Significant Figures Calculator

What are significant figures in maths?

Significant figures refer to the key digits in a number that convey its precision. For example, the number 6.658 has four significant figures. These figures help ensure accuracy in numbers. They are also called significant digits. When counting significant figures, the first non-zero digit is the starting point. The following digits then represent the second, third, and so on. Significant figures can appear both before and after a decimal point. Much like rounding to a certain number of decimal places, rounding significant figures involves identifying the correct figure to maintain the desired precision.

What are the rules for significant figures?

Every non-zero digit is significant. For instance, the number 33.2 has three significant figures.
Zeros between non-zero digits are also significant. In the number 2051, there are four significant figures, as the zero is between 2 and 5.
Leading zeros (those before the first non-zero digit) are not significant. They merely serve as placeholders. For example, 0.54 has two significant figures, and 0.0032 also has two.
Trailing zeros (those after the decimal point) are significant. The number 92.00 has four significant figures.
For whole numbers with a decimal point, the trailing zeros are significant. For example, "540." contains three significant figures, as the decimal indicates the zero is meaningful.
However, trailing zeros in whole numbers without a decimal point are not significant. For instance, "540" has only two significant figures.
Exact numbers (like definitions) have an infinite number of significant figures. For example, 1 metre could be written as 1.00 metres, with an infinite level of precision.

Why use significant figures?

Significant figures are widely used in science and measurement to express the precision of a result. Not all measurements are equally precise. For instance, consider two scales: one measures to the nearest gram, while the other measures to the nearest hundredth of a gram. Both might give a reading of 3 grams, but the meaning differs in terms of precision. The first scale would record the measurement as 3 grams, while the second might record it as 3.00 grams. This additional precision is reflected by the number of significant figures in the result.

How does rounding to significant figures work?

Rounding significant figures involves reducing a number to the required level of precision by eliminating some digits. If the first digit to be dropped is less than 5, the last retained digit stays the same. If the first digit to be dropped is 5 or greater, the last retained digit is rounded up. When there is a trailing 5, the digit is rounded to the nearest even number. Importantly, rounding is done to the entire number, not one digit at a time.

There are two main rules for rounding significant figures:

Identify which digit needs to be rounded.
If the digit after it is less than 5, drop the extra digits. If the next digit is 5 or more, add 1 to the rounded digit and ignore the remaining digits.

What are three significant figures?

The third significant figure of a number is simply the third digit when counting from the first non-zero digit. This is true even if the third figure is a zero. For example, in 20,499, the third significant figure is 4, and in 0.0020499, it’s 9. Rounding to three significant figures follows the same principle as rounding to three decimal places. If there are empty spaces to the right of the decimal point, they are filled with zeros to maintain the place value of the significant figures.

How to apply significant figures in addition, subtraction, multiplication, and division?

When performing operations on numbers with significant figures, the result should not be more precise than the least precise number in the calculation. For example, if one number has three significant figures and another has four, the result must be rounded to three significant figures.

For example:

7.9391 + 6.263 + 11.1 = 25.3021 → Round to 25.3 (since 11.1 has the fewest significant figures).
12.50 × 169.1 = 2113.75 → Round to 2114 (as both numbers have four significant figures).

Examples of rounding to three significant figures

654.389 becomes 654 (because the fourth digit is less than 5).
65.4389 rounds to 65.4.
654,389 rounds to 654,000 (zeros maintain place value).
56.7688 becomes 56.8 (rounding the third figure up).
0.03542110 rounds to 0.0354.
0.0041032 rounds to 0.00410 (zeros are necessary to hold place value).
45.989 rounds to 46.0 (because rounding the third figure affects the next digit).