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The exercise you shared has two parts. The first part (question 21) deals with a real number \( x \), and the second part (question 22) is about the approximation of \(\pi\) by the Chinese mathematician Zu Chongzhi. Let's address each part:

### Question 21:
You are given a real number \( x \) such that \( 2.45876 < x < 2.45877 \).

- **a. Donner un encadrement de \( x \) à \( 10^{-3} \) près.**
  
  To provide an interval (encadrement) for \( x \) to the nearest \( 10^{-3} \) (which is 0.001), you need to round the lower and upper bounds to three decimal places:

  - Lower bound: \( 2.45876 \) rounds to \( 2.459 \).
  - Upper bound: \( 2.45877 \) rounds to \( 2.459 \).
  
  So the interval is:
  \[
  x \approx [2.459, 2.459]
  \]
  
  This suggests that \( x \) would be approximately 2.459 to \( 10^{-3} \) precision.

- **b. Donner la troncature de \( x \) à \( 10^{-4} \) près.**

  Truncating \( x \) to \( 10^{-4} \) (which is 0.0001) means cutting off after the fourth decimal place:

  - Truncation of \( x \): \( 2.4587 \) (since we remove the digits after the fourth decimal place).

- **c. Donner l'arrondi de \( x \) à \( 10^{-2} \) près.**

  Rounding \( x \) to the nearest \( 10^{-2} \) (which is 0.01):

  - Both \( 2.45876 \) and \( 2.45877 \) round to \( 2.46 \).

### Question 22:
- **Au \( V^{\text{e}} \) siècle, le mathématicien chinois Zu Chongzhi proposait le nombre rationnel \(\frac{355}{113}\) comme approximation de \(\pi\). Était-ce une valeur approchée par excès ou par défaut?**

  Zu Chongzhi's approximation for \(\pi\) is given as \(\frac{355}{113}\).

  First, let's calculate the decimal value of \(\frac{355}{113}\):

  \[
  \frac{355}{113} \approx 3.1415929203539825
  \]

  The actual value of \(\pi\) is approximately:

  \[
  \pi \approx 3.141592653589793
  \]

  Comparing the two:

  \[
  3.1415929203539825 > 3.141592653589793
  \]

  Therefore, Zu Chongzhi's approximation is slightly greater than the true value of \(\pi\), meaning it is an **overestimate** (par excès).

Would you like more details or have any questions?

Here are some related questions:
1. What would the truncation of \( x \) be if rounded to \( 10^{-2} \)?
2. How do you calculate a decimal approximation of a fraction?
3. What other historical approximations of \(\pi\) exist?
4. Can you express \(\pi\) using continued fractions?
5. How accurate is Zu Chongzhi's approximation compared to modern approximations?

**Tip:** For rounding, always consider the next digit to the right to determine whether to round up or down.

Au Vᵉ siècle, le mathématicien chinois Zu Chongzhi proposait le nombre rationnel 355/113 comme approximation de π. Était-ce une valeur approchée par excès ou par défaut?

Explore how the mathematician Zu Chongzhi approximated the value of Pi (π) in the 5th century using the rational number 355/113. Understand if this value is an overestimate or an underestimate compared to the true value of π. Suitable for Grades 10-12.

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