8.1 Approximation and Estimation

Approximation and estimation are useful techniques in numerical computations to simplify calculations while maintaining accuracy.

Key Concepts:

• Significant Figures:

  Significant figures include all nonzero digits, any zeros between nonzero digits, and trailing zeros in a decimal number.
  • Examples of significant figures:
    • 123 has 3 significant figures.
    • 0.004567 has 4 significant figures (leading zeros are not significant).
    • 50.00 has 4 significant figures (trailing zeros in a decimal are significant).
  • Rounding to a given number of significant figures:
    • Identify the required number of significant figures.
    • Look at the next digit after the last significant figure:
      • If the next digit is 5 or greater, round up the last significant digit.
      • If the next digit is less than 5, leave the last significant digit unchanged.
  • Example: 0.00723456 rounded to 3 significant figures is **0.00723**.
  • Example: 456.78 rounded to 2 significant figures is **460** (since 6 is greater than 5, round up).

  • Which zeros are significant?

    • Zeros between nonzero digits

      (Captive Zeros) are significant.
      • Example: 105 has 3 significant figures.
      • Example: 20.08 has 4 significant figures.

    • Leading zeros

      (before the first nonzero digit) are not significant.
      • Example: 0.0047 has 2 significant figures.
      • Example: 0.000230 has 3 significant figures.

    • Trailing zeros in a decimal number

      are significant.
      • Example: 50.00 has 4 significant figures.
      • Example: 2.500 has 4 significant figures.

    • Trailing zeros in a whole number without a decimal

      are not significant.
      • Example: 1500 has 2 significant figures.
      • Example: 42000 has 2 significant figures.
      • However, 1500.0 has 5 significant figures (because of the decimal point).

• Decimal Places:

  The number of digits after the decimal point.
  • Example: 12.3456 rounded to 2 decimal places is 12.35.

• Rounding:

  Adjusting a number to a given decimal place or significant figure.
  • Example: 457 rounded to the nearest ten is 460.

• Absolute Error:

  The absolute error is the difference between the measured (or estimated) value and the true value. It represents the size of the error without considering its direction (positive or negative). \[ \text{Absolute Error} = |\text{Measured Value} - \text{True Value}| \]

• Percentage Error:

  The relative difference between an estimated value and the actual value. \[ \text{Percentage Error} = \left( \frac{| \text{Approximate Value} - \text{Exact Value} |}{\text{Exact Value}} \right) \times 100 \]

• Estimation Techniques:

  Approximating calculations to simplify operations, such as using rounding to estimate sums and products.

• Solving Equations Correct to n Decimal Places:

  Finding numerical solutions to equations with a specified accuracy.

Examples:

Example 1: Rounding a Number to Significant Figures

Round 0.00723456 to 3 significant figures.

Solution:

The first three significant figures are 7, 2, and 3. Therefore, rounding to 3 significant figures: \[ 0.00723456 \approx 0.00723. \]

Example 2: Calculating Percentage Error

An estimated value for a quantity is 48.3, while the actual value is 50. Find the percentage error.

Solution:

Step 1: Compute the absolute error. \[ | 48.3 - 50 | = 1.7. \] Step 2: Compute the percentage error. \[ \text{Percentage Error} = \left( \frac{1.7}{50} \right) \times 100 = 3.4\%. \] Thus, the percentage error is **3.4\%**.

Example 3: Solving an Equation Correct to 2 Decimal Places

Solve \(x^2 - 2x - 3 = 0\) correct to 2 decimal places.

Solution:

Using the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} \] \[ x = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} \] \[ x = \frac{2 \pm 4}{2}. \] Thus, the two roots are: \[ x = \frac{2 + 4}{2} = 3.00, \quad x = \frac{2 - 4}{2} = -1.00. \] The solutions correct to **2 decimal places** are **\(x = 3.00\) and \(x = -1.00\)**.