Let's answer each part step-by-step: --- ### **1. Underline all significant digits in the following numbers:** #### (a) 1230000370000 **Significant digits** are all nonzero digits and any zeros between or after nonzero digits in the decimal part. Trailing zeros in a whole number with no decimal are NOT significant unless specified. **Here:** 1, 2, 3, 0, 0, 0, 0, 3, 7 are significant. The final four zeros are not significant. **Underline:** **1 2 3 0 0 0 0 3 7** 0 0 0 0 --- #### (b) 0.000036500000 **Leading zeros** (zeros before the first nonzero digit) are NOT significant. Zeros **after** nonzero digits in the decimal are significant. **Here:** 3, 6, 5, 0, 0, 0, 0, 0 are significant. **Underline:** 0.0000**3 6 5 0 0 0 0 0** --- ### **2. Round the following numbers to 5 decimal places:** #### (a) 0.999999999999 First 5 decimals: 0.99999 The next digit is 9, so we round up. **Rounded:** **1.00000** --- #### (b) 0.7844250000000 First 5 decimals: 0.78442 The next digit is 5, so we round up. **Rounded:** **0.78443** --- ### **3. Fill in the missing words:** Suppose that a number \( x \) is **approximated** by another number \( y \) (where \( x, y \in R \)). Then \( |x-y| \) is called **the absolute error**. --- ### **4. Complete the sentence:** Loss of significant digits occurs when subtracting **nearly equal** numbers. --- ### **5. Fill in the missing mathematical symbols:** Algorithm \( \mathcal{A} \) is called iterative if there is a sequence of operations \( \rightarrow \) that generates a sequence of points \( x_k \), \( k = 1, 2, \ldots \), such that \( x_{k+1} = \mathcal{A}(x_k) \). Or more simply: ...generates a sequence of points \( x_k, k = 1, 2, \ldots \), such that \( x_{k+1} \rightarrow x_k \). --- ### **6. Complete the sentence:** A computational problem is well-conditioned if **small** changes in the data cause **small** changes in the results. --- ## **Summary Table of Answers:** | # | Answer | |----|--------| | 1a | **1 2 3 0 0 0 0 3 7** 0 0 0 0 | | 1b | 0.0000**3 6 5 0 0 0 0 0** | | 2a | 1.00000 | | 2b | 0.78443 | | 3 | approximated, the absolute error | | 4 | nearly equal | | 5 | \( x_k, k=1,2,\ldots \), such that \( x_{k+1} \) | | 6 | small, small | --- **Let me know if you want explanations for any step!**

Certainly! Here's a step-by-step approach with explanations for each part, followed by the final answers: --- ### **Part 1: Underline all significant digits in the following numbers** #### Step 1: Identify significant digits: - For **(a) 1230000370000**: Significant digits are all nonzero digits and zeros between nonzero digits. Zeros at the end of a number without a decimal point are generally not significant unless specified. - For **(b) 0.000036500000**: Leading zeros are not significant. The digits after the first nonzero digit in the decimal are significant, including zeros between nonzero digits and zeros at the end. #### Step 2: Underline the significant digits: - **(a)**: 1 2 3 0 0 0 0 3 7 **(zeros after 7 are not significant in a whole number without decimal)** **Answer:** **1 2 3 0 0 0 0 3 7** - **(b)**: 0.0000**3 6 5 0 0 0 0 0** **Answer:** 0.0000**3 6 5 0 0 0 0 0** --- ### **Part 2: Round the numbers to 5 decimal places** #### Step 1: Identify the digits up to 5 decimal places: - **(a) 0.999999999999**: First five decimals: 0.99999. The next digit is 9, so we round up. - **(b) 0.7844250000000**: First five decimals: 0.78442. Next digit is 5, so we round up. #### Step 2: Perform rounding: - **(a)**: 0.99999 rounds up to **1.00000** - **(b)**: 0.78442 rounds up to **0.78443** --- ### **Part 3: Fill in the missing words in the definitions** #### Step 1: Understand the context: - "Suppose that a number \( z \) is **approximated** by another number \( y \) ..." - The absolute difference \( |x - y| \) measures the error in approximation. #### Step 2: Complete the sentence: - "Suppose that a number \( z \) is **approximated** by another number \( y \) (where \( z, y \in R \)). Then \( |z - y| \) is called **the absolute error**." --- ### **Part 4: Complete the sentence about loss of significant digits** #### Step 1: Recall that loss of significant digits occurs when subtracting nearly equal numbers, because the significant digits cancel out, leaving fewer meaningful digits. #### Step 2: Complete the sentence: - "Loss of significant digits occurs when subtracting **nearly equal** numbers." --- ### **Part 5: Fill in the missing mathematical symbols** #### Step 1: Understand the definition: - An **iterative algorithm** repeatedly applies a process to generate a sequence \( x_k \). - The sequence should satisfy the relation \( x_{k+1} = \mathcal{A}(x_k) \), where \( \mathcal{A} \) is the algorithm. #### Step 2: Fill in the blank: - "Algorithm \( \mathcal{A} \) is called iterative if there is a sequence of operations **\( \rightarrow \)** that generates a sequence of points **\( x_k \), \( k=1,2,\ldots \)** such that **\( x_{k+1} = \mathcal{A}(x_k) \)**." --- ### **Part 6: Complete the sentence about well-conditioned problems** #### Step 1: Recall that a problem is **well-conditioned** if small changes in the data cause small changes in the results. #### Step 2: Finish the sentence: - "A computational problem is well-conditioned if **small** changes in the data cause **small** changes in the results." --- ## **Final summarized answers:** | Part | Final Answer | |--------|--------------| | 1a | **1 2 3 0 0 0 0 3 7** | | 1b | 0.0000**3 6 5 0 0 0 0 0** | | 2a | **1.00000** | | 2b | **0.78443** | | 3 | approximated, the absolute error | | 4 | nearly equal | | 5 | \( x_k, k=1,2,\ldots \), such that \( x_{k+1} = \mathcal{A}(x_k) \) | | 6 | small, small | --- **Let me know if you'd like further clarification or additional steps!**