Pour z
𝑧
 et z′
𝑧
′
 complexes :

 

Si  z≠z′
𝑧
≠
𝑧
′
 alors Arg(z)≠Arg(z′)
𝐴
𝑟
𝑔
(
𝑧
)
≠
𝐴
𝑟
𝑔
(
𝑧
′
)
 et |z|≠|z′|
|
𝑧
|
≠
|
𝑧
′
|
 ?

Question 3Veuillez choisir une réponse.
Vrai
Faux

The statement in the question is asking whether for two different complex numbers \( z \) and \( z' \), the arguments (\( \text{Arg}(z) \) and \( \text{Arg}(z') \)) and the magnitudes (\( |z| \) and \( |z'| \)) must also be different.

Let's break this down:

1. **Different Complex Numbers \( z \) and \( z' \)**: Two complex numbers \( z \) and \( z' \) are considered different if either their magnitudes or their arguments differ. 

2. **Magnitude \( |z| \) and \( |z'| \)**: It is possible for two different complex numbers to have the same magnitude but different arguments. For example, \( z = 1 \) and \( z' = -1 \) both have a magnitude of 1, but their arguments are different (\( 0 \) and \( \pi \), respectively).

3. **Argument \( \text{Arg}(z) \) and \( \text{Arg}(z') \)**: It is also possible for two different complex numbers to have the same argument but different magnitudes. For example, \( z = 1 \) and \( z' = 2 \) both have an argument of 0, but their magnitudes are 1 and 2, respectively.

Given these points, the statement:

"If \( z \neq z' \), then \( \text{Arg}(z) \neq \text{Arg}(z') \) and \( |z| \neq |z'| \)" 

is **False** because two different complex numbers can have either the same magnitude or the same argument but not both simultaneously.

### Would you like more details on this topic or have any questions? 

Here are 5 related questions you might find interesting:

1. What is the geometric interpretation of the argument and magnitude of a complex number?
2. How does the polar form of a complex number relate to its Cartesian form?
3. Can two different complex numbers have the same modulus and argument? Why or why not?
4. How do you calculate the argument of a complex number in different quadrants?
5. What is the relationship between complex conjugates and their arguments?

**Tip**: Always remember that the magnitude of a complex number is its distance from the origin in the complex plane, while the argument is the angle it makes with the positive real axis.

Pour z et z′ complexes : Si z≠z′ alors Arg(z)≠Arg(z′) et |z|≠|z′| ?

Explore whether two different complex numbers must have different arguments and magnitudes. Learn key concepts of complex number theory, including Arg(z) and |z|, with step-by-step explanations. Suitable for students in Grades 11-12.

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