{"id":8235,"date":"2023-09-29T14:19:56","date_gmt":"2023-09-29T14:19:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8235"},"modified":"2025-08-28T03:47:20","modified_gmt":"2025-08-28T03:47:20","slug":"rational-and-irrational-numbers-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/rational-and-irrational-numbers-background-youll-need-1\/","title":{"raw":"Rational and Irrational Numbers: Background You\u2019ll Need 1","rendered":"Rational and Irrational Numbers: Background You\u2019ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Recognize different number types, such as whole numbers, integers, and counting numbers&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4929,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Recognize different number types, such as whole numbers, integers, and counting numbers<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Identify Counting Numbers and Whole Numbers<\/h2>\r\n<p>In elementary mathematics, we frequently use the most fundamental set of numbers, which we typically employ for counting objects: [latex]1, 2, 3, 4, 5, ...[\/latex] and so forth. These numbers are referred to as the <strong>counting numbers<\/strong>.<\/p>\r\n<p>The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the <strong>whole numbers<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Whole and Counting Numbers<\/h3>\r\n<p><strong>Counting numbers<\/strong> start with [latex]1[\/latex] and continue.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]1,2,3,4,5\\dots[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<strong>Whole numbers<\/strong> are the counting numbers and zero.\r\n\r\n<p style=\"text-align: center;\">[latex]0,1,2,3,4,5\\dots[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">The notation \u201c\u2026\u201d is called an ellipsis, which is another way to show \u201cand so on\u201d, or that the pattern continues endlessly.<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3165[\/ohm2_question]<\/section>\r\n<h2>Opposites and Integers<\/h2>\r\n<p>On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers [latex]2[\/latex] and [latex]-2[\/latex] are the same distance from zero, they are called <strong>opposites<\/strong>. The opposite of [latex]2[\/latex] is [latex]-2[\/latex], and the opposite of [latex]-2[\/latex] is [latex]2[\/latex] as shown in figure(a). Similarly, [latex]3[\/latex] and [latex]-3[\/latex] are opposites as shown in figure(b).<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220126\/CNX_BMath_Figure_03_01_016.png\" alt=\"This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line, the statement is the numbers negative 2 and 2 are opposites. The second number line has the points negative 3 and 3 labeled. Below the number line is the statement negative 3 and 3 are opposites.\" width=\"440\" height=\"323\" \/> Figure 1. Number line a has the opposite integers -2 and 2 labeled. Number line b has the opposite integers -3 and 3 labeled[\/caption]\r\n<\/center>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Opposite<\/h3>\r\n<p>The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The notation [latex]-a[\/latex] is read <em>the opposite of<\/em> [latex]a[\/latex].<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">Just as the same word in English can have different meanings, the same symbol in math can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol \"[latex]-[\/latex]\" in three different ways.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]10 - 4[\/latex]<\/td>\r\n<td>Between two numbers, the symbol indicates the operation of subtraction.We read [latex]10 - 4[\/latex] as [latex]10[\/latex] <em>minus<\/em> [latex]4[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-8[\/latex]<\/td>\r\n<td>In front of a number, the symbol indicates a negative number.We read [latex]-8[\/latex] as <em>negative eight<\/em>.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-x[\/latex]<\/td>\r\n<td>In front of a variable or a number, it indicates the opposite.We read [latex]-x[\/latex] as <em>the opposite of<\/em> [latex]x[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-\\left(-2\\right)[\/latex]<\/td>\r\n<td>Here we have two signs. The sign in the parentheses indicates that the number is negative [latex]2[\/latex]. The sign outside the parentheses indicates the opposite.We read [latex]-\\left(-2\\right)[\/latex] as <em>the opposite of<\/em> [latex]-2[\/latex], which is [latex]2[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Integers<\/h3>\r\n<p><strong>Integers<\/strong> are counting numbers, their opposites, and zero.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dots{-3,-2,-1,0,1,2,3}\\dots [\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13821[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Recognize different number types, such as whole numbers, integers, and counting numbers&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4929,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Recognize different number types, such as whole numbers, integers, and counting numbers<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Identify Counting Numbers and Whole Numbers<\/h2>\n<p>In elementary mathematics, we frequently use the most fundamental set of numbers, which we typically employ for counting objects: [latex]1, 2, 3, 4, 5, ...