Operations with Complex Numbers in Binomial Form

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

• $i^{45}$

• $i^{102}$

• $i^{2024}$

• $i^{-13}$

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.

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.

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

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

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

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 $z' = z \times i + (-1 + 3i)$. 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 $( (1-i)^2 )^2$.

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

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 $a + bi$:

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

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):

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

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:

$x^2 + 25 = 0$

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 $\mathbb{R}$.

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 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.

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

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

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 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 $z_3$ working in Cartesian form.

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

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

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:

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

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

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 ($\psi \times \bar{\psi} = 1$).