# Ordering numbers: Whole numbers and decimals

## In a nutshell

Ordering both whole numbers and decimal numbers requires the use of place values.

## Ordering whole numbers

To order whole numbers, follow this procedure:

#### procedure

- Group the numbers by how many digits each number has.

- Sort the numbers in each group by comparing the leftmost digits first.

- Write the sorted numbers in a list without the groups.

##### Example 1

*Sort the following numbers from largest to smallest:*

$13$ | $528$ | $99$ | $428$ | $3$ | $91$ | $5142$ | $5$ | $3125$ |

*First, group the numbers by number of digits. Because the question asks from largest to smallest, put the numbers with the most digits to the left.*

#### 4 digits | #### 3 digits | #### 2 digits | #### 1 digit |

| | | |

*Then, sort the numbers in each group from largest the smallest by comparing the leftmost digit:*

*For the **4 digit** group:*

*$5142$ is bigger than $3125$ because $5$ is bigger than $3$.*

*For the **3 digit** group: *

*$528$ is bigger than $428$ because $5$ is bigger than $4$.*

*For the **2 digit** group: *

*$13$ is the smallest. $99$ and $91$ have the same tens digit.*

*So, move on to the next digit and compare: *

*$99$ is bigger than $91$ because $9$ is bigger than $1$.*

*For the **1 digit** group: *

*$5$ is bigger than $3$.*

*Hence, the table should now look like this:*

#### 4 digits | #### 3 digits | #### 2 digits | #### 1 digit |

| | | |

*Remove the groups to get the ordered list:*

$\underline{5142}$ | $\underline{3125}$ | $\underline{528}$ | $\underline{428}$ | $\underline{99}$ | $\underline{91}$ | $\underline{13}$ | $\underline{5}$ | $\underline3$ |

## Ordering numbers with decimals

To order numbers that have decimals, use this method:

#### procedure

- Sort the whole number parts.

- Group the numbers that have the same "whole number parts" by how many initial zeroes are to the right of the decimal point.

- Sort each group by comparing the first non zero digits.

- Write the sorted numbers without the groups.

##### Example 2

*Sort the following numbers from smallest to largest:*

$0.0028$ | $4.01$ | $0.01$ | $0.005$ | $0.089$ | $12.0001$ | $4.2$ | $0.0023$ |

*First, sort the numbers by their integer parts:*

#### between $0$ and $1$ | #### Between $1$ and $9$ | #### between $10$ and $99$ |

$0.0028$ | $0.01$ | $0.005$ | $0.089$ | $0.0023$ | | | |

*There are only two numbers that begin with $4$, so they can be compared easily:*

* $4.01$ is smaller than $4.2$ because $0$ is smaller than $2$.*

*Next, sort the numbers between $0$* *and $1$ by how many zeroes come after the decimal point:*

#### 2 initial zeroes | #### 1 initial zero |

$0.0028$ | $0.005$ | $0.0023$ | | |

**Note:** Because this is from smallest to largest, put the numbers with the most zeroes to the left as they're inherently smaller.

*Sort each smaller group:*

**2 initial zeroes:**

*$0.005$ is the largest because $5$ is bigger than $2$. *

*$0.0023$ is smaller than $0.0028$ because $3$ is smaller than $8$*.

**1 initial zero:**

$0.01$ *is smaller than $0.089$ because $1$ is less than $8$.*

*Hence, the table becomes:*

#### between 0 and 1 | #### between 1 and 9 | #### between 10 and 99 |

#### 2 initial #### zeroes | #### 1 initial zero | $0.0023$ | $0.0028$ | $0.005$ | | | | | |

*Therefore, the numbers sorted from smallest to biggest are: *

$\underline{0.0023}$ | $\underline{0.0028}$ | $\underline{0.005}$ | $\underline{0.01}$ | $\underline{0.089}$ | $\underline{4.01}$ | $\underline{4.2}$ | $\underline{12.0001}$ |