{"id":8126,"date":"2023-09-20T17:48:23","date_gmt":"2023-09-20T17:48:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8126"},"modified":"2024-10-18T20:56:48","modified_gmt":"2024-10-18T20:56:48","slug":"real-numbers-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/real-numbers-fresh-take\/","title":{"raw":"Real Numbers: Fresh Take","rendered":"Real Numbers: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Categorize real numbers into counting, whole, rational, irrational, or integers<\/li>\r\n\t<li>Recognize and use the properties of real numbers<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Defining and Identifying Real Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Real numbers<\/strong> are numbers that are either rational or irrational.<\/p>\r\n<p>Real numbers are the backbone of our numerical universe, encompassing both rational and irrational numbers. Think of real numbers as a big family gathering where everyone from counting numbers to irrational numbers shows up.<\/p>\r\n<\/div>\r\n<section class=\"textbox recall\">\r\n<ul>\r\n\t<li><strong>Natural Numbers or Counting Numbers:<\/strong> Start with [latex]1[\/latex] and continue. [latex] 1,2,3,4,5\u2026[\/latex]<\/li>\r\n\t<li><strong>Whole Numbers:<\/strong> Counting numbers plus zero. [latex] 0,1,2,3,4,5\u2026[\/latex]<\/li>\r\n\t<li><strong>Integers:<\/strong> Whole numbers and their negative counterparts.<\/li>\r\n\t<li><strong>Rational Numbers:<\/strong> Numbers that can be written in the form [latex]{\\Large\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne o[\/latex]. In decimal form, the numbers terminate or repeat.<\/li>\r\n\t<li><strong>Irrational Numbers:<\/strong> Numbers that can't be expressed as a simple fraction. In decimal form, the numbers do not repeat or terminate.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>The following mini-lesson provides more examples of how to classify real numbers.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/htP2goe31MM?si=SBkOgZ731CkZmnWH\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Identifying+Sets+of+Real+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Sets of Real Numbers\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Recognizing Properties of Real Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n\r\nReal numbers have a set of rules that they play by, and these rules are called properties. Think of these properties as the \"grammar\" of math, setting the stage for how numbers interact with each other. Whether it's addition, multiplication, or even the use of parentheses, these properties ensure that numbers behave in a predictable way.\r\n\r\n<ul>\r\n\t<li><strong>Distributive Property:<\/strong> Imagine you're sharing a pizza equally among friends. If you have [latex]5[\/latex] friends and [latex]3[\/latex] pizzas, each friend gets a share from each pizza. Mathematically, [latex]a\u00d7(b+c)=a\u00d7b+a\u00d7c[\/latex].<\/li>\r\n\t<li><strong>Commutative Properties:<\/strong> In music, a playlist on shuffle plays songs in any order but the music is still enjoyable. Similarly, in math, whether it's addition or multiplication, the order doesn't matter. For addition, [latex]a+b=b+a[\/latex], and for multiplication, [latex]a\u00d7b=b\u00d7a[\/latex].<\/li>\r\n\t<li><strong>Associative Properties:<\/strong> Think of a team huddle in sports. It doesn't matter how players are grouped; the huddle remains the same. In math, whether you're adding or multiplying, the numbers stick together like a team. For addition, [latex]a+(b+c)=(a+b)+c[\/latex], and for multiplication, [latex]a\u00d7(b\u00d7c)=(a\u00d7b)\u00d7c[\/latex].<\/li>\r\n\t<li><strong>Identity Properties:<\/strong> Zero and one are like the superheroes of the number world. Zero, when added to any number, doesn't change its identity ([latex]a+0=a[\/latex]). One, when multiplied with any number, keeps it the same ([latex]a\u00d71=a[\/latex]).<\/li>\r\n\t<li><strong>Inverse Properties:<\/strong> These are your \"undo\" buttons in math. For addition, every number has a negative that will bring it back to zero ([latex]a+(\u2212a)=0[\/latex]). For multiplication, every non-zero number has a reciprocal that will bring it back to one ([latex]a\u00d7\\frac{1}{a}=1[\/latex], provided [latex]a \\ne 0[\/latex]).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>In each of the following, identify which property of the real numbers is being applied.<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]5\u00d7(6+19)=5\u00d76+5\u00d719[\/latex]<\/li>\r\n\t<li>[latex]41.7+(\u221241.7)=0[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>distributive property<\/li>\r\n\t<li>additive inverse property<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Watch the following video for more information on the properties of real numbers.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8SFm8Os_4C8?si=RjmMkYbkCBEgI5qi\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Properties+of+Real+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProperties of Real Numbers\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Categorize real numbers into counting, whole, rational, irrational, or integers<\/li>\n<li>Recognize and use the properties of real numbers<\/li>\n<\/ul>\n<\/section>\n<h2>Defining and Identifying Real Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Real numbers<\/strong> are numbers that are either rational or irrational.<\/p>\n<p>Real numbers are the backbone of our numerical universe, encompassing both rational and irrational numbers. Think of real numbers as a big family gathering where everyone from counting numbers to irrational numbers shows up.