{"id":1441,"date":"2023-04-07T17:47:12","date_gmt":"2023-04-07T17:47:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1441"},"modified":"2025-08-26T23:22:50","modified_gmt":"2025-08-26T23:22:50","slug":"fractals-background-youll-need","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-background-youll-need\/","title":{"raw":"Fractals: Background You'll Need 1","rendered":"Fractals: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>How to break down numbers<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"title\">Prime Factorization<\/h2>\r\n<p>The prime factorization of a number is the product of prime numbers that equals the number. You may want to refer to the following list of prime numbers less than [latex]50[\/latex] as you work through this section.<\/p>\r\n<p style=\"text-align: center;\">[latex]2,3,5,7,11,13,17,19,23,29,31,37,41,43,47[\/latex]<\/p>\r\n<section class=\"textbox proTip\">Memorizing the first five prime numbers --[latex]2, 3, 5, 7, 11[\/latex]-- will coming in handy when reducing fractions.<\/section>\r\n<h3>Prime Factorization Using the Factor Tree Method<\/h3>\r\n<p>One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment\u2014a \"branch\" of the factor tree. If a factor is prime, we circle it (like a bud on a tree), and do not factor that \"branch\" any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree. We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.<\/p>\r\n<p>For example, let\u2019s find the prime factorization of [latex]36[\/latex]. We can start with any factor pair such as [latex]3[\/latex] and [latex]12[\/latex]. We write [latex]3[\/latex] and [latex]12[\/latex] below [latex]36[\/latex] with branches connecting them.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"189\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220021\/CNX_BMath_Figure_02_05_018_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.\" width=\"189\" height=\"102\" \/> Figure 1. 36 can factor into 12 and 3[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]12[\/latex] is composite, so we need to find its factors. Let\u2019s use [latex]3[\/latex] and [latex]4[\/latex]. We write these factors on the tree under the [latex]12[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"180\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220022\/CNX_BMath_Figure_02_05_019_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.\" width=\"180\" height=\"146\" \/> Figure 2. 12 can be factored into 3 and 4[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]4[\/latex] is composite, and it factors into [latex]2\\cdot 2[\/latex]. We write these factors under the [latex]4[\/latex]. Since [latex]2[\/latex] is prime, we circle both [latex]2\\text{s}[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"185\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220023\/CNX_BMath_Figure_02_05_009_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.\" width=\"185\" height=\"211\" \/> Figure 3. 4 can be factored into 2 and 2[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.<\/p>\r\n<p style=\"text-align: center;\">[latex]3\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/p>\r\n<p>In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2\\cdot 2\\cdot 3\\cdot 3\\\\ \\\\ {2}^{2}\\cdot {3}^{2}\\end{array}[\/latex]<\/p>\r\n<p>Note that we could have started our factor tree with any factor pair of [latex]36[\/latex]. We chose [latex]12[\/latex] and [latex]3[\/latex], but the same result would have been the same if we had started with [latex]2[\/latex] and [latex]18,4[\/latex] and [latex]9[\/latex], or [latex]6[\/latex] and [latex]6[\/latex].<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p class=\"title\"><strong>How to: Find the prime factorization of a composite number using the tree method<\/strong><\/p>\r\n<ol id=\"eip-id1168469875559\" class=\"stepwise\">\r\n\t<li>Find any factor pair of the given number, and use these numbers to create two branches.<\/li>\r\n\t<li>If a factor is prime, that branch is complete. Circle the prime.<\/li>\r\n\t<li>If a factor is not prime, write it as the product of a factor pair and continue the process.<\/li>\r\n\t<li>Write the composite number as the product of all the circled primes.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Find the prime factorization of [latex]48[\/latex] using the factor tree method.[reveal-answer q=\"714411\"]Show Answer[\/reveal-answer] [hidden-answer a=\"714411\"]\r\n\r\n<table id=\"eip-id1168466026521\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The figure shows multiple factor trees with the number 48 at the top. In the first tree two branches are splitting out from under 48. The branches use the factor pair 2 and 24 with 24 at the end of the right branch and 2 at the end of the left branch. Two has a circle around it to show that it is prime and that branch is complete. In the next tree the previous tree is repeated, but now with two branches splitting out from under 24. The branches use the factor pair 4 and 6 with 6 at the end of the right branch and 4 at the end of the left branch. Neither of these factors is circled because they are not prime. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 6. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 6 use the factor pair 2 and 3. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 48 is made up of all of the circled numbers from the factor tree which is 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.\">\r\n<tbody>\r\n<tr>\r\n<td>We can start our tree using any factor pair of [latex]48[\/latex]. Let's use [latex]2[\/latex] and [latex]24[\/latex]. We circle the [latex]2[\/latex] because it is prime and so that branch is complete.