📘 complex numbers

1. **State the problem:** We are given a complex number $z = x + iy$ that satisfies the equation $$\frac{z}{z+2} = 2 - i.$$ We want to find the values of $x$ and $y$. 2. **Write th

1. **Stating the problem:** Given the complex number $z=x-iy$, we want to find $x$ such that $$|z+3|-|z|=0.$$ 2. **Rewrite the problem:** This means $$|z+3| = |z|.$$

1. **Stating the problem:** We want to find the form of a complex number $z$ if $z$ is equal to the multiplicative inverse of its conjugate $\overline{z}$. That is, $$z = \frac{1}{

1. The problem asks us to find the complex number obtained by rotating the complex number $2+i$ by 90° anticlockwise. 2. A rotation of 90° anticlockwise in the complex plane corres

1. **State the problem:** Find all complex solutions to the equation $$(z+1)^3 = (\overline{z} - 2 + \sqrt{3}i)^3.$$\n\n2. **Rewrite the equation:** Since both sides are cubes and

1. **State the problem:** Express the complex number $1 + i$ in polar form. 2. **Recall the polar form:** A complex number $z = x + yi$ can be represented in polar form as $$z = r(

1. Stating the problem: Simplify the complex numbers $Z_1 = \frac{-1}{i}$ and $Z_2 = \frac{1}{2} + i\frac{\sqrt{3}}{2} \div -\sqrt{3} - i$.\n\n2. Simplify $Z_1$: Recall that $\frac

1. **State the problem:** We want to express the roots of $\left(4-3i\right)^{-\frac{2}{3}}$ in polar form. 2. **Convert the base to polar form:** The complex number $4-3i$ has mod

1. **Énoncé du problème** : Nous devons déterminer l'ensemble des points $M$ d'affixe $z$ tels que :