Exercises: Operations with complex numbers in Cartesian form

Carry out the following addition and subtraction operations with complex numbers: given $z_1 = 3 - 4i$, $z_2 = -1 + 2i$ and $z_3 = 5i$, calculate the result of the expression $z_1 - z_2 + z_3$ in Cartesian form.

Multiply the following complex numbers expressed in Cartesian form and simplify the result by grouping the real part and the imaginary part: $(2 + 5i)(-3 + i)$.

Perform the following division of complex numbers by multiplying the numerator and denominator by the conjugate of the denominator. Express the final result in the form $a + bi$:

$\frac{4 - 3i}{1 + 2i}$

Simplify the following powers of the imaginary unit $i$ applying the periodicity of its values:

• $i^{45}$

• $i^{102}$

• $i^{2024}$

• $i^{-13}$

Given the complex number $z = -7 + 8i$, determine its conjugate $\bar{z}$ and its opposite $-z$. Then, perform the operation $z + \bar{z}$ and check algebraically that the result is a real number.

Solve the following quadratic equation in the set of complex numbers $\mathbb{C}$ and express its two solutions:

$x^2 + 25 = 0$

Determine the values of the real numbers $x$ and $y$ such that the following equality between complex numbers holds:

$(x + 2) + (y - 3)i = 5 + 4i$

Calculate the value of the polynomial $P(z) = z^2 - 4z + 5$ when evaluated at the complex number $z = 2 + i$.

Lucas is trying to simplify the fraction of complex numbers $\dfrac{5+2i}{2-i}$ to express it in standard form. On his exam, he writes down the following steps:

1. a) Identify, with reasoning, the conceptual error Lucas made in the denominator and explain why his calculation of $(2-i)(2+i)$ contradicts the rule for multiplying by the conjugate.
2. b) Solve the operation correctly and express the final result in standard form $a + bi$.

The goal is to construct a quadratic polynomial $P(x) = x^2 + bx + c$ with real coefficients ($b, c \in \mathbb{R}$) such that one of its roots is the complex number $z_1 = 2 - 3i$.

1. a) Without performing algebraic operations, justify what the second root $z_2$ of this polynomial must necessarily be, relying on the properties explained regarding equations with real coefficients.
2. b) Obtain the exact values of the coefficients $b$ and $c$ by expanding the product of factors $(x - z_1)(x - z_2)$ in standard form and substituting $i^2 = -1$.

In an alternating current electrical circuit, two components are connected in parallel. Their complex impedances are given by $Z_1 = 4 + 3i \ \Omega$ and $Z_2 = 2 - i \ \Omega$. Calculate the equivalent impedance of the circuit, expressed in Cartesian form, using the physics formula:

$Z_{eq} = \frac{Z_1 \cdot Z_2}{Z_1 + Z_2}$

Determine the value of the real parameter $k$ so that the result of the following quotient is a pure real number (that is, its imaginary part is zero):

$\frac{k + 2i}{3 - 4i}$

In 2D video game development, the position of a character in the plane is represented by the complex number $z = 5 + 2i$. To apply an anticlockwise rotation of $90^\circ$ about the origin and, subsequently, a translation defined by the vector $(-1, 3)$, the graphics engine performs the operation $z' = z \cdot i + (-1 + 3i)$. Calculate the new position $z'$ of the character.

Expand and simplify the expression $(1 - i)^4$ working exclusively in Cartesian form. You may use the binomial expansion or apply the properties of powers $( (1-i)^2 )^2$.

In the study of ideal fluid flow around a cylindrical obstacle, the complex potential function is given by the expression $W(z) = V \cdot z + \frac{m}{z}$. Given that the fluid velocity is $V = 2$ and the parameter $m = 8$, evaluate the potential at the point $z = 2 + 2i$. Express the final result $W(2+2i)$ in the form $a + bi$.

Perform the combined operations indicated below, respecting the order of operations, and express the final result in Cartesian form:

$\frac{(3 - i)(1 + 2i)}{2 + i} - (1 - 3i)$

Find the value of the real number $x$ so that the product of the complex numbers $(x - 3i)$ and $(2 + 5i)$ results in a pure imaginary number (that is, with a real part equal to zero).

In a simplified mathematical model of quantum mechanics, the state of a particle is described by the probability amplitude $\psi = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i$. Check whether this state satisfies the normalisation condition, which requires the product of the complex number and its conjugate to be exactly one ($\psi \cdot \bar{\psi} = 1$).

To calculate the reciprocal of the complex number $z = \dfrac{1+2i}{3-i}$ in standard form, two classmates propose different approaches:

• Alejandro proposes first computing the division $\dfrac{1+2i}{3-i}$ by multiplying by the conjugate of the denominator and, once the result $A + Bi$ is obtained, finding its reciprocal as $\dfrac{1}{A+Bi}$ by multiplying again by its conjugate.

• Berta proposes first algebraically inverting the fraction, writing $z^{-1} = \dfrac{3-i}{1+2i}$, and then performing a single division by multiplying by the conjugate of this new denominator.

1. a) Compare both approaches: justify whether both are mathematically valid and which of them minimizes arithmetic operations and the risk of error.
2. b) Apply the chosen method to calculate $z^{-1}$ and express the result in the form $a + bi$.

