📘 complex numbers

1. **Problem (a):** Given $u = 3 - 7i$ and $\overline{u}$ is the complex conjugate of $u$, find $2\overline{u} + 5iu$ in the form $a + bi$. 2. The complex conjugate $\overline{u}$

1. **State the problem:** We are given two complex numbers $Z_1 = -1 + 2i$ and $Z_2 = 2 + 3i$. We need to find their sum $Z_1 + Z_2$. 2. **Formula used:** The sum of two complex nu

1. **State the problem:** Express the complex number $-8 - 8\sqrt{3}i$ in polar form $r(\cos\theta + i\sin\theta)$, then use de Moivre's theorem to find the four roots of the equat

1. **Problem statement:** Find the two values of $$\sqrt{2(1 - \sqrt{3}i)}$$ using De Moivre's theorem, expressing each solution in the form $$a + bi$$ where $$a,b \in \mathbb{R}$$

1. **State the problem:** Simplify the complex fraction $$\frac{3+2i}{4-i}$$ by dividing. 2. **Formula and rule:** To divide complex numbers, multiply numerator and denominator by

1. **Énoncé du problème :** Démontrer que l’ensemble des racines 3-ièmes de l’unité est $U_3 = \{1, j, j^2\}$ avec $j = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$.

1. **State the problem:** We want to express the complex number $1 \pm \frac{i}{2}$ in the form $\frac{\sqrt{5}}{2}(\cos\varphi + i\sin\varphi)$ and find the angle $\varphi$. 2. **

1. **State the problem:** Express $$\frac{\sqrt{3} + i}{1 + \sqrt{3}i}$$ in the form $$r(\cos \theta + i \sin \theta)$$ and then evaluate $$\left(\frac{\sqrt{3} + i}{1 + \sqrt{3}i}

1. **State the problem:** Given the complex number $u=2(\cos(\frac{\pi}{5})+i\sin(\frac{\pi}{5}))$, express it in exponential form and understand its meaning.

1. **State the problem:** We have complex numbers $z_1 = 2 + i$, $z_2 = 3 + 4i$, and $z_3 = \overline{z_1} = 2 - i$ (the conjugate of $z_1$).

1. **Stating the problem:** We are asked to find the value of the expression $$\frac{|z_2|}{|z_1|}$$ where $z_1$ and $z_2$ are complex numbers.

1. 문제를 이해하기: 복소수 $z$에 대해 $\frac{1 - i}{z} = \frac{1}{\sqrt{2}} i$가 주어졌습니다. 여기서 $i = \sqrt{-1}$입니다. 2. 주어진 식을 $z$에 대해 풀기 위해 양변에 $z$를 곱합니다:

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

1. **Énoncé du problème :** Montrer que pour tout réel $\theta$, on a $$1 - e^{i\theta} = -2i \sin\left(\frac{\theta}{2}\right) e^{i\frac{\theta}{2}}.$$ 2. **Formule utilisée :** O

1. **Énoncé du problème :** Résoudre les équations complexes données dans \(\mathbb{C}\) et exprimer certains nombres complexes sous forme exponentielle. ---

1. **Énoncé du problème** : Trouver la forme exponentielle du nombre complexe $z_3 = 1 - \frac{1}{2} - i \frac{\sqrt{3}}{2}$. 2. **Simplification de $z_3$** :

1. **Énoncé du problème** : Montrer que pour tout réel $\theta$, on a $$1 - e^{i\theta} = -2i \sin \left(\frac{\theta}{2}\right) e^{i\frac{\theta}{2}}.$$ 2. **Formule utilisée** :

1. **Calculate the expression** $$\frac{7 - 5i}{129 + 6i} + \frac{1111 + 47i}{20031}$$ 2. **Find the square roots of** $$1 + i\sqrt{3}$$ in the form $$a + bi$$ where $$a,b \in \mat

1. **Énoncé du problème :** Nous devons donner une forme trigonométrique des nombres complexes $a=2i$, $b=\sqrt{3}+i$ et $c=\sqrt{2}+\sqrt{2}i$.

1. **Placer les points A, B et C sur le repère** On place les points selon leurs affixes complexes dans le plan complexe muni du repère \( (O, \vec{u}, \vec{v}) \).

1. **Énoncé du problème :** Montrer que $$\left(\frac{z_4}{4}\right)^{2016} \in \mathbb{R}$$ avec $$z_4 = (\sqrt{6} + \sqrt{2}) + i(\sqrt{6} - \sqrt{2})$$.