📘 complex numbers

1. **State the problem:** Find the cube roots of the complex number $i$. 2. **Formula and explanation:** To find the cube roots of a complex number, express it in polar form $z = r

1. **Problem Statement:** Find the multiplicative identity in complex numbers. 2. **Concept:** The multiplicative identity is the complex number which, when multiplied by any compl

1. **State the problem:** Given $\frac{1}{z_1} = 2 + 3i$ and $z_2 = 4 + 2i$, verify that $\frac{z_1}{z_2} = \frac{z_1}{z_2}$. This means we need to find $z_1$ from the first equati

1. **State the problem:** We are given two complex numbers in polar form: $z = 2 \operatorname{cis}(-\frac{\pi}{4})$ and $w = \sqrt{3} \operatorname{cis}(\frac{\pi}{6})$. We need t

1. **State the problem:** We are given two complex numbers in polar form:

1. **Stel het probleem vast:** We hebben een complex getal met modulus 5 en argument $\frac{\pi}{2}$. We moeten de correcte notatie van dit complex getal vinden in de vorm $a + bj$

1. **State the problem:** Express $$3 \frac{e^{j400} + e^{-2j100}}{e^{j00}}$$ in terms of cosine only.

1. The problem is to simplify the expression $|i \times \sqrt{3}|$. 2. Recall that $i$ is the imaginary unit with magnitude $|i|=1$.

1. **Stating the problem:** We are given the complex number $Z = 1 - i$ and we want to analyze it. 2. **Formula and rules:** A complex number is generally written as $Z = a + bi$,

1. مسئله: ساده کردن عبارت $$z = \frac{(1 + i\sqrt{3})^4 (1 - i)^5}{2\sqrt{3} (1 - i\sqrt{3})^3 (1 + i)^4}$$

1. مسئله: ساده کنید $$z = \frac{(1 + i\sqrt{3})^4 (1 - i)^5}{2\sqrt{2} (1 - i\sqrt{3})^3 (1 + i)^4}$$

1. **Énoncé du problème** : Résoudre dans $\mathbb{C}$ l'équation quadratique $(E_1) : z^2 - (3 - 2i) z + 5 - zi = 0$. 2. **Formule utilisée** : Pour une équation quadratique $az^2

1. **Énoncé du problème :** Déterminer le module et un argument des nombres complexes suivants :

1. The problem is to evaluate the magnitude of the complex number $\frac{5i}{3 - i}$.\n\n2. Recall that the magnitude of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2

1. **State the problem:** Given a complex number $z = 4 + 5i$, find the absolute value $|z|$ and the product $z \cdot z^*$, where $z^*$ is the conjugate of $z$. 2. **Recall formula

1. Problem statement: Given the complex number $z_1 = 2 + i$, find complex numbers $z = x + iy$ satisfying the given conditions. 2. Recall that for complex numbers $z = x + iy$ and

1. **State the problem:** We are given the complex number $$-2 - i\sqrt{3}$$ and it is expressed as $$x + iy$$ where $$x$$ and $$y$$ are real numbers. We need to find the value of

1. **State the problem:** Evaluate $\left( \sin \frac{\pi}{9} + i \sin \frac{7\pi}{18} \right)^{-6}$ using De Moivre's theorem. 2. **Recall De Moivre's theorem:** For a complex num

1. **State the problem:** Convert the complex number $0 + j80$ to its polar form. 2. **Recall the polar form of a complex number:** A complex number $z = x + jy$ can be expressed i

1. **State the problem:** Convert each complex number $z_k$ into the form $\cos\theta + i\sin\theta$ by finding the angle $\theta$.

1. The problem is to express each complex number $z_k = x + yi$ in the form $\cos \theta + i \sin \theta$, where $\theta$ is the argument (angle) of the complex number. 2. Recall t