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

$3125$

$5142$

$528$

$428$

$13$

$99$

$91$

$3$

$5$

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

$5142$

$3125$

$528$

$428$

$99$

$91$

$13$

$5$

$3$

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 $99$

$0.0028$

$0.01$

$0.005$

$0.089$

$0.0023$

$4.01$

$4.2$

$12.0001$

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

$0.0028$

$0.005$

$0.0023$

$0.01$

$0.089$

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

$0.0023$

$0.0028$

$0.005$

$0.01$

$0.089$

$4.01$

4.2

$12.0001$

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}