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220024\/CNX_BMath_Figure_02_05_022_img-01.png\" alt=\"A factor tree with the number 48 at the top. Two branches are splitting out from under 48. The right branch has a number 24 at the end. The left branch has the number 2 with a circle around it.\" width=\"293\" height=\"108\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now we will factor [latex]24[\/latex]. Let's use [latex]4[\/latex] and [latex]6[\/latex].<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220025\/CNX_BMath_Figure_02_05_022_img-02.png\" alt=\"A factor tree with the number 48 at the top. Two branches are splitting out from under 48.  The left branch has the number 2 with a circle around it. The right branch has a number 24 at the end. Two branches split out from under the 24. The left one ends at the number 4 and the right one ends at the number 6.\" width=\"293\" height=\"181\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Neither factor is prime, so we do not circle either. We factor the [latex]4[\/latex], using [latex]2[\/latex] and [latex]2[\/latex]. We factor [latex]6[\/latex], using [latex]2[\/latex] and [latex]3[\/latex]. We circle the [latex]2 [\/latex]s and the [latex]3[\/latex] since they are prime. Now all of the branches end in a prime.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220026\/CNX_BMath_Figure_02_05_022_img-03.png\" alt=\"A factor tree with the number 48 at the top. Two branches are splitting out from under 48.  The left branch has the number 2 with a blue circle around it. The right branch has a number 24 at the end. Two branches split out from under the 24. The left one ends at the number 4 and the right one ends at the number 6. Two branches split out from under the 4, each ending in a 2 in a blue circle. Two branches also split out from beneath the 6, one ending in a 2 in a blue circle and the other in a 3 in a red circle.\" width=\"293\" height=\"266\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the product of the circled numbers.<\/td>\r\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write in exponential form.<\/td>\r\n<td>[latex]{2}^{4}\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Check this on your own by multiplying all the factors together. The result should be [latex]48[\/latex].<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3804[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3805[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>How to break down numbers<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"title\">Prime Factorization<\/h2>\n<p>The prime factorization of a number is the product of prime numbers that equals the number. You may want to refer to the following list of prime numbers less than [latex]50[\/latex] as you work through this section.<\/p>\n<p style=\"text-align: center;\">[latex]2,3,5,7,11,13,17,19,23,29,31,37,41,43,47[\/latex]<\/p>\n<section class=\"textbox proTip\">Memorizing the first five prime numbers &#8212;[latex]2, 3, 5, 7, 11[\/latex]&#8212; will coming in handy when reducing fractions.<\/section>\n<h3>Prime Factorization Using the Factor Tree Method<\/h3>\n<p>One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment\u2014a &#8220;branch&#8221; of the factor tree. If a factor is prime, we circle it (like a bud on a tree), and do not factor that &#8220;branch&#8221; any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree. We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.<\/p>\n<p>For example, let\u2019s find the prime factorization of [latex]36[\/latex]. We can start with any factor pair such as [latex]3[\/latex] and [latex]12[\/latex]. We write [latex]3[\/latex] and [latex]12[\/latex] below [latex]36[\/latex] with branches connecting them.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 189px\" 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\/24220021\/CNX_BMath_Figure_02_05_018_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.\" width=\"189\" height=\"102\" \/><figcaption class=\"wp-caption-text\">Figure 1. 36 can factor into 12 and 3<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]12[\/latex] is composite, so we need to find its factors. Let\u2019s use [latex]3[\/latex] and [latex]4[\/latex]. We write these factors on the tree under the [latex]12[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 180px\" 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\/24220022\/CNX_BMath_Figure_02_05_019_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.\" width=\"180\" height=\"146\" \/><figcaption class=\"wp-caption-text\">Figure 2. 12 can be factored into 3 and 4<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]4[\/latex] is composite, and it factors into [latex]2\\cdot 2[\/latex]. We write these factors under the [latex]4[\/latex]. Since [latex]2[\/latex] is prime, we circle both [latex]2\\text{s}[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 185px\" 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\/24220023\/CNX_BMath_Figure_02_05_009_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.\" width=\"185\" height=\"211\" \/><figcaption class=\"wp-caption-text\">Figure 3. 4 can be factored into 2 and 2<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.<\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/p>\n<p>In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2\\cdot 2\\cdot 3\\cdot 3\\\\ \\\\ {2}^{2}\\cdot {3}^{2}\\end{array}[\/latex]<\/p>\n<p>Note that we could have started our factor tree with any factor pair of [latex]36[\/latex]. We chose [latex]12[\/latex] and [latex]3[\/latex], but the same result would have been the same if we had started with [latex]2[\/latex] and [latex]18,4[\/latex] and [latex]9[\/latex], or [latex]6[\/latex] and [latex]6[\/latex].