<\/p>\n<\/div>\n<section class=\"textbox recall\">\n<ul>\n<li><strong>Natural Numbers or Counting Numbers:<\/strong> Start with [latex]1[\/latex] and continue. [latex]1,2,3,4,5\u2026[\/latex]<\/li>\n<li><strong>Whole Numbers:<\/strong> Counting numbers plus zero. [latex]0,1,2,3,4,5\u2026[\/latex]<\/li>\n<li><strong>Integers:<\/strong> Whole numbers and their negative counterparts.<\/li>\n<li><strong>Rational Numbers:<\/strong> Numbers that can be written in the form [latex]{\\Large\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne o[\/latex]. In decimal form, the numbers terminate or repeat.<\/li>\n<li><strong>Irrational Numbers:<\/strong> Numbers that can&#8217;t be expressed as a simple fraction. In decimal form, the numbers do not repeat or terminate.<\/li>\n<\/ul>\n<\/section>\n<p>The following mini-lesson provides more examples of how to classify real numbers.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/htP2goe31MM?si=SBkOgZ731CkZmnWH\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Identifying+Sets+of+Real+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Sets of Real Numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Recognizing Properties of Real Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Real numbers have a set of rules that they play by, and these rules are called properties. Think of these properties as the &#8220;grammar&#8221; of math, setting the stage for how numbers interact with each other. Whether it&#8217;s addition, multiplication, or even the use of parentheses, these properties ensure that numbers behave in a predictable way.<\/p>\n<ul>\n<li><strong>Distributive Property:<\/strong> Imagine you&#8217;re sharing a pizza equally among friends. If you have [latex]5[\/latex] friends and [latex]3[\/latex] pizzas, each friend gets a share from each pizza. Mathematically, [latex]a\u00d7(b+c)=a\u00d7b+a\u00d7c[\/latex].<\/li>\n<li><strong>Commutative Properties:<\/strong> In music, a playlist on shuffle plays songs in any order but the music is still enjoyable. Similarly, in math, whether it&#8217;s addition or multiplication, the order doesn&#8217;t matter. For addition, [latex]a+b=b+a[\/latex], and for multiplication, [latex]a\u00d7b=b\u00d7a[\/latex].<\/li>\n<li><strong>Associative Properties:<\/strong> Think of a team huddle in sports. It doesn&#8217;t matter how players are grouped; the huddle remains the same. In math, whether you&#8217;re adding or multiplying, the numbers stick together like a team. For addition, [latex]a+(b+c)=(a+b)+c[\/latex], and for multiplication, [latex]a\u00d7(b\u00d7c)=(a\u00d7b)\u00d7c[\/latex].<\/li>\n<li><strong>Identity Properties:<\/strong> Zero and one are like the superheroes of the number world. Zero, when added to any number, doesn&#8217;t change its identity ([latex]a+0=a[\/latex]). One, when multiplied with any number, keeps it the same ([latex]a\u00d71=a[\/latex]).<\/li>\n<li><strong>Inverse Properties:<\/strong> These are your &#8220;undo&#8221; buttons in math. For addition, every number has a negative that will bring it back to zero ([latex]a+(\u2212a)=0[\/latex]). For multiplication, every non-zero number has a reciprocal that will bring it back to one ([latex]a\u00d7\\frac{1}{a}=1[\/latex], provided [latex]a \\ne 0[\/latex]).<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>In each of the following, identify which property of the real numbers is being applied.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]5\u00d7(6+19)=5\u00d76+5\u00d719[\/latex]<\/li>\n<li>[latex]41.7+(\u221241.7)=0[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214538\">Show Solution<\/button> <\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>distributive property<\/li>\n<li>additive inverse property<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more information on the properties of real numbers.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8SFm8Os_4C8?si=RjmMkYbkCBEgI5qi\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Properties+of+Real+Numbers.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProperties of Real Numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Identifying Sets of Real Numbers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/htP2goe31MM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Properties of Real Numbers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/8SFm8Os_4C8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":53,"module-header":"fresh_take","content_attributions":[{"type":"cc-attribution","description":"Identifying Sets of Real Numbers","author":"James Sousa (Mathispower4u.com)","organization":"","url":"https:\/\/youtu.be\/htP2goe31MM","project":"","license":"cc-by","license_terms":""},{"type":"cc-attribution","description":"Properties of Real Numbers","author":"James Sousa (Mathispower4u.com)","organization":"","url":"https:\/\/youtu.be\/8SFm8Os_4C8","project":"","license":"cc-by","license_terms":""}],"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\/8126"}],"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":14,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8126\/revisions"}],"predecessor-version":[{"id":12836,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8126\/revisions\/12836"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/53"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8126\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8126"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8126"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8126"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}