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Study Guides > Prealgebra

Locating and Ordering Decimals With a Number Line

Learning Outcomes

  • Locate decimals on a number line
  • Order decimals using inequality notation

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Exercises

Locate 0.40.4 on a number line. Solution The decimal 0.40.4 is equivalent to 410\frac{4}{10}, so 0.40.4 is located between 00 and 11. On a number line, divide the interval between 00 and 11 into 1010 equal parts and place marks to separate the parts. Label the marks 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.00.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0. We write 00 as 0.00.0 and 11 as 1.01.0, so that the numbers are consistently in tenths. Finally, mark 0.40.4 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 at 0.4.
 

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[ohm_question]146228[/ohm_question] Locate 0.60.6 on a number line.

Answer: This image shows a number line from 0.0 to 1.0 and segmented into tenths. A point is plotted at 0.6 on the number line.

 

example

Locate −0.74-0.74 on a number line.

Answer: Solution The decimal −0.74-0.74 is equivalent to −74100-\frac{74}{100}, so it is located between 00 and −1-1. On a number line, mark off and label the multiples of −0.10-0.10 in the interval between 00 and −1-1 ( −0.10-0.10 , −0.20-0.20 , etc.) and mark −0.74-0.74 between −0.70-0.70 and −0.80-0.80, a little closer to −0.70-0.70 . A number line is shown with negative 1.00, negative 0.90, negative 0.80, negative 0.70, negative 0.60, negative 0.50, negative 0.40, negative 0.30, negative 0.20, negative 0.10, and 0.00 labeled. There is a red dot between negative 0.80 and negative 0.70 labeled as negative 0.74.

 

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[ohm_question]146577[/ohm_question] [ohm_question]146578[/ohm_question]
In the next video we show more examples of how to locate a decimal on teh number line. https://youtu.be/F3LAKsOBdNA

Order Decimals

Which is larger, 0.040.04 or 0.40?0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So, 0.40>0.040.40>0.04 In previous chapters, we used the number line to order numbers. a<b a is less than b when a is to the left of b on the number linea>b a is greater than b when a is to the right of b on the number line\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} Where are 0.040.04 and 0.400.40 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 0.400.40 is to the right of 0.040.04. So we know 0.40>0.040.40>0.04. How does 0.310.31 compare to 0.308?0.308? This doesn’t translate into money to make the comparison easy. But if we convert 0.310.31 and 0.3080.308 to fractions, we can tell which is larger.
0.310.31 0.3080.308
Convert to fractions. 31100\frac{31}{100} 3081000\frac{308}{1000}
We need a common denominator to compare them. 31⋅10100⋅10\frac{31\cdot\color{red}{10}}{100\cdot\color{red}{10}} 3081000\frac{308}{1000}
3101000\frac{310}{1000} 3081000\frac{308}{1000}
Because 310>308310>308, we know that 3101000>3081000\frac{310}{1000}>\frac{308}{1000}. Therefore, 0.31>0.3080.31>0.308. Notice what we did in converting 0.310.31 to a fraction—we started with the fraction 31100\frac{31}{100} and ended with the equivalent fraction 3101000\frac{310}{1000}. Converting 3101000\frac{310}{1000} back to a decimal gives 0.3100.310. So 0.310.31 is equivalent to 0.3100.310. Writing zeros at the end of a decimal does not change its value. 31100=3101000 and 0.31=0.310\frac{31}{100}=\frac{310}{1000}\text{ and }0.31=0.310 If two decimals have the same value, they are said to be equivalent decimals. 0.31=0.3100.31=0.310 We say 0.310.31 and 0.3100.310 are equivalent decimals.

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.

Order decimals

  1. 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.
  2. Compare the numbers to the right of the decimal point as if they were whole numbers.
  3. Order the numbers using the appropriate inequality sign.
   

example

Order the following decimals using <or>:<\text{or}\text{>:} ⓐ 0.64__0.6 ⓑ 0.83__0.803

Answer: Solution

ⓐ
0.64__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 0.60.6. 0.64__0.60
Compare the numbers to the right of the decimal point as if they were whole numbers. 64>6064>60
Order the numbers using the appropriate inequality sign. 0.64>0.600.64>0.60 0.64>0.60.64>0.6
ⓑ
0.83__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 0.830.83. 0.830__0.803
Compare the numbers to the right of the decimal point as if they were whole numbers. 830>803830>803
Order the numbers using the appropriate inequality sign. 0.830>0.8030.830>0.803 0.83>0.8030.83>0.803

 

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[ohm_question]146232[/ohm_question] [ohm_question]146237[/ohm_question] [ohm_question]146238[/ohm_question]  
  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 −2-2 lies to the right of −3-3 on the number line, we know that −2>−3-2>-3. Similarly, smaller numbers lie to the left on the number line. For example, because −9-9 lies to the left of −6-6 on the number line, we know that −9<−6-9<-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 00 and −1-1, we would see in the same way that −0.2>−0.3and−0.9<−0.6-0.2>-0.3\text{and}-0.9<-0.6.

example

Use <or><\text{or}>; to order. -0.1__ - 0.8.

Answer: Solution

-0.1__ - 0.8
Write the numbers one under the other, lining up the decimal points. −0.1-0.1 −0.8-0.8
They have the same number of digits.
Since −1>−8,−1-1>-8,-1 tenth is greater than −8-8 tenths. −0.1>−0.8-0.1>-0.8

 

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[ohm_question]146239[/ohm_question]
In teh following video lesson we show how to order decimals using inequality notation by comparing place values, and by using fractions. https://youtu.be/fjO3fnt3ABA

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