{"id":1079,"date":"2023-03-29T16:24:06","date_gmt":"2023-03-29T16:24:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1079"},"modified":"2025-08-24T04:05:02","modified_gmt":"2025-08-24T04:05:02","slug":"decimals-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/decimals-learn-it-3\/","title":{"raw":"Decimals: Learn It 3","rendered":"Decimals: Learn It 3"},"content":{"raw":"<h2>Locating and Ordering Decimals With a Number Line<\/h2>\r\n<p>Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.<\/p>\r\n<section class=\"textbox recall\">Let's take a moment to refresh our memories on how to order fractions on a number line.\r\n\r\n<ol>\r\n\t<li><strong>Determine the range of the number line<\/strong>: Look at the numbers given and determine the lowest and highest values that need to be included on the number line.<\/li>\r\n\t<li><strong>Divide the number line into equal parts<\/strong>: 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.<\/li>\r\n\t<li><strong>Locate the whole numbers<\/strong>: If the fraction includes a whole number, locate that number on the number line first.<\/li>\r\n\t<li><strong>Locate the numerator<\/strong>: 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.<\/li>\r\n\t<li><strong>Mark the point on the number line<\/strong>: Once you have found the correct division of the number line, mark the point with the fraction you are trying to locate.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Locate [latex]0.4[\/latex] on a number line.[reveal-answer q=\"157184\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"157184\"]<br \/>\r\nThe decimal [latex]0.4[\/latex] is equivalent to [latex]{\\Large\\frac{4}{10}}[\/latex], so [latex]0.4[\/latex] is located between [latex]0[\/latex] and [latex]1[\/latex]. On a number line, divide the interval between [latex]0[\/latex] and [latex]1[\/latex] into [latex]10[\/latex] equal parts and place marks to separate the parts. Label the marks [latex]0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0[\/latex]. We write [latex]0[\/latex] as [latex]0.0[\/latex] and [latex]1[\/latex] as [latex]1.0[\/latex], so that the numbers are consistently in tenths. Finally, mark [latex]0.4[\/latex] on the number line.<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221458\/CNX_BMath_Figure_05_01_010_img.png\" alt=\"A number line 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.\" width=\"900\" height=\"64\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<h2>Order Decimals<\/h2>\r\n<p>Which is larger, [latex]0.04[\/latex] or [latex]0.40?[\/latex]<\/p>\r\n<p>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:<\/p>\r\n<p style=\"text-align: center;\">[latex]0.40&gt;0.04[\/latex]<\/p>\r\n<p>In previous chapters, we used the number line to order numbers.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ a &lt; 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 &gt; 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]<\/p>\r\n<p>Where are [latex]0.04[\/latex] and [latex]0.40[\/latex] located on the number line?<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221458\/CNX_BMath_Figure_05_01_010_img.png\" alt=\"A number line 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.\" width=\"900\" height=\"64\" \/> Figure 1. Use the number line locate 0.04 and 0.4. Which is larger?[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>We see that [latex]0.40[\/latex] is to the right of [latex]0.04[\/latex]. So we know [latex]0.40&gt;0.04[\/latex].<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Order Decimals<\/strong><\/p>\r\n<ol id=\"eip-id1168468520524\" class=\"stepwise\">\r\n\t<li>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.<\/li>\r\n\t<li>Compare the numbers to the right of the decimal point as if they were whole numbers.<\/li>\r\n\t<li>Order the numbers using the appropriate inequality sign.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p>How does [latex]0.31[\/latex] compare to [latex]0.308?[\/latex] This doesn\u2019t 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.<\/p>\r\n<table id=\"eip-id1168466250128\" class=\"unnumbered unstyled\" summary=\"The numbers 0.31 and 0.308 are shown. The first step says, \">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]0.31[\/latex]<\/td>\r\n<td>[latex]0.308[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Convert to fractions.<\/td>\r\n<td>[latex]{\\Large\\frac{31}{100}}[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{308}{1000}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We need a common denominator to compare them.<\/td>\r\n<td>[latex]{\\Large\\frac{31\\cdot\\color{red}{10}}{100\\cdot\\color{red}{10}}}[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{308}{1000}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]{\\Large\\frac{310}{1000}}[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{308}{1000}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Because [latex]310&gt;308[\/latex], we know that [latex]{\\Large\\frac{310}{1000}}&gt;{\\Large\\frac{308}{1000}}[\/latex]. Therefore, [latex]0.31&gt;0.308[\/latex].<\/p>\r\n<p>Notice what we did in converting [latex]0.31[\/latex] to a fraction\u2014we 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.<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{31}{100}}={\\Large\\frac{310}{1000}}\\text{ and }0.31=0.310[\/latex]<\/p>\r\n<p>If two decimals have the same value, they are said to be <strong>equivalent decimals<\/strong>. Remember, writing zeros at the end of a decimal does not change its value.<\/p>\r\n<p style=\"text-align: center;\">[latex]0.31=0.310[\/latex]<\/p>\r\n<p>We say [latex]0.31[\/latex] and [latex]0.310[\/latex] are <strong>equivalent decimals<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>equivalent decimals<\/h3>\r\n<p>Two decimals are <strong>equivalent decimals<\/strong> if they convert to equivalent fractions.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6768[\/ohm2_question]<\/section>\r\n<p>When we order negative decimals, it is important to remember how to order negative integers.\u00a0<\/p>\r\n<section class=\"textbox recall\">\r\n<p>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&amp;gt;-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&amp;lt;-6[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3695[\/ohm2_question]<\/section>","rendered":"<h2>Locating and Ordering Decimals With a Number Line<\/h2>\n<p>Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.<\/p>\n<section class=\"textbox recall\">Let&#8217;s take a moment to refresh our memories on how to order fractions on a number line.<\/p>\n<ol>\n<li><strong>Determine the range of the number line<\/strong>: Look at the numbers given and determine the lowest and highest values that need to be included on the number line.