Consider the multiplication of the complex numbers $z_1 = 5 + 0,2i$ and $z_2 = 4 - 0,1i$.

1. a) Before operating exactly, reasonably estimate the approximate value of the real part and the imaginary part of $z_1 \cdot z_2$. Justify your estimate by analyzing which of the four terms of the distributive product $(ac - bd) + (ad + bc)i$ will have a significant impact and which will be nearly negligible.
2. b) Perform the complete multiplication in standard form, substituting $i^2 = -1$.
3. c) Compare the exact value with your initial estimate, calculate the difference in the real part, and explain in which practical situations (such as calculations with measurement tolerances) it may be acceptable to neglect the term $-bd$.

Solve the following system of linear equations where the unknowns $z$ and $w$ are complex numbers. Express both solutions in Cartesian form:

$\begin{cases} (1+i)z - w = 2 + i \\ 2z + (1-i)w = 3 - i \end{cases}$

Find the complex numbers $z = x + yi$ that are square roots of the complex number $-5 + 12i$. To do this, set up the equation $(x + yi)^2 = -5 + 12i$, equate the real parts and the imaginary parts to form a system of non-linear equations, and solve it in $\mathbb{R}$.

Determine the locus of the affixes of the complex numbers $z = x + yi$ such that the expression $\frac{z - 2}{z + i}$ is a purely imaginary number (that is, with zero real part and nonzero imaginary part).

1. a) Obtain the equation of the locus in terms of $x$ and $y$.

1. b) Identify the curve: state its center and its radius.

1. c) State which points of that curve must be excluded and why.

Prove algebraically, defining the complex numbers in Cartesian form as $z = a + bi$ and $w = c + di$, that the conjugate of the product of two complex numbers is identical to the product of their respective conjugates. That is, prove step by step that $\overline{z \cdot w} = \overline{z} \cdot \overline{w}$.

In dynamical systems control theory, a system is considered stable if all roots of its characteristic polynomial have a negative real part. Given the characteristic equation $z^3 - 4z^2 + 6z - 4 = 0$:

1. Check by algebraic substitution whether $z_1 = 1 + i$ is a solution to the equation.

2. Knowing that polynomials with real coefficients have complex conjugate roots, find the other two solutions to the equation.

3. Conclude whether the described system is stable or unstable.

Find all complex numbers $z$ satisfying the non-linear equation $z^2 + 3\bar{z} = 0$. Consider $z = x + yi$ and solve the resulting system by analysing the different possible cases for $x$ and $y$.

In a fractal generation algorithm, the position of each new node in the complex plane is calculated from the sum of consecutive powers. Calculate the position of the node given by the sum of the first 10 terms of the geometric progression:

$S = \sum_{k=0}^{9} (1+i)^k$

Apply the formula for the sum of terms of a geometric progression and express the final result in Cartesian form.

In the study of fractals and the Mandelbrot set, the behaviour of the iterative sequence defined by $z_{n+1} = z_n^2 + c$ is analysed, starting from the initial condition $z_0 = 0$. For the complex constant $c = -1 + i$:

1. Calculate the terms $z_1$, $z_2$ and $z_3$ working in Cartesian form.

2. Determine the modulus $|z_3|$ and check whether it exceeds the escape threshold value (that is, whether $|z_3| > 2$).

In an industrial engineering project at a photovoltaic plant in Sevilla, two branches are connected in parallel, whose equivalent impedance is given by $Z_p = \dfrac{Z_1 \cdot Z_2}{Z_1 + Z_2}$. The first branch has an impedance of $Z_1 = 3 + i\ \Omega$, while the second branch has an impedance of $Z_2 = R - 2i\ \Omega$, where $R > 0$ is an adjustable ohmic resistance in ohms.

The technicians need to adjust $R$ so that the circuit enters parallel resonance, which means that the overall impedance $Z_p$ must be a purely real number (its imaginary part must be zero), avoiding phase shifts between voltage and current.

1. a) Expand the expression of $Z_p$ in standard form in terms of the parameter $R$, multiplying the numerator and denominator by the appropriate conjugate.
2. b) Set the resulting imaginary part equal to zero to calculate the exact value of the resistance $R$ that must be chosen.
3. c) With that value of $R$, find the final value of the equivalent resistance $Z_p$ of the circuit.

A student makes the following claim in class: "Since for any quadratic equation with complex solutions, these come in conjugate pairs, then in the equation $x^2 - (3+i)x + (4+3i) = 0$, knowing that one solution is $x_1 = 1+2i$, the other solution must necessarily be its conjugate $x_2 = 1-2i$."

1. a) Check by direct substitution and operations in standard form whether $x_1 = 1+2i$ and its supposed conjugate $1-2i$ satisfy the given quadratic equation.
2. b) Explain the student's error in reasoning: what essential condition indicated in the notes does this equation fail to meet for the conjugate root theorem to apply?
3. c) Using the relations between coefficients and roots for the sum ($x_1 + x_2 = 3+i$), calculate the true second root $x_2$ in standard form and verify that the product $x_1 \cdot x_2$ equals the constant term $4+3i$.