular $n$-gon with circumradius $R$, prove: $\ 1^\circ$ the sum of the squares of all sides and diagonals is $n^2R^2$; $\ 2^\circ$ their sum is $nR\cot\dfrac\pi{2n}$; $\ 3^\circ$ their product is $n^{n/2}R^{n(n-1)/2}$.
Show that $\mathbb C\cup\{\infty\}$ (the extended complex plane $\overline{\mathbb C}$) is both an open and a closed set — in itself, as a topological space.
Taking $e^z=e^x(\cos y+i\sin y)$ ($z=x+iy$) as the definition, prove: $\ 1^\circ\ e^{z_1}e^{z_2}=e^{z_1+z_2}$; $\ 2^\circ\ e^{z+2k\pi i}=e^z$ for $k\in\mathbb Z$; $\ 3^\circ$ if $e^{z+w}=e^z$ for all $z$, then $w=2k\pi i$ for some $k\in\mathbb Z$.
Find the modulus and principal argument of: $\ 1^\circ\ e^{\pi i/2};\quad2^\circ\ e^{2-3i};\quad3^\circ\ e^{-3-4i};\quad4^\circ\ e^{-i\varphi}\ (|\varphi|<\pi)$.
Investigate the change of $\arg(\sqrt[n]{z-a})$ as $z$ traces a closed Jordan curve around $a$ once in the positive direction. What kind of point is $z=a$ for $w=\sqrt[n]{z-a}$? Is $z=\infty$ a critical point of this function?
Investigate the change of $w=\sqrt{1-z^2}$ as $z$ traces a closed Jordan curve $c$ in the cases: $\ 1^\circ\ c$ encloses one of $\pm1$; $\ 2^\circ\ c$ encloses both $\pm1$; $\ 3^\circ\ \pm1$ lie outside $c$.
Let $f(z)=z^{1/2}$. Find $f^*(-e)$ where $f^*(z)$ is a single-valued branch with $f^*(1)=1$, extended continuously as $z$ moves through: $1^\circ$ the upper half-plane; $2^\circ$ the lower half-plane.
For $z_n=\dfrac{1+i^n}2$: $\ 1^\circ$ plot several terms; $\ 2^\circ$ find its accumulation points; $\ 3^\circ$ is the sequence bounded; $\ 4^\circ$ accumulation points of $|z_n|$; $\ 5^\circ$ find $\varlimsup|z_n|$ and $\varliminf|z_n|$.
If $z_{n+1}=\dfrac12\left(z_n+\dfrac1{z_n}\right)$ for $n=0,1,2,\ldots$ and $-\dfrac\pi2<\arg z_0<\dfrac\pi2$, then $\lim\limits_{n\to\infty}z_n=1$. Prove this.
For $\lim\limits_{n\to\infty}z_n=a\neq0$ to exist, it is necessary and sufficient that $\lim\limits_{n\to\infty}|z_n|$ (nonzero) and $\lim\limits_{n\to\infty}\arg z_n$ both exist.
For every sequence $(z_n)$ of nonzero complex numbers, show that:
$$ \varliminf_{n\to\infty}\left|\frac{z_{n+1}}{z_n}\right|\le\varliminf_{n\to\infty}\sqrt[n]{|z_n|}\le\varlimsup_{n\to\infty}\sqrt[n]{|z_n|}\le\varlimsup_{n\to\infty}\left|\frac{z_{n+1}}{z_n}\right| $$
If $(z_n)$ is bounded, prove that the following are also bounded: $\ 1^\circ\ \left(1+\dfrac{z_n}n\right)^n;\quad2^\circ\ \dfrac{z_1+\cdots+z_n}n;\quad3^\circ\ \dfrac{z_1+2z_2+\cdots+nz_n}{n^2};\quad4^\circ\ \dfrac{p_1z_1+\cdots+p_nz_n}{p_1+\cdots+p_n}\ (p_i>0);\quad5^\circ\ \sqrt[n]{z_1z_2\cdots z_n}$.
The following functions are not defined at $z=0$: $\ 1^\circ\ \dfrac{\operatorname{Re}(z)}z;\quad2^\circ\ \dfrac z{|z|};\quad3^\circ\ \dfrac{z\operatorname{Re}(z)}{|z|^2}$. (a) Determine whether they have limits as $z\to0$. (b) Can they be extended to $z=0$ to become continuous?
Let $f(z)$ be continuous on all of $\mathbb C$ and satisfy $\ f(z+z')=f(z)+f(z')\ $ and $\ f(zz')=f(z)f(z')\ $ for all $z,z'$. Prove that $f\equiv0$, or $f(z)=z$ for all $z$, or $f(z)=\bar z$ for all $z$.