[\/latex] and so forth. These numbers are referred to as the <strong>counting numbers<\/strong>.<\/p>\n<p>The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the <strong>whole numbers<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Whole and Counting Numbers<\/h3>\n<p><strong>Counting numbers<\/strong> start with [latex]1[\/latex] and continue.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]1,2,3,4,5\\dots[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Whole numbers<\/strong> are the counting numbers and zero.<\/p>\n<p style=\"text-align: center;\">[latex]0,1,2,3,4,5\\dots[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">The notation \u201c\u2026\u201d is called an ellipsis, which is another way to show \u201cand so on\u201d, or that the pattern continues endlessly.<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3165\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3165&theme=lumen&iframe_resize_id=ohm3165&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Opposites and Integers<\/h2>\n<p>On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers [latex]2[\/latex] and [latex]-2[\/latex] are the same distance from zero, they are called <strong>opposites<\/strong>. The opposite of [latex]2[\/latex] is [latex]-2[\/latex], and the opposite of [latex]-2[\/latex] is [latex]2[\/latex] as shown in figure(a). Similarly, [latex]3[\/latex] and [latex]-3[\/latex] are opposites as shown in figure(b).<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 440px\" 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\/24220126\/CNX_BMath_Figure_03_01_016.png\" alt=\"This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line, the statement is the numbers negative 2 and 2 are opposites. The second number line has the points negative 3 and 3 labeled. Below the number line is the statement negative 3 and 3 are opposites.\" width=\"440\" height=\"323\" \/><figcaption class=\"wp-caption-text\">Figure 1. Number line a has the opposite integers -2 and 2 labeled. Number line b has the opposite integers -3 and 3 labeled<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Opposite<\/h3>\n<p>The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.<\/p>\n<p>&nbsp;<\/p>\n<p>The notation [latex]-a[\/latex] is read <em>the opposite of<\/em> [latex]a[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">Just as the same word in English can have different meanings, the same symbol in math can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol &#8220;[latex]-[\/latex]&#8221; in three different ways.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]10 - 4[\/latex]<\/td>\n<td>Between two numbers, the symbol indicates the operation of subtraction.We read [latex]10 - 4[\/latex] as [latex]10[\/latex] <em>minus<\/em> [latex]4[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>[latex]-8[\/latex]<\/td>\n<td>In front of a number, the symbol indicates a negative number.We read [latex]-8[\/latex] as <em>negative eight<\/em>.<\/td>\n<\/tr>\n<tr>\n<td>[latex]-x[\/latex]<\/td>\n<td>In front of a variable or a number, it indicates the opposite.We read [latex]-x[\/latex] as <em>the opposite of<\/em> [latex]x[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>[latex]-\\left(-2\\right)[\/latex]<\/td>\n<td>Here we have two signs. The sign in the parentheses indicates that the number is negative [latex]2[\/latex]. The sign outside the parentheses indicates the opposite.We read [latex]-\\left(-2\\right)[\/latex] as <em>the opposite of<\/em> [latex]-2[\/latex], which is [latex]2[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Integers<\/h3>\n<p><strong>Integers<\/strong> are counting numbers, their opposites, and zero.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\dots{-3,-2,-1,0,1,2,3}\\dots[\/latex]<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13821\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13821&theme=lumen&iframe_resize_id=ohm13821&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"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":53,"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\/8235"}],"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":18,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8235\/revisions"}],"predecessor-version":[{"id":15795,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8235\/revisions\/15795"}],"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\/8235\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8235"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8235"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8235"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}