1. **Problem:** Describe the basic concepts of errors using $a=\sqrt{2}$ (exact) and $\bar{a}=1.414$ (approximate).\n\n2. **Error Concepts:** The **absolute error** is $|a - \bar{a}| = |\sqrt{2} - 1.414|$. The **relative error** is $\frac{|a - \bar{a}|}{|a|}$. These measure how far the approximate value is from the exact value.\n\n3. **Problem:** Explain two ways to reduce the number of digits in a numerical value.\n\n4. **Digit Reduction Methods:** (a) **Rounding:** Adjust digits to the nearest value at a certain decimal place. (b) **Truncation:** Cut off digits beyond a certain decimal place without rounding.\n\n5. **Problem:** What are the constituents of a machine number? Give 2 examples.\n\n6. **Machine Number Constituents:** A machine number typically has three parts: sign bit, mantissa (or significand), and exponent.\n\n7. **Examples:** (a) $-1.101_2 \times 2^3$ (binary floating-point) (b) $+3.14 \times 10^2$ (decimal floating-point)\n\n8. **Problem:** State the formula to eliminate cancellation error in $y=\sqrt{x+\delta} - \sqrt{x}$ for $x>0$ and small $|\delta|$.\n\n9. **Cancellation Error Elimination:** Multiply numerator and denominator by the conjugate to get\n$$y = \frac{(\sqrt{x+\delta} - \sqrt{x})(\sqrt{x+\delta} + \sqrt{x})}{\sqrt{x+\delta} + \sqrt{x}} = \frac{(x+\delta) - x}{\sqrt{x+\delta} + \sqrt{x}} = \frac{\delta}{\sqrt{x+\delta} + \sqrt{x}}.$$\n\nThis form avoids subtracting two nearly equal numbers, reducing cancellation error.