# Significant Figure: Definition, Examples & Practice Problems

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• 0:02 What Are Significant Figures?
• 1:59 Determining…
• 3:22 Equations Using Sig Figs
• 5:37 Lesson Summary
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Lesson Transcript
Instructor
Richard Cardenas

Richard Cardenas has taught Physics for 15 years. He has a Ph.D. in Physics with a focus on Biological Physics.

Expert Contributor
Sarah Pierce

Sarah has a doctorate in chemistry, and 12 years of experience teaching high school chemistry & biology, as well as college level chemistry.

In this lesson, you'll learn about the concept of significant figures and apply it to addition, subtraction, multiplication, and division. You'll learn what numbers are significant, why they are significant, and how to deal with zeroes in a number.

## What Are Significant Figures?

A significant figure is a figure or a digit that contributes to how accurately something can be measured. Measuring anything is limited by the measuring device you use. For example, a standard meter stick with centimeters as the smallest division can be used to accurately measure to a tenth of the smallest unit, which is a millimeter. Anything smaller than that will be meaningless, since your measuring device only measures to the nearest millimeter. Therefore, if you use this meter stick then measurements like 0.567 meters, or 0.600 meters, are good measurements since they both indicate that you can measure to the nearest millimeter. A measurement like 0.6 meters does not accurately depict that the meter stick can measure to the nearest millimeter. Similarly, a measurement like 0.6123 meters goes beyond the accuracy of the meter stick.

The digits up to the thousandths place then are all significant figures, or sig figs, because together they maximize the accuracy of the measurement being indicated. Given a measuring stick like this, which measures only as small as centimeters, let's consider the process of trying to measure the length of an object. You can only estimate the length of the object to the nearest tenth of a centimeter, or a millimeter; any further isn't possible with the information the ruler is giving us. Let's say the object looks to be about 4.8 centimeters (cm) long, with only two significant figures. The exact length may be different, but with our two significant figures, that's as close as we can get.

But what if the ruler had millimeter markings in addition to centimeters? You could now estimate the length of the object to the nearest tenth of a millimeter. That could give us a measurement like 4.84 cm, with three significant figures. The more significant figures are available to us when measuring, the more exact the measurement will be.

## Determining Significant Figures

It can be tricky to decide which digits in a complex number are significant figures. Because significant figures are used to increase the accuracy of measurements, numbers which use them will often include long strings of extra zeroes, making it even more difficult to identify the sig figs. Luckily, we have some tools to help.

The most useful one is the Atlantic-Pacific rule, a mnemonic device for identifying significant figures. The rule states that if a decimal point is absent, for example in the case of whole numbers, zeroes on the right side of the number (the Atlantic) are not significant. If a decimal point is present and there are non-integer digits involved in the measurement, then the zeroes on the Pacific side (the left side) are not significant. In both cases, the opposite side contains the significant zeroes.

For example, let's take an imaginary measurement of:

0015.00120

The nonzero digits are all significant figures, but are all of the four zeroes? Notice that a decimal point is present, meaning that there are significant digits on the right side of the decimal. According to the Atlantic-Pacific rule, that means that zeroes on the decimal side are significant, and we can ignore the extra zeroes on the left side of the number. So this number has seven significant figures.

## Equations Using Sig Figs

When performing mathematical actions, such as addition and multiplication on numbers with the same number of significant digits, the solution will always have the same number of significant figures as the original terms. But what about when the numbers being added, subtracted, divided, or multiplied have different amounts of significant figures?

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## Practice Problems Using Significant Figures

Answer the following problems using the correct number of significant figures in the solution.

#### Practice Problem 1

The density of an unknown metal alloy is 5.20 g/mL. If the volume of the metal is 5.214 mL, what is the mass? Remember, density equals mass divided by volume.

#### Practice Problem 2

You have a piece of fabric that is 4.35 m wide. The pattern you are following requires you to trim off 1.1 m. How much fabric does the pattern require?

#### Practice Problem 1

To solve the problem, multiply the density by the volume.

5.20 g/mL x 5.214 mL = 27.1 g

When you multiply these numbers in a calculator, you get 27.1128 g. This number does not reflect the correct number of significant figures. When performing multiplication and division, the answer must have the same number of significant figures as the least specific number. For example, 5.20 g/mL has three significant figures, while 5.214 mL has four significant figures. 5.20 g/mL is the least specific number, so the answer needs to also have three significant figures. Therefore, you need to round 27.1128 to three significant figures.

#### Practice Problem 2

To solve the question, subtract 1.1 m from 4.35 m.

4.35 m - 1.1 m = 3.3 m

When adding and subtracting numbers, you need to round the answer to the same number of decimal places as the number with the fewest decimal places. The number 1.1 only goes to the tenths place, while 4.35 goes to the hundredths place. Therefore, the answer must also be to the tenths place. When you subtract these numbers using a calculator, the answer is 3.25. Look at the hundredths place to determine how to round to achieve the final answer. If the number is five or greater, you round up. This is the case in this problem, so the answer is rounded to 3.3m.

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