Significant Figures in Mathematical Operations: Multiplication, Division, Addition, and Subtraction

Significant Figures in Mathematical Operations

Multiplication and Division

When multiplying or dividing numbers in chemistry, the result should be reported with the same number of significant figures as the measurement with the fewest significant figures.

• Significant figures are the digits in a measurement that are known with certainty plus one digit that is estimated.

• To determine the correct number of significant figures in the final answer, identify the number in the calculation with the least number of significant figures.

Example:

• Calculate: $(3.16) \times (0.003027) \times (5.7 \times 10^3)$

• Step 1: Count significant figures in each number: - 3.16 (3 sig figs) - 0.003027 (4 sig figs) - $5.7 \times 10^3$ (2 sig figs)

• Step 2: The answer should have 2 significant figures (from $5.7 \times 10^3$).

• Calculation: $3.16 \times 0.003027 \times 5.7 \times 10^3 = 5.4522324 \times 10^1$

• Final answer (2 sig figs): $5.5 \times 10^1$

Addition and Subtraction

When adding or subtracting numbers, the result should be reported with the same number of decimal places as the measurement with the fewest decimal places.

• Decimal places refer to the number of digits to the right of the decimal point.

• Identify the number in the calculation with the least number of decimal places.

Example:

• Calculate: $402.09 - 212.2 + 2.671$

• Step 1: Decimal places: - 402.09 (2 decimal places) - 212.2 (1 decimal place) - 2.671 (3 decimal places)

• Step 2: The answer should have 1 decimal place (from 212.2).

• Calculation: $402.09 - 212.2 + 2.671 = 192.561$

• Final answer (1 decimal place): $192.6$

Mixed Operations

When calculations involve a mixture of multiplication, division, addition, and subtraction, follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) and apply the significant figure rules at each step.

• PEMDAS stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

• Apply significant figure rules after each operation, not just at the end.

Example:

• Calculate: $[(1.89 \times 10^4) \div (3.005)] \cdot (5.21^2) \div [(8.829 - 6.5) \div (2.920)]$

• Step 1: Calculate each part according to PEMDAS.

• Step 2: Apply significant figure rules after each operation.

• Sample intermediate results: - $1.89 \times 10^4 \div 3.005 = 6.290$ (3 sig figs) - $5.21^2 = 27.1441$ (3 sig figs, round to 3: 27.1) - $8.829 - 6.5 = 2.329$ (1 decimal place, round to 1: 2.3) - $2.3 \div 2.920 = 0.7877$ (2 sig figs, round to 2: 0.79)

• Final answer: Apply sig fig rules to the final result.

Practice Problems

Practice applying significant figure rules to the following calculations:

• Multiplication/Division Practice: $[(1.7 \times 10^4) \div (2.63 \times 10^4)] \cdot 6.96$

• Mixed Operations Practice: $(6.404 \times 2.91) \div (18.7 - 17.1)$

• Addition/Subtraction Practice: $(42.00 - 40.914) \div (25.739 - 25.729)$ $(11.50 - 1.001) + (0.00710 - 700)$

Note: Always express your final answer with the correct number of significant figures or decimal places, as appropriate for the operation.

Summary Table: Significant Figures Rules

Operation	Rule	Example
Multiplication/Division	Result has the same number of significant figures as the measurement with the fewest significant figures.	$3.16 \times 0.003027 \times 5.7 \times 10^3$ (2 sig figs)
Addition/Subtraction	Result has the same number of decimal places as the measurement with the fewest decimal places.	$402.09 - 212.2 + 2.671$ (1 decimal place)
Mixed Operations	Follow order of operations (PEMDAS) and apply sig fig rules at each step.	See example above