{"id":8678,"date":"2024-01-08T21:56:45","date_gmt":"2024-01-08T16:26:45","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8678"},"modified":"2024-01-17T23:11:23","modified_gmt":"2024-01-17T17:41:23","slug":"integration-of-fractional-part-of-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/","title":{"rendered":"Integration of fractional part of x"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Integration of fractional part of x<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The integration of the fractional part function, often denoted as <code>{x}<\/code> or <code>frac(x)<\/code>, is a bit more complex than standard functions because the fractional part function is not continuous everywhere. The fractional part of <code>x<\/code> is defined as <code>x - [x]<\/code>, where <code>[x]<\/code> is the floor function, representing the greatest integer less than or equal to <code>x<\/code>.<\/li>\n\n\n\n<li>The fractional part function has a sawtooth wave pattern and is periodic with a period of 1. To integrate this function over an interval, you typically break the interval into subintervals where the function is continuous.<\/li>\n<\/ul>\n\n\n\n<p>For example, to integrate the fractional part function from <code>a<\/code> to <code>b<\/code>, where <code>a<\/code> and <code>b<\/code> are real numbers and <code>a &lt; b<\/code>, you would do the following:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Identify the Intervals: Break the interval <code>[a, b]<\/code> into subintervals at each integer point. For example, if <code>a = 1.5<\/code> and <code>b = 4.5<\/code>, the subintervals would be <code>[1.5, 2]<\/code>, <code>[2, 3]<\/code>, <code>[3, 4]<\/code>, and <code>[4, 4.5]<\/code>.<\/li>\n\n\n\n<li>Integrate Over Each Subinterval: The integral of the fractional part function over each subinterval <code>[n, n+1)<\/code> (where <code>n<\/code> is an integer) is straightforward, as the function is simply <code>x - n<\/code> over this interval. So, the integral over <code>[n, n+1)<\/code> is <code>(1\/2)(n+1)^2 - (1\/2)n^2<\/code>.<\/li>\n\n\n\n<li>Sum the Integrals: Add up the integrals over each subinterval.<\/li>\n<\/ol>\n\n\n\n<p>For a general formula, if <code>a<\/code> and <code>b<\/code> are not integers, you would need to calculate the integrals over the first partial interval <code>[a, ?a?]<\/code> and the last partial interval <code>[?b?, b]<\/code> separately, and then sum the integrals over the full intervals in between.<\/p>\n\n\n\n<p><strong>Example 1<\/strong><br>$\\int_{1.5}^{4.5} \\left\\{ x \\right \\} dx$<\/p>\n\n\n\n<p>Let&#8217;s integrate the fractional part of <code>x<\/code> from 1.5 to 4.5. We break this into intervals:<\/p>\n\n\n\n<p>From 1.5 to 2:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The fractional part function is <code>x - 1<\/code>.<\/li>\n\n\n\n<li>Integral: $\\int_{1.5}^{2} (x &#8211; 1) dx$.<\/li>\n<\/ul>\n\n\n\n<p>From 2 to 3:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The fractional part function is <code>x - 2<\/code>.<\/li>\n\n\n\n<li>Integral: $\\int_{2}^{3} (x &#8211; 2) dx$.<\/li>\n<\/ul>\n\n\n\n<p>From 3 to 4:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The fractional part function is <code>x - 3<\/code>.<\/li>\n\n\n\n<li>Integral: $\\int_{3}^{4} (x &#8211; 3) dx$.<\/li>\n<\/ul>\n\n\n\n<p>From 4 to 4.5:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The fractional part function is <code>x - 4<\/code>.<\/li>\n\n\n\n<li>Integral: $\\int_{4}^{4.5} (x &#8211; 4) dx$.<\/li>\n<\/ul>\n\n\n\n<p>Now, calculate each integral:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>$\\int_{1.5}^{2} (x &#8211; 1) dx = \\left[\\frac{1}{2}x^2 &#8211; x\\right]_{1.5}^{2} = \\left(\\frac{1}{2}(2)^2 &#8211; 2\\right) &#8211; \\left(\\frac{1}{2}(1.5)^2 &#8211; 1.5\\right) = .375$.<\/li>\n\n\n\n<li>$\\int_{2}^{3} (x &#8211; 2) dx = \\left[\\frac{1}{2}x^2 &#8211; 2x\\right]_{2}^{3} = \\left(\\frac{1}{2}(3)^2 &#8211; 2(3)\\right) &#8211; \\left(\\frac{1}{2}(2)^2 &#8211; 2(2)\\right) =.5$.<\/li>\n\n\n\n<li>$\\int_{3}^{4} (x &#8211; 3) dx = \\left[\\frac{1}{2}x^2 &#8211; 3x\\right]_{3}^{4} = \\left(\\frac{1}{2}(4)^2 &#8211; 3(4)\\right) &#8211; \\left(\\frac{1}{2}(3)^2 &#8211; 3(3)\\right)= .5$.<\/li>\n\n\n\n<li>$\\int_{4}^{4.5} (x &#8211; 4) dx = \\left[\\frac{1}{2}x^2 &#8211; 4x\\right]_{4}^{4.5} = \\left(\\frac{1}{2}(4.5)^2 &#8211; 4(4.5)\\right) &#8211; \\left(\\frac{1}{2}(4)^2 &#8211; 4(4)\\right) =.125 $.<\/li>\n<\/ol>\n\n\n\n<p>Finally, Sum is 1.5 units<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Integration of the fractional part function using Periodic method<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The fractional part function is periodic with a period of 1. This means that the function repeats its values every interval of length 1. Mathematically, this is expressed as {x + n} = {x} for any integer n.