# Rounding: Decimal places and significant figures

## In a nutshell

Rounding numbers is very useful when tasked with presenting information clearly and simply. There are two ways to round numbers: decimal places and significant figures.

## Decimal places

To round a number to a certain amount of decimal places, follow this procedure.

#### procedure

1. | Go to the digit on the specified decimal place. Underline this digit. |

2. | Circle the digit to the right of this number. |

3. | If the circled digit is between $5$ and $9$, then add $1$ to the underlined digit. If the circled digit is between $0$ and $4$, then don't change the underlined digit. |

4. | Rewrite the number, ignoring everything past the underlined digit. |

**Note: **If you have to add $1$* to the underlined digit, and the underlined digit is a *$9$*, then add *$1$* to the digit to the left of the underlined digit and write the underlined digit as *$0$*.*

##### Example 1

*Round $3.089$ to two decimal places.*

*Underline the second digit after the decimal point and circle the number to the right. In this case, underline $8$* *and circle $9$*:

$3.0\underline{8}\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {9}}}$

*The circled digit is greater than $4$*, .*so increase the underlined digit by $1$: *

*$3.0\underline{9}\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {9}}}$*

*Rewrite the number ignoring everything past the underlined $9$:*

$3.09$

*Therefore, *$3.089$ to two decimal places is $\underline{3.09}$.

## Significant figures

To round a number to a certain amount of significant figures, follow this procedure.

#### procedure

1. | Starting from the left, go to the $n^{th}$ digit, where $n$ is the number of significant figures. If the number is a decimal between $0$ and $1$, then start counting only when you reached the first non-zero digit. |

2. | Underline this digit and circle the digit to the right. |

3. | If the circled digit is between $5$ and $9$, add $1$ to the underlined digit. Otherwise, leave the digit unchanged. |

4. | Rewrite the number, changing every digit to the right of the underlined digit to a zero and ignoring decimals (if any). |

**Note:** The same rule applies for significant figures if the underlined digit is a $9$ and needs to be increased by $1$.

##### Example 2

*What is $184732.58$ when rounded to two significant figures?*

*First, underline the second digit from the left and circle the third digit. In this case, underline $8$ and circle $4$:*

$1\underline{8}\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {4}}}732.58$

*The circled digit is less than $5$, so don't change the underlined digit. Rewrite the number, changing every digit to the right of $8$ to a zero, and ignoring the decimals:*

$180000$

*Therefore, *$184732.58$* rounded to two significant figures is *$\underline{180000}$*.*

##### Example 3

*What is $0.003097232$ to three significant figures?*

*First, underline the third digit from the left, but start counting from the first **non-zero** digit, which is $3$. In this case, underline $9$ and circle $7$.*

$0.0030\underline{9}\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {7}}}232$

*The circled digit is greater than $4$, so increase the underlined digit by $1$. However, the underlined digit is $9$, so increase the number **to the left** by $1$ and write $9$ as $0$*:

$0.0031\underline{0}\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {7}}}232$

*Rewrite the number, ignoring any decimals to the right of the underlined digit:*

$0.00310$

*Therefore, *$0.003097232$* rounded to three significant figures is *$\underline{0.00310}$.

**Note: **You can see in this example that the zero on the end of the number counts as a significant figure. $0.0031$ would not be the correct answer, as this is only given to two significant figures.