<\/li>\n<li><strong>Divide the number line into equal parts<\/strong>: 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.<\/li>\n<li><strong>Locate the whole numbers<\/strong>: If the fraction includes a whole number, locate that number on the number line first.<\/li>\n<li><strong>Locate the numerator<\/strong>: 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.<\/li>\n<li><strong>Mark the point on the number line<\/strong>: Once you have found the correct division of the number line, mark the point with the fraction you are trying to locate.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Locate [latex]0.4[\/latex] on a number line.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q157184\">Show Solution<\/button><\/p>\n<div id=\"q157184\" class=\"hidden-answer\" style=\"display: none\">\nThe decimal [latex]0.4[\/latex] is equivalent to [latex]{\\Large\\frac{4}{10}}[\/latex], so [latex]0.4[\/latex] is located between [latex]0[\/latex] and [latex]1[\/latex]. On a number line, divide the interval between [latex]0[\/latex] and [latex]1[\/latex] into [latex]10[\/latex] equal parts and place marks to separate the parts. Label the marks [latex]0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0[\/latex]. We write [latex]0[\/latex] as [latex]0.0[\/latex] and [latex]1[\/latex] as [latex]1.0[\/latex], so that the numbers are consistently in tenths. Finally, mark [latex]0.4[\/latex] on the number line.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221458\/CNX_BMath_Figure_05_01_010_img.png\" alt=\"A number line 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.\" width=\"900\" height=\"64\" \/><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Order Decimals<\/h2>\n<p>Which is larger, [latex]0.04[\/latex] or [latex]0.40?[\/latex]<\/p>\n<p>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:<\/p>\n<p style=\"text-align: center;\">[latex]0.40>0.04[\/latex]<\/p>\n<p>In previous chapters, we used the number line to order numbers.<\/p>\n<p style=\"text-align: center;\">[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]<\/p>\n<p>Where are [latex]0.04[\/latex] and [latex]0.40[\/latex] located on the number line?<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 900px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221458\/CNX_BMath_Figure_05_01_010_img.png\" alt=\"A number line 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.\" width=\"900\" height=\"64\" \/><figcaption class=\"wp-caption-text\">Figure 1. Use the number line locate 0.04 and 0.4. Which is larger?<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>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].<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Order Decimals<\/strong><\/p>\n<ol id=\"eip-id1168468520524\" class=\"stepwise\">\n<li>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.<\/li>\n<li>Compare the numbers to the right of the decimal point as if they were whole numbers.<\/li>\n<li>Order the numbers using the appropriate inequality sign.<\/li>\n<\/ol>\n<\/section>\n<p>How does [latex]0.31[\/latex] compare to [latex]0.308?[\/latex] This doesn\u2019t 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.<\/p>\n<table id=\"eip-id1168466250128\" class=\"unnumbered unstyled\" summary=\"The numbers 0.31 and 0.308 are shown. The first step says,\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]0.31[\/latex]<\/td>\n<td>[latex]0.308[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert to fractions.<\/td>\n<td>[latex]{\\Large\\frac{31}{100}}[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{308}{1000}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We need a common denominator to compare them.<\/td>\n<td>[latex]{\\Large\\frac{31\\cdot\\color{red}{10}}{100\\cdot\\color{red}{10}}}[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{308}{1000}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]{\\Large\\frac{310}{1000}}[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{308}{1000}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>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].<\/p>\n<p>Notice what we did in converting [latex]0.31[\/latex] to a fraction\u2014we 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.<\/p>\n<p style=\"text-align: center;\">[latex]{\\Large\\frac{31}{100}}={\\Large\\frac{310}{1000}}\\text{ and }0.31=0.310[\/latex]<\/p>\n<p>If two decimals have the same value, they are said to be <strong>equivalent decimals<\/strong>. Remember, writing zeros at the end of a decimal does not change its value.<\/p>\n<p style=\"text-align: center;\">[latex]0.31=0.310[\/latex]<\/p>\n<p>We say [latex]0.31[\/latex] and [latex]0.310[\/latex] are <strong>equivalent decimals<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>equivalent decimals<\/h3>\n<p>Two decimals are <strong>equivalent decimals<\/strong> if they convert to equivalent fractions.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6768\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6768&theme=lumen&iframe_resize_id=ohm6768&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>When we order negative decimals, it is important to remember how to order negative integers.\u00a0<\/p>\n<section class=\"textbox recall\">\n<p>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&gt;-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&lt;-6[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3695\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3695&theme=lumen&iframe_resize_id=ohm3695&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"Lynn Marecek & MaryAnne Anthony-Smith\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/prealgebra\/pages\/5-1-decimals\",\"project\":\"5.1 Decimals\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":54,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Prealgebra","author":"Lynn Marecek & MaryAnne Anthony-Smith","organization":"OpenStax","url":"https:\/\/openstax.org\/books\/prealgebra\/pages\/5-1-decimals","project":"5.1 Decimals","license":"cc-by","license_terms":"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1079"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":25,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1079\/revisions"}],"predecessor-version":[{"id":15602,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1079\/revisions\/15602"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1079\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1079"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1079"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1079"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}