# Significant Figures

## What Are Significant Figures

Whenever we measure something, we can only list as many digits as our measuring tool allows us. For instance, if you were to measure the length of a string using a ruler that can measure length to the nearest millimeter, then you might measure its length to be 42.5cm. However, you could not express this value as 42.51cm since your measuring tool cannot measure a hundredth of a centimeter. **Significant figures** are used to indicate the reliable digits, with acceptable uncertainty, of a measurement. For example, the measured value of 42.5cm has three significant figures.

Note that the last digit in a measurement is estimated in some way by the person taking the measurement. For instance, in our previous example, the length of the string could've been somewhere between 42.4cm and 42.5cm, and the person taking the measurement estimated the last value to be 42.5cm. When using significant figures, the rule is that the last digit written down in a measurement is the first digit with some uncertainty.

Significant figures are sometimes referred to as significant digits.

## Determining The Number of Significant Figures

Let's take a look at the number 0.00700, which has three significant figures (7, 0, and 0). You might be wondering what happened to the other zeros in the number! Well, we did not count them, as they were not significant. To make things a bit more clear, let's presume the value 0.00700 represents the distance in kilometers from your house to the mailbox. You realize the number is not that big, and you decide to convert it to meters, then you will end up with 7.00 meters. The two zeros after the number 7 tell you how precise your measurement is. However, the preceding zeros disappeared, as they were only there as place-keepers for the digits when they were in kilometers and not an indication of the accuracy of the measurement.

Whenever you think about significant figures, try to think of which digits are actually giving you information on how precise your measurement is.

**The Rules of Significant Figures**

- All non-zero numbers are significant. The number 32.5 has three significant figures.
- Any zeros between two non-zero digits are significant. The number 40105 has five significant figures.
- Leading zeros are not significant. The number 0.0025 has two significant figures.
- Trailing zeros after a decimal are significant. The number 14.00 has four significant figures.
- Trailing zeros in a whole number with a decimal are significant. The number 350. has three significant figures.
- Trailing zeros in a whole number without a decimal are not significant. The number 6100 has two significant figures.

## Significant Figures in Calculations

The idea behind significant figures is that when you do some computation, you are not misrepresenting the amount of precision you had and that the results are not more precise than the values you initially measured. When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can't be greater than the number of significant digits in the least precise measured value.

The way we determine the least precise value is slightly different for multiplication and division than for addition and subtraction. The two methods are described below.

**For multiplication and division**, the result should have the same number of significant figures as the quantity having the least significant figures used in the calculation.

### Significant Figures in multiplication and division Example

Let's say you wanted to find the area of a rectangle. You measured one side and found it to be 8.3cm, and for some reason, you measured the other side with more precision and found it as 3.25cm.

First, you multiply the two sides of the rectangle to find the area:

8.3cm x 3.25cm = 26.975cm^{2}

However, this result is quite misleading, it kind of indicates that your precision of measuring the area was to the nearest 0.001cm, which is not true. Because one side has only two significant figures, it limits the result's precision to two significant figures.

Therefore, the right representation of the results would be: 27cm^{2}

**For addition and subtraction**, the answer can't contain more decimal places than the least precise measurement.

### Significant Figures in Addition And Subtraction Example

Suppose you wanted to add the weight of three bags of vegetables. You measured the first bag using a laboratory balance, which has a precision of 0.01kg, and found that the first bag weighs 4.25kg. For the second bag, you used your scale at home, which has a precision of 0.1kg, and found the weight of the second bag to be 2.6kg. The weight of the third bag had already been measured and was labeled to weigh 3.055kg.

You can find out the total weight of the three bags and the number of significant figures you should have to get a reliable result as follows:

Step 1) You add the three values together:

4.25 + 2.6 + 3.055 = 9.905kgStep 2) Then, find the least precise measurement, which in this case is:

2.6Step 3) Finally, round your final answer to match the value with the least number of decimals used in the calculation:

9.905 9.9kg### Frequently Asked Questions

- Why are significant figures important?
- Significant figures are important because they indicate the precision of a measurement.

- What is the idea behind significant figures?
- To ensure that the result of a calculation does not misrepresent the amount of precision of the measured values.

**References:**Paul Peter Urone. “College Physics - Accuracy, Precision, and Significant Figures.” OpenStax, 2012. June 21. https://openstax.org/books/college-physics/pages/1-3-accuracy-precision-and-significant-figures.