2.4 Significant figures and rounding

Significant figures and rounding are crucial in chemical engineering calculations. They help maintain accuracy and precision in measurements and results. Understanding these concepts ensures reliable data interpretation and prevents errors from propagating through complex calculations.

Proper use of significant figures reflects the limitations of measuring tools and experimental methods. Rounding rules standardize how to simplify numbers while preserving their essential meaning. These skills are fundamental for all chemical engineers to master.

Significant Figures in Values

Identifying Significant Figures

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• Significant figures are the digits in a value that are known with certainty plus one estimated digit
• Significant figures indicate the precision of a measurement or calculation
• The more significant figures a value has, the more precise it is considered to be

Rules for Determining Significant Figures

• All non-zero digits (1-9) are always significant (1.23, 45.6, 789)
• Zeros between non-zero digits are always significant (101, 1.001, 3.0204)
• Leading zeros (to the left of the first non-zero digit) are never significant as they are only placeholders (0.0123, 0.05)
• Trailing zeros (to the right of the last non-zero digit) are significant only if the number contains a decimal point (1.00, 45.0, 0.0200)
• Exact numbers have an infinite number of significant figures (12 eggs, 5 people)

Rounding to Significant Figures

Rounding Rules

• Rounding is a method used to reduce the number of significant figures in a value while maintaining a similar magnitude
• If the digit to the right of the last significant figure is less than 5, the last significant figure remains unchanged (4.732 rounded to 3 significant figures is 4.73)
• If the digit to the right of the last significant figure is 5 or greater, the last significant figure is increased by 1 (4.736 rounded to 3 significant figures is 4.74)
• All digits to the right of the last significant figure are dropped after rounding (4.7385 rounded to 3 significant figures is 4.74)

Rounding Examples

• 1.2345 rounded to 4 significant figures is 1.235
• 0.00987 rounded to 2 significant figures is 0.0099
• 45.999 rounded to 3 significant figures is 46.0
• 8,765,000 rounded to 3 significant figures is 8,770,000

Significant Figures in Calculations

Multiplication and Division

• When multiplying or dividing measured values, the result should have the same number of significant figures as the value with the least number of significant figures used in the calculation
• Example: $3.14 \times 2.7 = 8.478$, rounded to 2 significant figures is $8.5$
• Example: $\frac{12.5}{1.4} = 8.928571429$, rounded to 2 significant figures is $8.9$

Addition and Subtraction

• When adding or subtracting measured values, the result should have the same number of decimal places as the value with the least number of decimal places used in the calculation
• Example: $12.1 + 3.45 + 0.678 = 16.228$, rounded to 1 decimal place is $16.2$
• Example: $1,000.0 - 23.47 = 976.53$, rounded to 1 decimal place is $976.5$

Exact Numbers and Intermediate Results

• Exact numbers do not affect the number of significant figures in a calculation (2 in $2\pi r$ is exact)
• Intermediate results within a multi-step calculation should be carried through with additional digits to avoid rounding errors, with the final result rounded to the appropriate number of significant figures
• Example: $\frac{2.3 \times (4.1 + 6.279)}{1.2} = \frac{2.3 \times 10.379}{1.2} = 19.8766$, rounded to 2 significant figures in the final result is $20$