2.1 Approximation and Errors

• Estimation and approximation; absolute, relative and percentage errors
• Working limits and maximum absolute error of a measurement
• Round-off errors (decimal places and significant figures) and truncation errors
• Propagation of errors in addition, subtraction, multiplication and division

• determine errors from measurements of quantities
• determine absolute, relative and percentage errors from measurements
• compute round off and truncation errors from measurements
• determine maximum and minimum errors from limits in measurements
• perform propagation of errors using limits of measurements
• appreciate the use of approximations and errors in real life situations

“Why are quantities approximated?”

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1. Absolute and relative error
   40 min · 1 practice tasks · quiz
   Name the positive gap as the absolute error, then divide by the true value to get the relative error and the percentage error.
2. Working limits of a reading
   40 min · 1 practice tasks · quiz
   Take the maximum absolute error of a recorded measurement as half the smallest unit, and write the two working limits.
3. Percentage error from a reading
   40 min · 1 practice tasks · quiz
   Divide the maximum absolute error by the recorded value to compare the precision of short and long measurements.
4. The error made by rounding
   40 min · 1 practice tasks · quiz
   Round a known number to decimal places or significant figures, then take the absolute error as the positive difference.
5. Truncation is not rounding
   40 min · 1 practice tasks · quiz
   Cut off extra digits without rounding, find the truncation error, and compare it with the round-off error of the same number.
6. Adding two measured lengths
   40 min · 1 practice tasks · quiz
   Add the upper working limits and the lower working limits, and see that the maximum absolute errors add.
7. Subtracting measured quantities
   40 min · 1 practice tasks · quiz
   Find the greatest difference from the upper of the first limit and the lower of the second, and plan a remaining distance from the safe end.
8. The limits of a rectangular area
   40 min · 1 practice tasks · quiz
   Multiply the upper limits for the greatest area and the lower limits for the least area, for two positive measurements.
9. The limits of a speed
   40 min · 1 practice tasks · quiz
   Divide the upper distance by the lower time for the greatest speed, and the lower distance by the upper time for the least speed.
10. A margin for a journey
    40 min · 1 practice tasks · quiz
    Add the working limits of two journey times and use the maximum, not the recorded total, when arriving late would matter.