##### What are Significant Figures?

**Significant figures** are the digits of number that are used to express its accuracy. There are __four rules__ that determine the how many significant figures a number has. They are used in science to make sure that the numbers reported and used have the correct level of accuracy. Each measurement tool used in a chemistry experiment may have a different level of accuracy. A scale may be able to measure down to the thousandths of a gram while the graduated cylinder may only have increments of 5 ml. The scientific community needs an agreed upon system to determine how accurate our measurements are if we combine information from these two pieces of equipment.

We also cannot generate accuracy just by completing mathematical operations. If we had a 5.0 g object with a volume of 3.0 mL, when we calculate density as grams per mL we will have 1.666666... g/mL. This presents a problem. If we were unable to measure less than a tenth of a gram, then how can we be confident about my value in to infinitely small decimal places? Significant figures are a consistent system that allow us to be consistent with the level of accuracy across different pieces of equipment and operations.

##### Rules for Significant Figures

##### 1. Non-zero digits

Any digit that is not a zero is significant. For example, in the number **156**0 the 1, 5 and 6 are all significant figures because they are all non-zero numbers. These digits count towards accuracy because if the digit is non-zero it indicates we were able to measure the value at that place and found it to be the reported number.

##### Rules for Zeros

Zeros present and interesting problem because they are used for two different purposes in a number. They are either place holders for values that do not exist or we do not know (not significant), or they are values that were measured and found to be zero (significant).

##### 2. Sandwiched Zeros

Zeros that are between non-zero digits in a number are always significant. In the number 8**0**94 zero is significant. Sandwiched zeros count as significant because the non-zero digits on either side mean that we were able to measure values above and below this place, so we logically must have had the accuracy to measure the value where the 0 is and found it to be zero. For example, if you measure a mass of 205 kilograms on a scale, you know that you were able to measure hundreds of kilograms and found it to be two hundred, and you know that you were able to measure single kilograms and found there to be five. Logically, if the scale can measure up to hundreds and down to single kilograms then it must have measured tens of pounds and the value measured was 0.

##### 3. Trailing zeros

Trailing zeros are zeros that follow a non-zero digit. If there is a decimal point, trailing zeros are significant because this means that we were able to measure down to that level and found the value to be zero. In the number 5**.**4**00 **Trailing zeros the two trailing zeros are significant. This number communicates that we were able to measure down to the hundredths and thousandths places, and we measure the value as 0.

However, if there is no decimal place, then trailing zeros are being used as place holders for values we could not measure or do not know. In the number 2**000** all three of the trailing zeros are not significant. This number communicates that the equipment was only able to measure the thousands place and we simply do not know the values in the other places.

##### 4. Leading Zeros

Leading zeros are zeros that come before a non-zero digit and are not sandwiched between another non-zero digit. In the number **0.0**68, all of the zeros are not significant because they all come before a non-zero digit. Whatever we are measuring was so small that it only had values in the hundredths and thousands place, there was nothing to measure at the larger places. It makes sense then that the we should not count the lack of something existing towards accuracy.

##### Examples:

The number **508****000** has 3 significant figures. The non-zero digits count as well as the sandwiched zero, but the trailing zeros do not count because there is no decimal point.

The number **0.0****6070 **has 4 significant figures. The two leading zeros do not count, but the two non-zero digits do count as well as the sandwiched zero and the trailing zero because there is a decimal point.

The number **10.00** has 4 significant figures because all of the trailing zeros count because of the decimal point. However, the number **1****0** has only 1 significant figure because the the trailing zero does not count as there is no decimal point.

##### Want to learn more?

For printable science worksheets, including significant figures, with step by step examples and practice problems, see our resources on __Significant Figures and Dimensional Analysis__.

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