Locating and Ordering Decimals With a Number Line
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.
- Determine the range of the number line: Look at the numbers given and determine the lowest and highest values that need to be included on the number line.
- Divide the number line into equal parts: Decide how many equal parts you need to divide the number line into based on the denominator of the fraction. For example, if the denominator is [latex]4[/latex], divide the number line into [latex]4[/latex] equal parts.
- Locate the whole numbers: If the fraction includes a whole number, locate that number on the number line first.
- Locate the numerator: Look at the numerator of the fraction and find the corresponding division of the number line. For example, if the numerator is [latex]2[/latex] and the denominator is [latex]4[/latex], locate the second division on the number line.
- Mark the point on the number line: Once you have found the correct division of the number line, mark the point with the fraction you are trying to locate.
Order Decimals
Which is larger, [latex]0.04[/latex] or [latex]0.40?[/latex]
If you think of this as money, you know that [latex]$0.40[/latex] (forty cents) is greater than [latex]$0.04[/latex] (four cents). So:
[latex]0.40>0.04[/latex]
In previous chapters, we used the number line to order numbers.
[latex]\begin{array}{}\\ a < b\text{ , }a\text{ is less than }b\text{ when }a\text{ is to the left of }b\text{ on the number line}\hfill \\ a > b\text{ , }a\text{ is greater than }b\text{ when }a\text{ is to the right of }b\text{ on the number line}\hfill \end{array}[/latex]
Where are [latex]0.04[/latex] and [latex]0.40[/latex] located on the number line?
We see that [latex]0.40[/latex] is to the right of [latex]0.04[/latex]. So we know [latex]0.40>0.04[/latex].
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.
How does [latex]0.31[/latex] compare to [latex]0.308?[/latex] This doesn’t translate into money to make the comparison easy. But if we convert [latex]0.31[/latex] and [latex]0.308[/latex] to fractions, we can tell which is larger.
| [latex]0.31[/latex] | [latex]0.308[/latex] | |
| Convert to fractions. | [latex]{\Large\frac{31}{100}}[/latex] | [latex]{\Large\frac{308}{1000}}[/latex] |
| We need a common denominator to compare them. | [latex]{\Large\frac{31\cdot\color{red}{10}}{100\cdot\color{red}{10}}}[/latex] | [latex]{\Large\frac{308}{1000}}[/latex] |
| [latex]{\Large\frac{310}{1000}}[/latex] | [latex]{\Large\frac{308}{1000}}[/latex] |
Because [latex]310>308[/latex], we know that [latex]{\Large\frac{310}{1000}}>{\Large\frac{308}{1000}}[/latex]. Therefore, [latex]0.31>0.308[/latex].
Notice what we did in converting [latex]0.31[/latex] to a fraction—we started with the fraction [latex]\Large\frac{31}{100}[/latex] and ended with the equivalent fraction [latex]\Large\frac{310}{1000}[/latex]. Converting [latex]\Large\frac{310}{1000}[/latex] back to a decimal gives [latex]0.310[/latex]. So [latex]0.31[/latex] is equivalent to [latex]0.310[/latex]. Writing zeros at the end of a decimal does not change its value.
[latex]{\Large\frac{31}{100}}={\Large\frac{310}{1000}}\text{ and }0.31=0.310[/latex]
If two decimals have the same value, they are said to be equivalent decimals. Remember, writing zeros at the end of a decimal does not change its value.
[latex]0.31=0.310[/latex]
We say [latex]0.31[/latex] and [latex]0.310[/latex] are equivalent decimals.
equivalent decimals
Two decimals are equivalent decimals if they convert to equivalent fractions.
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 [latex]-2[/latex] lies to the right of [latex]-3[/latex] on the number line, we know that [latex]-2>-3[/latex]. Similarly, smaller numbers lie to the left on the number line. For example, because [latex]-9[/latex] lies to the left of [latex]-6[/latex] on the number line, we know that [latex]-9<-6[/latex].