sig figs rounding

[2025-12-10 Wed]

1. Significant Figures

Significant figures (often abbreviated sig figs) indicate the precision of a measured or calculated value. They represent the digits in a number that are known with certainty, plus one estimated digit.

1.1. Rules for Counting Significant Figures

Non-zero digits are always significant.
- Example: 347 has three significant figures.
Zeros between non-zero digits are significant.
- Example: 4052 has four significant figures.
Leading zeros are not significant. They simply locate the decimal point.
- Example: 0.00480 has three significant figures (4, 8, and the trailing zero).
Trailing zeros are significant if the number is written with a decimal point.
- Example: 25.00 has four significant figures.
Trailing zeros in a whole number without a decimal point are ambiguous.
- Example: 1500 may have two, three, or four significant figures unless additional notation (e.g., scientific notation) is used.

1.2. Scientific Notation and Clarity

Writing numbers in scientific notation removes ambiguity. For example:

\( 1.500 \times 10^3 \) → four significant figures
\( 1.5 \times 10^3 \) → two significant figures

2. Rounding to Significant Figures

To round a number to a specified number of significant figures:

Identify the last significant figure you want to keep.
Look at the next digit (the rounding digit).
Apply the standard rounding rules:
- If the rounding digit is \( \geq 5 \), round the last kept digit up.
- If the rounding digit is \( < 5 \), leave the last kept digit unchanged.

2.1. Examples

2.1.1. Example 1: Rounding 0.004863 to three significant figures

Number: \( 0.004863 \)

Significant digits: 4, 8, 6 Next digit = 3 (less than 5)

So: \[ 0.004863 \rightarrow 0.00486 \]

2.1.2. Example 2: Rounding 12.499 to three significant figures

Digits: 1, 2, 4 Next digit = 9 (≥ 5)

Thus: \[ 12.499 \rightarrow 12.5 \]

2.1.3. Example 3: Rounding 3.9999 to two significant figures

Digits to keep: 3, 9 Next digit = 9 (≥ 5)

\[ 3.9999 \rightarrow 4.0 \]

Note that the trailing zero is significant because it reflects precision to two significant figures.

3. Arithmetic with Significant Figures

3.1. Multiplication and Division

The result must have the same number of significant figures as the factor with the fewest significant figures.

Example: \( (4.62)(3.1) = 14.322 \) → two significant figures (from 3.1): \[ 14.322 \rightarrow 14 \]

3.2. Addition and Subtraction

The result is limited by the least precise decimal place, not by the number of significant figures.

Example: \( 12.11 + 0.3 = 12.41 \) 0.3 is precise to the tenths place → result must be to tenths: \[ 12.41 \rightarrow 12.4 \]