<\/p>\n<section class=\"textbox questionHelp\">\n<p class=\"title\"><strong>How to: Find the prime factorization of a composite number using the tree method<\/strong><\/p>\n<ol id=\"eip-id1168469875559\" class=\"stepwise\">\n<li>Find any factor pair of the given number, and use these numbers to create two branches.<\/li>\n<li>If a factor is prime, that branch is complete. Circle the prime.<\/li>\n<li>If a factor is not prime, write it as the product of a factor pair and continue the process.<\/li>\n<li>Write the composite number as the product of all the circled primes.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Find the prime factorization of [latex]48[\/latex] using the factor tree method.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q714411\">Show Answer<\/button> <\/p>\n<div id=\"q714411\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168466026521\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The figure shows multiple factor trees with the number 48 at the top. In the first tree two branches are splitting out from under 48. The branches use the factor pair 2 and 24 with 24 at the end of the right branch and 2 at the end of the left branch. Two has a circle around it to show that it is prime and that branch is complete. In the next tree the previous tree is repeated, but now with two branches splitting out from under 24. The branches use the factor pair 4 and 6 with 6 at the end of the right branch and 4 at the end of the left branch. Neither of these factors is circled because they are not prime. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 6. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 6 use the factor pair 2 and 3. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 48 is made up of all of the circled numbers from the factor tree which is 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.\">\n<tbody>\n<tr>\n<td>We can start our tree using any factor pair of [latex]48[\/latex]. Let&#8217;s use [latex]2[\/latex] and [latex]24[\/latex]. We circle the [latex]2[\/latex] because it is prime and so that branch is complete.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220024\/CNX_BMath_Figure_02_05_022_img-01.png\" alt=\"A factor tree with the number 48 at the top. Two branches are splitting out from under 48. The right branch has a number 24 at the end. The left branch has the number 2 with a circle around it.\" width=\"293\" height=\"108\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now we will factor [latex]24[\/latex]. Let&#8217;s use [latex]4[\/latex] and [latex]6[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220025\/CNX_BMath_Figure_02_05_022_img-02.png\" alt=\"A factor tree with the number 48 at the top. Two branches are splitting out from under 48.  The left branch has the number 2 with a circle around it. The right branch has a number 24 at the end. Two branches split out from under the 24. The left one ends at the number 4 and the right one ends at the number 6.\" width=\"293\" height=\"181\" \/><\/td>\n<\/tr>\n<tr>\n<td>Neither factor is prime, so we do not circle either. We factor the [latex]4[\/latex], using [latex]2[\/latex] and [latex]2[\/latex]. We factor [latex]6[\/latex], using [latex]2[\/latex] and [latex]3[\/latex]. We circle the [latex]2[\/latex]s and the [latex]3[\/latex] since they are prime. Now all of the branches end in a prime.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220026\/CNX_BMath_Figure_02_05_022_img-03.png\" alt=\"A factor tree with the number 48 at the top. Two branches are splitting out from under 48.  The left branch has the number 2 with a blue circle around it. The right branch has a number 24 at the end. Two branches split out from under the 24. The left one ends at the number 4 and the right one ends at the number 6. Two branches split out from under the 4, each ending in a 2 in a blue circle. Two branches also split out from beneath the 6, one ending in a 2 in a blue circle and the other in a 3 in a red circle.\" width=\"293\" height=\"266\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the product of the circled numbers.<\/td>\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write in exponential form.<\/td>\n<td>[latex]{2}^{4}\\cdot 3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Check this on your own by multiplying all the factors together. The result should be [latex]48[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3804\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3804&theme=lumen&iframe_resize_id=ohm3804&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3805\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3805&theme=lumen&iframe_resize_id=ohm3805&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"background_you_need","content_attributions":[],"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\/1441"}],"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\/16"}],"version-history":[{"count":32,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1441\/revisions"}],"predecessor-version":[{"id":15688,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1441\/revisions\/15688"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1441\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1441"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1441"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1441"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1441"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}