<\/li>\n\n\n\n<li>When integrating the fractional part function over a full period (or multiple periods), the integral can be simplified due to its periodic nature. For example, the integral of {x} from 0 to 1 (a full period) can be calculated directly, and this result can be used to find integrals over longer intervals that span multiple periods.<\/li>\n<\/ul>\n\n\n\n<p>$\\int_{0}^{n}  \\left\\{ x \\right \\}  \\; dx =n \\int_{0}^{1} \\left\\{ x \\right \\} \\; dx$<\/p>\n\n\n\n<p>Now Since {x} equals x in this interval (as there are no integers between 0 and 1), the integral is straightforward<\/p>\n\n\n\n<p>$\\int_{0}^{n}  \\left\\{ x \\right \\}  \\; dx =n \\int_{0}^{1}  \\left\\{ x \\right \\}  \\; dx = n \\int_{0}^{1} x \\; dx = n\\left[ \\frac{x^2}{2} \\right]_{0}^{1} = \\frac{n}{2} $<\/p>\n\n\n\n<p><strong>Example<\/strong><br>Evaluate<\/p>\n\n\n\n<p>$ \\int_{0}^{100}  \\left\\{ x \\right \\}  \\, dx $<br><strong>Solution<\/strong><br>$$ \\int_{0}^{100}  \\left\\{ x \\right \\}  \\, dx= 100 \\times \\int_{0}^{1}  \\left\\{ x \\right \\}  \\, dx = 100 \\times \\frac{1}{2} = \\frac{100}{2}= 50 $$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved Examples<\/h2>\n\n\n\n<p>Question 1<\/p>\n\n\n\n<p>$$ \\int_{0}^{1}  \\left\\{ x \\right \\} [x+1] \\, dx $$ <br>here {x} denotes fractional part and [.] denotes greatest integer function<\/p>\n\n\n\n<p>Solution<\/p>\n\n\n\n<p>$ \\int_{0}^{1} \\left\\{   x  \\right\\}  [x+1] \\, dx = \\int_{0}^{1} (x &#8211; [x]) [x+1] \\, dx$<\/p>\n\n\n\n<p>now for $x \\in (0,1)$, we have [x]=0 and [x+1] =1<\/p>\n\n\n\n<p>Hence<\/p>\n\n\n\n<p>$\\int_{0}^{1} (x &#8211; [x]) [x+1] \\, dx = \\int_{0}^{1}  x \\, dx = \\frac {1}{2} $<\/p>\n\n\n\n<div class=\"wp-block-group has-ast-global-color-4-background-color has-background is-layout-constrained wp-container-core-group-is-layout-ca99af60 wp-block-group-is-layout-constrained\" style=\"border-radius:17px;margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:var(--wp--preset--spacing--20);padding-right:var(--wp--preset--spacing--50);padding-bottom:var(--wp--preset--spacing--20);padding-left:var(--wp--preset--spacing--50)\"><div class=\"wp-block-group__inner-container\">\n<p class=\"has-medium-font-size\">Other Integration Related Articles<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\" style=\"padding-right:0;padding-left:0;font-size:16px\"><table style=\"border-style:none;border-width:0px\"><tbody><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-modulus-function\/\" data-type=\"post\" data-id=\"8632\">Integration of modulus function<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-tan-x\/\" data-type=\"post\" data-id=\"8622\">Integration of tan x<\/a><\/td><td><\/td><td><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-sec-x\/\" data-type=\"post\" data-id=\"8628\">Integration of sec x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-greatest-integer-function\/\" data-type=\"post\" data-id=\"8599\">Integration of greatest integer function<\/a><\/td><td><\/td><td><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-trigonometric-functions\/\" data-type=\"post\" data-id=\"8595\">Integration of trigonometric functions<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-cot-x\/\" data-type=\"post\" data-id=\"8591\">Integration of cot x<\/a><\/td><td><\/td><td><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-cosec-x\/\" data-type=\"post\" data-id=\"8548\">Integration of cosec x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-sinx\/\" data-type=\"post\" data-id=\"8530\">Integration of sinx<\/a><\/td><td><\/td><td><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-irrational-functions\/\">Integration of irrational functions<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-root-x\/\">integration of root x<\/a><\/td><td><\/td><td><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/\">Integration of fractional part of x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-cos-square-x\/\">Integration of cos square x<\/a><\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Integration of fractional part of x For example, to integrate the fractional part function from a to b, where a and b are real numbers and a &lt; b, you would do the following: For a general formula, if a and b are not integers, you would need to calculate the integrals over the first [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[506],"class_list":["post-8678","post","type-post","status-publish","format-standard","hentry","category-maths","tag-integration"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Integration of fractional part of x - physicscatalyst&#039;s Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of fractional part of x - physicscatalyst&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"Integration of fractional part of x For example, to integrate the fractional part function from a to b, where a and b are real numbers and a &lt; b, you would do the following: For a general formula, if a and b are not integers, you would need to calculate the integrals over the first [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/\" \/>\n<meta