Let's break down and solve the problems from the image.

Exercice 1

The complex number $j$ is given as:

$j = -\frac{1}{2} + \frac{\sqrt{3}}{2} i$

This is in polar form corresponding to $e^{i\theta}$, where $\theta = \frac{2\pi}{3}$.

1. Calculer $j^2$

   Use the binomial formula or polar form to calculate $j^2$.

2. En déduire que $j^3 = 1$

   If $j^2$ is calculated, multiplying by $j$ again should give 1.

3. Calculer $j^{10}$ et $j^{2024}$

   Using the periodicity property of powers of complex numbers on the unit circle (since $j^3 = 1$, then $j^{3k} = 1$ for any integer $k$), reduce the powers modulo 3.

Exercice 2

In both cases, determine if the propositions are true or false.

1. Proposition 1: Si $z + \overline{z} = 0$, alors $z = 0$

   Justify if the real part of $z$ is zero when this condition holds.

2. Proposition 2: Z est un nombre réel

   Use the expression $Z = z + \overline{z} - 3i$ and check whether it leads to a real number.

Exercice 3

Solve the equation $4z + (2 - i)\overline{z} = 3 + 2i$

• Express $z = x + yi$ and $\overline{z} = x - yi$, then solve by separating real and imaginary parts.

Exercice 4

Gaussian integers are of the form $a + bi$, where $a$ and $b$ are integers.

1. Show that the set of Gaussian integers is stable under addition.

2. Is it stable under multiplication? Justify.

3. Is the inverse of a Gaussian integer necessarily a Gaussian integer? Justify.

Gaussian integers are closed under addition, but their behavior under other operations needs verification.

Would you like detailed solutions for each question? Here are some related questions:

1. How does the polar form help simplify powers of complex numbers?
2. What properties of complex conjugates are important in solving proposition 1?
3. What methods are useful for solving systems of real and imaginary parts in complex equations?
4. How do Gaussian integers relate to integer factorization?
5. Why are Gaussian integers important in number theory?

Tip:

For complex numbers, expressing them in polar form often simplifies exponentiation tasks, leveraging Euler's formula.