Significant figures

Here you will learn about significant figures, how to identify significant figures within a given number and how to round to one significant figure, two significant figures, and three significant figures.

Students will first learn about significant figures as part of expression and equations in the 8th grade.

What are significant figures?

Significant figures are each of the digits in a number that are used to express it, to the required degree of accuracy, starting from the first non-zero digit. The first significant figure, or significant digit, of a number is the most important digit because it expresses the size of the number.

Significant figures are often used with measured values.

To locate a significant figure, you need to find the first non-zero digit. This is the first significant figure. The next digit is the second significant figure and so on.

A significant figure could be to the left of the decimal point or the right of the decimal point.

There are significant figure rules that will help you identify all the significant figures in a given number:

• Non-zero digits are always significant.
• Any zeros located between non-zero digits are significant.
• Trailing zeros are only significant if a decimal point is present.

For example,

Rounding numbers to significant figures (often abbreviated to sig figs or s.f.), is similar to rounding to a number of decimal places, such as the ones place, tens place, hundreds place, and thousands place.

What are significant figures?

Common Core State Standards

How does this relate to 8th grade math?

• Grade 8: Expressions and Equations (8.EE.A.4)
  Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (for example, use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

How to identify significant figures

In order to identify significant figures:

1. Find the first non-zero digit.
2. Starting at the first non-zero digit, count the number of digits that are non-zero numbers or zeros between non-zero digits.
3. If there is a decimal present, count trailing zeros.
4. Label the number of significant figures.

Significant figures examples

Example 1: significant figures

Identify the correct number of significant figures in the number 405.

1. Find the first non-zero digit.

The first non-zero digit is the 4 in the hundreds place.

2Starting at the first non-zero digit, count the number of digits that are non-zero numbers or zeros between non-zero digits.

All three digits are either non-zero numbers or have a zero between non-zero digits.

The 4 is the first significant digit, and the 5 is the last significant digit.

3If there is a decimal present, count trailing zeros.

There are no decimal points or trailing zeros present.

4Label the number of significant figures.

The number 405 has 3 significant figures.

Example 2: significant figures

Identify the correct number of significant figures in the number 0.037.

Find the first non-zero digit.

 

The first non-zero digit is the 3 in the hundredths place.

Starting at the first non-zero digit, count the number of digits that are non-zero numbers or zeros between non-zero digits.

 

The 3 and the 7 are non-zero digits. The 3 is the first significant digit, and the 7 is the last significant digit.

If there is a decimal present, count trailing zeros.

 

There is a decimal in this number, but no trailing zeros.

Example 3: significant figures

Calculate the number of significant figures for the number 0.004500.

Find the first non-zero digit.

 

The first non-zero digit is the 4 in the thousandths place.

Starting at the first non-zero digit, count the number of digits that are non-zero numbers or zeros between non-zero digits.

 

The 4 is the first significant digit, and the 5 is the last significant non-zero digit.

How to round to significant figures

In order to round to a given number of significant digits:

1. Locate the significant figure for the degree of accuracy required.
2. Look at the next digit immediately to the right. When the digit is \bf{5} or more, round up by adding \bf{1} to the previous digit. When the digit is less than \bf{5} , round down by keeping the previous digit the same.
3. When the degree of accuracy is \bf{10} or more, use zeros as place holders.

Example 4: rounding to one significant figure

Round 3,692 to one significant figure (1 \, s.f) .

Locate the significant figure for the degree of accuracy required.

 

The first non-zero digit is 3, therefore 3 is the first significant figure.

Look at the next digit immediately to the right. When the digit is \bf{5} or more, round up by adding \bf{1} to the previous digit. When the digit is less than \bf{5} , round down by keeping the previous digit the same.

 

6 is more than 5.

 

Because 6 is greater than 5, you will round up. Adding 1 to the 3 gives you 4.

Example 5: rounding to two significant figures

Round 0.07039 to two significant figures (2 \, s.f.) .

Locate the significant figure for the degree of accuracy required.

 

The first non-zero digit is the 7, which is the first significant figure; therefore, the 0, at the thousandths place, is the second significant figure.

Look at the next digit immediately to the right. When the digit is \bf{5} or more, round up by adding \bf{1} to the previous digit. When the digit is less than \bf{5} , round down by keeping the previous digit the same.

 

3 is immediately to the right.

 

3 is less than 5.

 

Because 3 is less than 5, you will round down.

Example 6: rounding to three significant figures

Round 24.753 to three significant figures (3 \, s.f.) .

Locate the significant figure for the degree of accuracy required.

 

The first non-zero digit is 2, which is the first significant figure. Therefore, the 7, in the tenths place, is the third significant figure.

Look at the next digit immediately to the right. When the digit is \bf{5} or more, round up by adding \bf{1} to the previous digit. When the digit is less than \bf{5} , round down by keeping the previous digit the same.

 

The digit to the right is 5.

 

Because the number is 5, you will round up. Adding 1 to the 7 gives you 8.

Teaching tips for significant figures

• When rounding to a given significant figure, students may struggle with numbers with more digits. Having students underline or color code the significant figure they are rounding to will help them visually see the place, and not get lost in all of the digits.

• Rounding errors are common when rounding to significant figures. For some students, posting of common rounding rules may help keep this from happening.

Easy mistakes to make

• Assuming the first zeros are significant
  A common error is to think the first zeros in a number like 0.00467 are significant. These are called leading zeros and do not tell you the size of the number. The first non-zero digit is 4, and tells you that the number is roughly 4 thousandths.

• Assuming all zeros are not significant
  A common error is to think that all the zeros in a number like 9,006 are not significant. Once the first significant figure has been located, all digits after are significant. Otherwise, you might round to the incorrect number of significant figures.

• Leaving out zeros to make the number the correct size
  It is important to remember to fill in the zeros if the degree of accuracy is greater than 10. A common mistake when rounding a number like 378 to two significant figures is to say it is 38 and not 380.

Related lessons on rounding numbers

This significant figures topic guide is part of our series on rounding decimals. You may find it helpful to start with the main rounding decimals topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

• Rounding numbers
• Rounding decimals
• Place value
• Estimation

The next lessons are

• Factors and multiples
• Fractions
• Fractions operations

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