property=\"og:site_name\" content=\"physicscatalyst&#039;s Blog\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/PhysicsCatalyst\" \/>\n<meta property=\"article:published_time\" content=\"2024-01-08T16:26:45+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-01-17T17:41:23+00:00\" \/>\n<meta name=\"author\" content=\"physicscatalyst\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"physicscatalyst\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Integration of fractional part of x - physicscatalyst&#039;s Blog","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/","og_locale":"en_US","og_type":"article","og_title":"Integration of fractional part of x - physicscatalyst&#039;s Blog","og_description":"Integration of fractional part of x For example, to integrate the fractional part function from a to b, where a and b are real numbers and a &lt; b, you would do the following: For a general formula, if a and b are not integers, you would need to calculate the integrals over the first [&hellip;]","og_url":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/","og_site_name":"physicscatalyst&#039;s Blog","article_publisher":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_author":"https:\/\/www.facebook.com\/PhysicsCatalyst","article_published_time":"2024-01-08T16:26:45+00:00","article_modified_time":"2024-01-17T17:41:23+00:00","author":"physicscatalyst","twitter_misc":{"Written by":"physicscatalyst","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/#article","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/"},"author":{"name":"physicscatalyst","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f"},"headline":"Integration of fractional part of x","datePublished":"2024-01-08T16:26:45+00:00","dateModified":"2024-01-17T17:41:23+00:00","mainEntityOfPage":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/"},"wordCount":595,"commentCount":0,"publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"keywords":["Integration"],"articleSection":["Maths"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/","url":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/","name":"Integration of fractional part of x - physicscatalyst&#039;s Blog","isPartOf":{"@id":"https:\/\/physicscatalyst.com\/article\/#website"},"datePublished":"2024-01-08T16:26:45+00:00","dateModified":"2024-01-17T17:41:23+00:00","breadcrumb":{"@id":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/physicscatalyst.com\/article\/"},{"@type":"ListItem","position":2,"name":"Maths","item":"https:\/\/physicscatalyst.com\/article\/maths\/"},{"@type":"ListItem","position":3,"name":"Integration of fractional part of x"}]},{"@type":"WebSite","@id":"https:\/\/physicscatalyst.com\/article\/#website","url":"https:\/\/physicscatalyst.com\/article\/","name":"physicscatalyst's Blog","description":"Learn free for class 9th, 10th science\/maths , 12th and IIT-JEE Physics and maths.","publisher":{"@id":"https:\/\/physicscatalyst.com\/article\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/physicscatalyst.com\/article\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/physicscatalyst.com\/article\/#organization","name":"physicscatalyst","url":"https:\/\/physicscatalyst.com\/article\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/","url":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","contentUrl":"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2024\/08\/cropped-logo-1.jpg","width":96,"height":96,"caption":"physicscatalyst"},"image":{"@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst","https:\/\/www.youtube.com\/user\/thephysicscatalyst","https:\/\/www.instagram.com\/physicscatalyst\/"]},{"@type":"Person","@id":"https:\/\/physicscatalyst.com\/article\/#\/schema\/person\/9b302efdc9b32e459cb1e61ab7506d3f","name":"physicscatalyst","sameAs":["https:\/\/physicscatalyst.com","https:\/\/www.facebook.com\/PhysicsCatalyst","https:\/\/x.com\/physicscatalyst"]}]}},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"shareaholic-thumbnail":false},"uagb_author_info":{"display_name":"physicscatalyst","author_link":"https:\/\/physicscatalyst.com\/article\/author\/physicscatalyst\/"},"uagb_comment_info":0,"uagb_excerpt":"Integration of fractional part of x For example, to integrate the fractional part function from a to b, where a and b are real numbers and a &lt; b, you would do the following: For a general formula, if a and b are not integers, you would need to calculate the integrals over the first&hellip;","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8678","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=8678"}],"version-history":[{"count":6,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8678\/revisions"}],"predecessor-version":[{"id":8791,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8678\/revisions\/8791"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=8678"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=8678"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=8678"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}