# Significant figures

Significant figures (also known as the significant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.

If a number expressing the result of a measurement (e.g., length, pressure, volume, or mass) has more digits than the number of digits allowed by the measurement resolution, then only as many digits as allowed by the measurement resolution are reliable, and so only these can be significant figures.

For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, showing 114 mm) are certain and so they are significant figures. Digits which are uncertain but reliable are also considered significant figures. In this example, the last digit (8, which adds 0.8 mm) is also considered a significant figure even though there is uncertainty in it.

Another example is a volume measurement of 2.98 L with an uncertainty of ± 0.05 L. The actual volume is somewhere between 2.93 L and 3.03 L. Even when some of the digits are not certain, as long as they are reliable, they are considered significant because they indicate the actual volume within the acceptable degree of uncertainty. In this example the actual volume might be 2.94 L or might instead be 3.02 L. And so all three are significant figures.

The following digits are not significant figures.

• All leading zeros. For example, 013 kg has two significant figures, 1 and 3, and the leading zero is not significant since it is not necessary to indicate the mass; 013 kg = 13 kg so 0 is not necessary. In the case of 0.056 m there are two insignificant leading zeros since 0.056 m = 56 mm and so the leading zeros are not necessary to indicate the length.
• Trailing zeros when they are merely placeholders. For example, the trailing zeros in 1500 m as a length measurement are not significant if they are just placeholders for ones and tens places as the measurement resolution is 100 m. In this case, 1500 m means the length to measure is close to 1500 m rather than saying that the length is exactly 1500 m.
• Spurious digits, introduced by calculations resulting in a number with a greater precision than the precision of the used data in the calculations, or in a measurement reported to a greater precision than the measurement resolution.

Of the significant figures in a number, the most significant is the digit with the highest exponent value (simply the left-most significant figure), and the least significant is the digit with the lowest exponent value (simply the right-most significant figure). For example, in the number "123", the "1" is the most significant figure as it counts hundreds (102), and "3" is the least significant figure as it counts ones (100).

Significance arithmetic is a set of approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as propagation of uncertainty.

Numbers are often rounded to avoid reporting insignificant figures. For example, it would create false precision to express a measurement as 12.34525 kg if the scale was only measured to the nearest gram. In this case, the significant figures are the first 5 digits from the left-most digit (1, 2, 3, 4, and 5), and the number needs to be rounded to the significant figures so that it will be 12.345 kg as the reliable value. Numbers can also be rounded merely for simplicity rather than to indicate a precision of measurement, for example, in order to make the numbers faster to pronounce in news broadcasts.

Radix 10 (base-10, decimal numbers) is assumed in the following.