Practice Questions

Explain the concept of the modulus of a complex number $z = x + iy$ and state its formula.

Express $(2i)^3$ in the standard form $a+ib$.

Critique the statement: "For any complex number $z$, the number $z + \bar{z}$ is always a real number." Is this statement always true? Justify your reasoning.

Define a complex number and provide one example.

What is the value of $i^2$?

Identify the real and imaginary parts of the complex number $z = -3 + 7i$.

What is the additive identity for complex numbers?

Define the conjugate of a complex number $z = a + ib$.

Explain the condition for two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ to be equal. Using this, find the real numbers $x$ and $y$ if $2x + i(y-1) = 4 + 5i$.

List the values of the first four positive integer powers of $i$ (i.e., $i^1, i^2, i^3, i^4$).

Given $z_1 = 2 + 3i$ and $z_2 = 1 - i$, explain how to find the sum $z_1 + z_2$.

Calculate the value of the expression $i^{49} + i^{68} + i^{89} + i^{110}$.

Find the multiplicative inverse of the complex number $z = 1 - i$.

Solve for the real numbers $x$ and $y$ if $(2x - 1) + 5i = 7 + (y + 3)i$.

Solve the quadratic equation $x^2 - 4x + 13 = 0$.

Justify the identity $|z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2 \text{Re}(z_1 \bar{z_2})$ for any two complex numbers $z_1$ and $z_2$.

If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1 + z_2| = |z_1| + |z_2|$, create an argument to determine the relationship between the arguments of $z_1$ and $z_2$.

A complex number $z$ satisfies the equation $|z - 5i| = |z + 5i|$. Justify that $z$ must lie on the real axis. Further, if it is also given that $|z|=5$, create the specific value(s) for $z$.

If $(1+i)^n = (1-i)^n$ for some integer $n$, formulate the general condition that $n$ must satisfy. Derive the smallest positive integer value for $n$.

Let $z = x+iy$. Derive a relationship between $x$ and $y$ if the complex number $w = \frac{z-2}{z+2}$ has a modulus of 2.

If $(x+iy)^3 = u+iv$, demonstrate that $\frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2)$.

State the commutative and associative laws for the addition of complex numbers.

Calculate the modulus of the complex number $z = (1+i)(2-i)$.

Express the complex number $(1-i)^4$ in the form $a+ib$.

Formulate a general rule for the value of $i^n + i^{n+1} + i^{n+2} + i^{n+3}$ for any integer $n$, and justify your answer.

Evaluate $\text{Im}(\frac{1}{z \bar{z}})$ for any non-zero complex number $z=x+iy$, and justify your conclusion.

Describe the multiplicative inverse of a non-zero complex number $z = a + ib$ and state its formula.

Define the Argand plane. What do the horizontal and vertical axes represent in it?

Find the conjugate of the complex number $z = \frac{(3-i)^2}{2+i}$.

Calculate the modulus of the complex number $z = \frac{1+i}{1-i}$.

List and explain the five fundamental properties of addition for complex numbers.

Explain the process of multiplying two complex numbers, $z_1 = a + ib$ and $z_2 = c + id$. State the general formula for the product $z_1 z_2$ and provide an example using $z_1 = 3+2i$ and $z_2 = 1-4i$.

Describe the relationship between a complex number $z$, its conjugate $\bar{z}$, and its modulus $|z|$. Show that $z \bar{z} = |z|^2$ for any complex number $z = a + ib$.

Formulate a quadratic equation with real coefficients, given that one of its roots is $3-2i$. Justify why the other root must be its conjugate.

If $z_1 = 2-i$ and $z_2 = -2+i$, calculate the value of $\operatorname{Re}\left(\frac{z_1 z_2}{\bar{z_1}}\right)$.

Given that $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 2x + 4 = 0$. Evaluate the expression $\alpha^3 + \beta^3$.

If $z_1, z_2, z_3$ are complex numbers representing the vertices of an equilateral triangle in the Argand plane, prove the relation $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$.

Propose a condition on the complex number $z$ such that $\frac{z-1}{z+1}$ is purely imaginary.

Express the complex number $z = \frac{i-1}{\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3})}$ in the standard form $a+ib$.

Prove that if $z$ is a complex number such that $|z|=1$, then $w = \frac{z-1}{z+1}$ (for $z \neq -1$) is a purely imaginary number.

Design a proof to show that for any two complex numbers $z_1$ and $z_2$, $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$. Interpret this result geometrically in the Argand plane.

Find the square roots of the complex number $-5 + 12i$.

If $z$ is a complex number such that $|z|=1$ and $z \neq -1$, analyze the expression $w = \frac{z-1}{z+1}$ and show that it is a purely imaginary number.

Convert the complex number $z = \frac{-16}{1+i\sqrt{3}}$ into polar form, $r(\cos\theta + i\sin\theta)$.

Critique the following statement: "The equation $z^2 + |z| = 0$ has only non-real solutions." Justify your conclusion by finding all solutions in the complex plane.