1. **Problem Q1 (A): Describe basic concepts of errors using $a=\sqrt{2}$ and $\bar{a}=1.414$.** 2. The exact value is $a=\sqrt{2} \approx 1.414213562$ and the approximate value is $\bar{a}=1.414$. 3. **Absolute error** is the difference between exact and approximate values: $$\text{Absolute error} = |a - \bar{a}| = |1.414213562 - 1.414| = 0.000213562.$$ 4. **Relative error** is the absolute error divided by the exact value: $$\text{Relative error} = \frac{|a - \bar{a}|}{|a|} = \frac{0.000213562}{1.414213562} \approx 0.000151.$$ 5. These errors measure how close the approximation is to the true value. 6. **Problem Q1 (B): Two ways to reduce digits in a numerical value.** 7. (i) **Rounding:** Adjust digits to the nearest value at a certain decimal place. 8. (ii) **Truncation:** Cut off digits beyond a certain decimal place without rounding. 9. Both reduce precision but simplify numbers for easier computation. 10. **Problem Q2 (A): Constituents of a machine number and examples.** 11. A machine number consists of three parts: sign bit, mantissa (or significand), and exponent. 12. Examples: - $1.101 \times 2^{3}$ (binary floating-point) - $-3.25 \times 10^{2}$ (decimal floating-point) 13. **Problem Q2 (B): Formula to eliminate cancellation error in $$y=\sqrt{x+\delta} - \sqrt{x}$$ where $x>0$ and $|\delta|$ is very small.** 14. Use the conjugate to avoid cancellation: $$y = \frac{(\sqrt{x+\delta} - \sqrt{x})(\sqrt{x+\delta} + \sqrt{x})}{\sqrt{x+\delta} + \sqrt{x}} = \frac{(x+\delta) - x}{\sqrt{x+\delta} + \sqrt{x}} = \frac{\delta}{\sqrt{x+\delta} + \sqrt{x}}.$$