Approximation and Estimation of Calculations

Estimating a Calculation

Estimating a calculation serves as a vital check on the accuracy of your work. It is particularly useful for identifying errors in magnitude, such as when a decimal point has been misplaced during a calculation. By using simpler, rounded numbers, you can determine whether a result is in the correct 'ballpark' before committing to more complex arithmetic.

Rules for Approximation

To produce an effective estimate, numbers are typically approximated to 1 or 2 significant figures. This allows for simple mental or written arithmetic while maintaining enough accuracy to check the magnitude of the final answer. For example, a value like $397,000$ should be approximated to $400,000$ for the purpose of estimation.

Approximating Mathematical Constants and Surds

When calculations involve $\pi$ or surds, specific approximations are used:

1. Approximating $\pi$: The constant $\pi$ is usually approximated to the integer $3$ or the fraction $\frac{22}{7}$ for quick estimates.
2. Approximating Surds: Surds are approximated by identifying the nearest square number. For example, $\sqrt{15.6} \approx \sqrt{16} = 4$. This provides a quick integer value that represents the scale of the irrational number.

Worked Example: Verifying Calculator Results

Suppose a calculator provides the value of $\sqrt{\frac{2 \times 26.37 \times \pi}{0.00389}}$ as $2063.8$ (correct to 1 decimal place). We can check if this result is likely to be correct using approximation in stages:

Step 1: Approximate the numerator. We approximate $26.37$ to $25$ and $\pi$ to $3$. $2 \times 26.37 \times \pi \approx 2 \times 25 \times 3 = 150$

Step 2: Approximate the denominator. We approximate $0.00389$ to $0.004$ (rounding to one significant figure).

Step 3: Combine and simplify the fraction. $\sqrt{\frac{2 \times 26.37 \times \pi}{0.00389}} \approx \sqrt{\frac{150}{0.004}}$ To divide by $0.004$, we can multiply the numerator and denominator by $1000$: $\sqrt{\frac{150,000}{4}} = \sqrt{37,500}$

Step 4: Final magnitude check. We can approximate $\sqrt{37,500}$ by looking for the nearest square number that is easy to calculate. Since $200^2 = 40,000$, we have: $\sqrt{37,500} \approx \sqrt{40,000} = 200$

Conclusion: Our estimate of $200$ suggests that the calculator answer of $2063.8$ is of the wrong magnitude. It is highly likely that the decimal point was placed incorrectly and the actual answer should be approximately $206.38$.