# Significant Figures Rules

Calculating the number of significant figures in a number is a relatively simple process, but it depends on a clear understanding of four main rules. The examples below set out the guidance for each rule and provide various examples to help you understand how the calculation is being made in each case. Should you wish to check your calculations at any point, you can use our significant figures calculator tool.

## Rule 1 - Non-zero digits are ALWAYS significant

This is a fundamental rule and the easiest to understand. Any non-zero digit in a number must be considered as significant.

Examples:

1. 1.1 has two significant figures (1, 1).
2. 13.55 has four significant figures (1, 3, 5, 5).

## Rule 2 - any zero contained between two non-zero numbers is significant

If a zero forms part of a string of digits AND falls between two non-zero digits then it must be counted as a significant figure.

Examples:

1. 1.05 has three significant figures (1, 0, 5); the zero is enclosed by non-zero digits and should be counted.
2. 100.45001 has eight significant figures (1, 0, 0, 4, 5, 0, 0, 1); all the zeroes in this number are enclosed by non-zero digits and therefore should be counted.

Where there are leading or trailing zeroes in a number, there are some other factors which come into play, specifically around the placement of the zeroes relative to any decimal point and other digits.

## Rule 3 - leading zeroes are never significant

Any leading zeroes are never significant, irrespective of a decimal point

Examples:

1. 0.05 - one significant figure (5); the leading zeroes are ignored.
2. 0.0501 - three significant figures (5, 0, 1); the leading zeroes are ignored, the third zero is enclosed by two non-zero digits and is therefore significant.

## Rule 4 - final or trailing zeroes are significant only after a decimal point

Zeroes contained before or after a decimal point are considered as significant figures if they fall between two non-zero digits, as per rule 2, OR potentially when they are trailing, not leading zeroes. Trailing zeroes are only counted when a decimal point is included in the number, as shown below.

This is the hardest rule to grasp and requires various examples to explain.

Examples:

1. 1.005 - four significant figures (1, 0, 0, 5); the zeroes are automatically counted as they fall between two non-zero digits.
2. 0.005 - one significant figure (5); the leading zeroes do not fall between two non-zero digits and only the 5 is considered significant.
3. 0.00500 - three significant figures (5, 0, 0); the leading zeroes do not count (as per Rule 3) but the two trailing zeroes are considered to be significant as they come after a decimal point. So in this example the three significant figures are the 5 and the final two zeroes.
4. 500 - one significant figure (5); there is no decimal point and therefore the two zeroes do not add anything to the "precision" of the number. However, if a decimal point is present then these zeroes will become significant, as per example 5).
5. 500.00 - five significant figures (5, 0, 0, 0, 0). The four trailing zeroes are all significant due to the decimal point which adds precision to the number.

These four basic rules will help determine the number of significant figures in any number. However if you wish to do mathematical calculations and still work out the number of significant figures then a new set of rules comes into play! These rules also differ depending on whether you are combining numbers in an addition or subtraction, or performing multiplication or division.

## Addition rules and subtraction rules for significant figures

1. Add up the number of significant figures to the right of the decimal part of each number used in the calculation.
2. Perform the calculation (addition or subtraction) as usual.
3. The answer must not contain more significant figures to the right of the decimal point than the fewest of any of the figures worked out in part 1. So for example if you are adding together two numbers with three and four significant figures to the right of the decimal point, the answer cannot have more than three significant figures to the right of the point.

• 1.13 has two significant figures to the right of the decimal point, whereas 2.2 has one in the decimal part. The answer therefore also cannot have more than one!
• 1.13 + 2.2 = 3.33
• As the answer cannot have more than 1 significant figure in the decimal part, you must round the result as appropriate, giving a final answer of 3.3.

### Example 2 - subtract 1.55 from 3.2

• 1.55 has two significant figures in the decimal part, while 3.2 has one - the answer must also therefore only have one.
• 3.2 - 1.55 = 1.65
• Round the number to give a final answer of 1.7

Note in this case that if we had been asked to subtract 1.55 from 3.20 (the trailing zero being significant) the answer can have two significant figures and would be correctly deemed to be 1.65!

## Multiplication rules and division rules for significant figure calculations

Unlike the addition and subtraction example, you now must calculate the number of significant figures in each number in its entirely before performing the calculation, not just the decimal part. Once you have performed the calculation, the answer now must contain the same number of significant figures as the smallest total of them in the initial numbers.

### Examples

a) multiply 3.1 by 3.5

Each number has two significant figures therefore the answer can have a maximum of two significant figures.

3.1 x 3.5 = 10.85

However 10.85 has four significant figures and therefore must be rounded to 11, which has two. So in this case the correct answer is 11.

b) multiply 3.10 by 3.50

Each number has three significant figures, so the answer can have a maximum of three.

3.10 x 3.50 = 10.85

Again, round this number appropriately to create an answer with three significant figures - in this case the answer is now 10.9!

c) multiply 3.100 by 3.500

Each number in the question has four significant figures and therefore the answer must too.

3.100 x 3.500 = 10.85

This answer (10.85) has four significant figures and therefore remains the correct one to use.

These examples cover many of the most common types of significant figure calculations and should help you answer questions on this topic. It's important to learn the rules around zeroes and particularly the difference between calculations involving addition/subtraction and multiplication/division as this is where most errors occur.

For help with calculating and counting significant figures, you can use the sig figs calculator on our homepage. We also have a growing collection of articles discussing individual numbers, such as the number of significant figures in 100.

Written by Paul Meehan, PhD chemistry

### References

1. 3.16: Significant Figures. LibreTexts libraries.
2. Rules for Significant Figures. Columbia University.