Overview
This lecture explains the concept of significant figures, their identification, significance in reporting measurements, rules for calculations, rounding, and their connection to uncertainty and precision.
Definition and Importance
- Significant figures (or digits) are the meaningful digits in a measured or calculated number, reflecting its reliability.
- Only digits within the resolution of the measuring instrument are considered significant.
- Reporting more digits than instrument resolution leads to false precision.
Identifying Significant Figures
- All non-zero digits are always significant.
- Zeros between significant digits (trapped zeros) are significant.
- Leading zeros (zeros before the first non-zero digit) are not significant.
- Trailing zeros after a decimal point are significant if within measurement resolution.
- Trailing zeros in whole numbers may or may not be significant, depending on measurement resolution.
- Exact numbers (e.g., counted items) have infinite significant figures.
Denoting Significant Figures
- Use scientific notation, unit prefixes, decimal points, or explicit statements to clarify significant digits in ambiguous cases.
Rounding to Significant Figures
- Round numbers to the desired significant figures according to specific rules based on the next digit.
- Do not round intermediate results—only round the final answer.
- Different tie-breaking rules exist for rounding numbers ending in 5.
Significant Figures in Calculations
- For multiplication/division: the result should have as many significant figures as the input with the fewest significant figures.
- For addition/subtraction: the result should match the least precise decimal place among the inputs.
- For logarithms: mantissa (decimal part) should have as many significant figures as the original number.
- For transcendental functions, use condition number to estimate the loss of significant figures.
Uncertainty and Implied Uncertainty
- Report uncertainties to one or two significant figures only.
- The position of the last significant digit in the value and its uncertainty should match.
- If uncertainty is not specified, it is implied from the last significant digit (typically half the last digit's value).
Accuracy, Precision, and Significant Figures
- Precision refers to repeatability; accuracy is closeness to the true value.
- Number of significant figures reflects measurement precision, not accuracy.
Key Terms & Definitions
- Significant Figures — Digits in a number that contribute to its measurement precision.
- Resolution — The smallest change a measurement instrument can detect.
- Leading Zeros — Zeros before the first non-zero digit; not significant.
- Trailing Zeros — Zeros to the right of the last non-zero digit; significant only if after a decimal or otherwise specified.
- Mantissa — The decimal part of a logarithm.
- Implied Uncertainty — The default uncertainty inferred from the last significant digit when none is stated.
Action Items / Next Steps
- Practice identifying and counting significant figures in sample values.
- Apply significant figure rules to rounding and calculations in homework problems.
- Review how to express uncertainty and significant digits in lab reports.