Significant Figures: Rules, Accuracy & Precision, and Calculations

Significant Figures

Introduction to Significant Figures

Significant figures (often abbreviated as sig figs) are the digits in a measurement that carry meaning regarding its precision. Understanding which digits are significant is essential for reporting scientific data accurately and for performing calculations in chemistry.

Rules for Determining Significant Figures

Rule 1: Non-Zero Digits and "Sandwiched Zeros"

All non-zero digits are always significant. Additionally, any zeros between non-zero digits (called sandwiched zeros) are also significant.

• Example: 9.00006 has 6 significant figures.

• Example: 523.065 has 6 significant figures.

• Example: 1.01 has 3 significant figures.

• Example: 970000001 (number of significant figures depends on context; if all digits are measured, then 9 significant figures).

Rule 2: Leading Zeros

Leading zeros (zeros that precede all non-zero digits) are not significant. They merely indicate the position of the decimal point.

• Example: 00600002 has 6 significant figures (the leading zeros do not count).

• Example: 2345 has 4 significant figures.

• Example: 0.00000000000000000000001 has 1 significant figure.

Rule 3: Trailing Zeros

Trailing zeros (zeros to the right of all non-zero digits) are only significant if a decimal point is shown.

• Example: 300.0 has 4 significant figures.

• Example: 0.000000 has 1 significant figure.

• Example: 2300 (without a decimal point) has 2 significant figures.

• Example: 2300. (with a decimal point) has 4 significant figures.

Additional info: Scientific notation can clarify the number of significant figures in ambiguous cases.

Practice: Identifying Significant Figures

Examples and Explanations

• 0.00342: 3 significant figures (leading zeros do not count).

• 1,009,650: 6 significant figures (sandwiched zeros count, trailing zero may or may not be significant depending on decimal point).

• 67,500: 3 significant figures (trailing zeros without decimal point are not significant).

• 108,400: 4 significant figures.

• 30,000,000: 1 significant figure.

Accuracy and Precision

Definitions

• Accuracy: How close a measured value is to the true or accepted value.

• Precision: How close repeated measurements are to each other, regardless of their closeness to the true value.

Practice Problems: Accuracy & Precision

Consider the following experimental results:

Name	Mass (iPhone 11)
Thanh	146.0 grams
Kelly	148.0 grams
Juan	140.0 grams

Actual mass: 137.0 grams. The results are neither accurate nor precise (they are not close to the true value and not close to each other).

Name	Mass (iPhone 4)
Thanh	136.0 grams
Kelly	138.0 grams
Juan	137.0 grams

Actual mass: 137.0 grams. The results are accurate but not precise (one value matches the true value, but the others are not close to each other).

Name	Mass (Droid)
Thanh	100.5 grams
Kelly	100.5 grams
Juan	100.5 grams

Actual mass: 169.5 grams. The results are not accurate but precise (all measurements are close to each other but far from the true value).

Mathematical Operations and Significant Figures

Multiplication and Division

When multiplying or dividing measurements, the answer must have the same number of significant figures as the measurement with the least number of significant figures.

• Example: $3\,\text{mL} \times 1.22\,\text{mL}$ Answer will have 2 significant figures.

• Example: $432\,\text{cm} / 12.5\,\text{cm}$ Answer will have 3 significant figures.

• Example: $100\,\text{g} \times 2.0000\,\text{g}$ Answer will have 1 significant figure.

• Example: $121 / 8.0$ Answer will have 2 significant figures.

Rules for Rounding

• If the digit to be removed is less than 5, the preceding digit stays the same; if it is equal to or greater than 5, round the preceding digit up.

• Carry extra digits through a series of calculations and round only the final result.

Examples:

• $3.04999$ rounds to $3.0$

• $3.05999$ rounds to $3.1$

• $3.05 \times 5.555 \times 3.0 = 50.82825$ rounds to $51$ (after all calculations)

Scientific Notation and Standard Form

Converting Between Forms

• Scientific Notation: Expresses numbers as a product of a coefficient and a power of ten.

• Standard Form: The usual decimal representation.

Examples:

• $2,340 = 2.34 \times 10^3$

• $0.369 = 3.69 \times 10^{-1}$

• $8.7 \times 10^{-5} = 0.000087$

• $3.7 \times 10^8 = 370,000,000$

Summary Table: Significant Figure Rules

Rule	Description	Example	Sig Figs
Non-zero digits	Always significant	123	3
Sandwiched zeros	Zeros between non-zero digits are significant	101	3
Leading zeros	Not significant	0.0025	2
Trailing zeros (with decimal)	Significant	100.0	4
Trailing zeros (no decimal)	Not significant	100	1

Additional info: For ambiguous cases, use scientific notation to clarify the number of significant figures.