Operations with Complex Numbers in Cartesian Form

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

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

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

Simplify the following powers of the imaginary unit $i$ by 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.

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 if this state satisfies the normalization condition, which requires that the product of the complex number and its conjugate is exactly unity ($\psi \cdot \bar{\psi} = 1$).

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 $\overline{z \cdot w} = \overline{z} \cdot \overline{w}$.

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

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

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

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

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

$x^2 + 25 = 0$

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

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

1. Check by substituting algebraically if $z_1 = 1 + i$ 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.

Determine the values of the real numbers $x$ and $y$ so that the following equality between complex numbers is satisfied:

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

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

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

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

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 physical formula:

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

In the development of a 2D video game, the position of a character on the plane is represented by the complex number $z = 5 + 2i$. To apply a counterclockwise rotation of $90^\circ$ with respect to 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.

In the study of the flow of ideal fluids around a cylindrical obstacle, the complex potential function is given by the expression $W(z) = V \cdot z + \frac{m}{z}$. Knowing 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 $a + bi$.

Determine the equation of the locus of the points corresponding to the complex numbers $z = x + yi$ that satisfy the condition that the expression $\frac{z - 2}{z + i}$ is a purely imaginary number.

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 non-linear system of equations, and solve it in $\mathbb{R}$.

Find all complex numbers $z$ that satisfy 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 on 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 the terms of a geometric progression and express the final result in Cartesian form.

In the study of fractals and the Mandelbrot set, the behavior 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 $z_3$ operating in Cartesian form.

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