Ordering Decimals

Which is larger, $0.04$ or $0.40?$

If you think of this as money, you know that $\text{\$0.40}$ (forty cents) is greater than $\text{\$0.04}$ (four cents). So,

$0.40>0.04$

In previous tutorials, we used the number line to order numbers.

$\begin{array}{} ab\phantom{\rule{0.5em}{0ex}}‘a\phantom{\rule{0.2em}{0ex}}\text{is greater than}\phantom{\rule{0.2em}{0ex}}b’\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is to the right of}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{on the number line}\hfill \end{array}$

Where are $0.04$ and $0.40$ located on the number line?

A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot between 0.0 and 0.1 labeled as 0.04. There is another red dot at 0.4.

We see that $0.40$ is to the right of $0.04.$ So we know $0.40>0.04.$

How does $0.31$ compare to $0.308?$ This doesn’t translate into money to make the comparison easy. But if we convert $0.31$ and $0.308$ to fractions, we can tell which is larger.

	$0.31$	$0.308$
Convert to fractions.	$\frac{31}{100}$	$\frac{308}{1000}$
We need a common denominator to compare them.		30810003081000
	$\frac{310}{1000}$	$\frac{308}{1000}$

Because $310>308,$ we know that $\frac{310}{1000}>\frac{308}{1000}.$ Therefore, $0.31>0.308.$

Notice what we did in converting $0.31$ to a fraction—we started with the fraction $\frac{31}{100}$ and ended with the equivalent fraction $\frac{310}{1000}.$ Converting $\frac{310}{1000}$ back to a decimal gives $0.310.$ So $0.31$ is equivalent to $0.310.$ Writing zeros at the end of a decimal does not change its value.

$\cfrac{31}{100}=\cfrac{310}{1000}\phantom{\rule{0.4em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}0.31=0.310$

If two decimals have the same value, they are said to be equivalent decimals.

$0.31=0.310$

We say $0.31$ and $0.310$ are equivalent decimals.

Definition: Equivalent Decimals

Two decimals are equivalent decimals if they convert to equivalent fractions.

Remember, writing zeros at the end of a decimal does not change its value.

How to Order decimals.

Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match.
Compare the numbers to the right of the decimal point as if they were whole numbers.
Order the numbers using the appropriate inequality sign.

Example

Order the following decimals using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\text{>:}$

$\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.6$
$\phantom{\rule{0.2em}{0ex}}0.83\phantom{\rule{0.2em}{0ex}}\_\_0.803$

Solution


	$\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.6$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.6.	$\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.60$
Compare the numbers to the right of the decimal point as if they were whole numbers.	$64>60$
Order the numbers using the appropriate inequality sign.	$0.64>0.60$ $0.64>0.6$


	$\phantom{\rule{0.2em}{0ex}}0.83\phantom{\rule{0.2em}{0ex}}\_\_0.803$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.83.	$\phantom{\rule{0.2em}{0ex}}0.830\phantom{\rule{0.2em}{0ex}}\_\_0.803$
Compare the numbers to the right of the decimal point as if they were whole numbers.	$830>803$
Order the numbers using the appropriate inequality sign.	$0.830>0.803$ $0.83>0.803$

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because $-2$ lies to the right of $-3$ on the number line, we know that $-2>-3.$ Similarly, smaller numbers lie to the left on the number line. For example, because $-9$ lies to the left of $-6$ on the number line, we know that $-9<-6.$

A number line is shown with integers from negative 10 to 0. Blue dots are placed on negative nine and negative six. Red dots are placed at negative two and negative three.

If we zoomed in on the interval between $0$ and $-1,$ we would see in the same way that $-0.2>-0.3\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-0.9<-0.6.$

Example

Use $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>$ to order. $-0.1\_\_-0.8.$

Solution

	$\phantom{\rule{0.2em}{0ex}}-0.1\phantom{\rule{0.2em}{0ex}}\_\_-0.8$
Write the numbers one under the other, lining up the decimal points.	$-0.1$ $-0.8$
They have the same number of digits.
Since $-1>-8,-1$ tenth is greater than $-8$ tenths.	$-0.1>-0.8$

	\(0.31\)	\(0.308\)
Convert to fractions.	\(\frac{31}{100}\)	\(\frac{308}{1000}\)
We need a common denominator to compare them.		30810003081000
	\(\frac{310}{1000}\)	\(\frac{308}{1000}\)


	\(\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.6\)
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.6.	\(\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.60\)
Compare the numbers to the right of the decimal point as if they were whole numbers.	\(64>60\)
Order the numbers using the appropriate inequality sign.	\(0.64>0.60\) \(0.64>0.6\)


	\(\phantom{\rule{0.2em}{0ex}}0.83\phantom{\rule{0.2em}{0ex}}\_\_0.803\)
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.83.	\(\phantom{\rule{0.2em}{0ex}}0.830\phantom{\rule{0.2em}{0ex}}\_\_0.803\)
Compare the numbers to the right of the decimal point as if they were whole numbers.	\(830>803\)
Order the numbers using the appropriate inequality sign.	\(0.830>0.803\) \(0.83>0.803\)

	\(\phantom{\rule{0.2em}{0ex}}-0.1\phantom{\rule{0.2em}{0ex}}\_\_-0.8\)
Write the numbers one under the other, lining up the decimal points.	\(-0.1\) \(-0.8\)
They have the same number of digits.
Since \(-1>-8,-1\) tenth is greater than \(-8\) tenths.	\(-0.1>-0.8\)

Ordering Decimals