{"id":204,"date":"2023-02-09T19:24:24","date_gmt":"2023-02-09T19:24:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=204"},"modified":"2025-08-29T21:04:18","modified_gmt":"2025-08-29T21:04:18","slug":"historical-counting-systems-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/historical-counting-systems-background-youll-need-1\/","title":{"raw":"Historical Counting Systems: Background You'll Need 1","rendered":"Historical Counting Systems: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand where and how to round numbers<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Identify the Place Value of a Digit<\/h2>\r\n<p>Identifying the place value of a digit is a foundational step in mastering number rounding techniques. By using a place value chart, we can effectively categorize digits into groups of three, or periods, such as ones, thousands, millions, and beyond. This organization is not just for clarity\u2014it directly informs where to round numbers for simplification, as rounding typically occurs at these period boundaries. In written form, we use commas to separate these periods, providing visual cues for where rounding can take place in larger numbers.<\/p>\r\n<p>The chart below\u00a0shows how the number [latex]5,278,194[\/latex] is written in a place value chart.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"431\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215214\/CNX_BMath_Figure_01_01_011.png\" alt=\"A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'.\" width=\"431\" height=\"331\" \/> Figure 1. A place value chart example with the number \"5,278,194\"[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<ul id=\"fs-id930962\">\r\n\t<li>The digit [latex]5[\/latex] is in the millions place. Its value is [latex]5,000,000[\/latex].<\/li>\r\n\t<li>The digit [latex]2[\/latex] is in the hundred thousands place. Its value is [latex]200,000[\/latex].<\/li>\r\n\t<li>The digit [latex]7[\/latex] is in the ten thousands place. Its value is [latex]70,000[\/latex].<\/li>\r\n\t<li>The digit [latex]8[\/latex] is in the thousands place. Its value is [latex]8,000[\/latex].<\/li>\r\n\t<li>The digit [latex]1[\/latex] is in the hundreds place. Its value is [latex]100[\/latex].<\/li>\r\n\t<li>The digit [latex]9[\/latex] is in the tens place. Its value is [latex]90[\/latex].<\/li>\r\n\t<li>The digit [latex]4[\/latex] is in the ones place. Its value is [latex]4[\/latex].<br \/>\r\n<section class=\"textbox example\">In the number [latex]63,407,218[\/latex]; find the place value of each of the following digits:\r\n\r\n<ol id=\"fs-id2218367\" class=\"circled\">\r\n\t<li>[latex]7[\/latex]<\/li>\r\n\t<li>[latex]0[\/latex]<\/li>\r\n\t<li>[latex]1[\/latex]<\/li>\r\n\t<li>[latex]6[\/latex]<\/li>\r\n\t<li>[latex]3[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"17862\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"17862\"]<\/p>\r\n<p>Write the number in a place value chart, starting at the right.<\/p>\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215216\/CNX_BMath_Figure_01_01_012_img.png\" alt=\"A chart with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 7 values in the next row are blank. Starting with the ninth column, the values are '6', '3', '4', '0', '7', '2' , '1' and '8'.\" width=\"431\" height=\"297\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<ul id=\"eip-id1168898159450\">\r\n\t<li>The [latex]7[\/latex] is in the thousands place.<\/li>\r\n\t<li>The [latex]0[\/latex] is in the ten thousands place.<\/li>\r\n\t<li>The [latex]1[\/latex] is in the tens place.<\/li>\r\n\t<li>The [latex]6[\/latex] is in the ten millions place.<\/li>\r\n\t<li>The [latex]3[\/latex] is in the millions place.<\/li>\r\n<\/ul>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2907[\/ohm2_question]<\/section>\r\n<h2>Rounding Numbers<\/h2>\r\n<p>Rounding numbers is a process used to simplify figures to make them easier to work with or understand. It involves adjusting a number to the nearest place value, which can be especially useful in estimating results or making numbers more readable. Here's a step-by-step guide to help you round whole numbers accurately to any specific place value.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Round a Whole Number to a Specific Place Value<\/strong><\/p>\r\n<ol id=\"eip-id1168287215567\" class=\"stepwise\">\r\n\t<li>Locate the given place value. All digits to the left of that place value do not change.<\/li>\r\n\t<li>Underline the digit to the right of the given place value.<\/li>\r\n\t<li>Determine if this digit is greater than or equal to [latex]5[\/latex].<br \/>\r\n<ul id=\"fs-id2486248\">\r\n\t<li>Yes\u2014add [latex]1[\/latex] to the digit in the given place value.<\/li>\r\n\t<li>No\u2014do not change the digit in the given place value.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Replace all digits to the right of the given place value with zeros.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">Round [latex]843[\/latex] to the nearest ten.<br \/>\r\n[reveal-answer q=\"17812\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"17812\"]\r\n\r\n<table id=\"eip-659\" class=\"unnumbered unstyled\" summary=\"This table is two columns. The first column states locate the tens place. Next to this is the number 843 with the label \">\r\n<tbody>\r\n<tr>\r\n<td>Locate the tens place.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215241\/CNX_BMath_Figure_01_01_034_img-01.png\" alt=\"The number 843 with the label &quot;tens place&quot; pointing to the 4.\" width=\"93\" height=\"78\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Underline the digit to the right of the tens place.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215242\/CNX_BMath_Figure_01_01_034_img-02.png\" alt=\"The number 843 with the 3 underlined.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]3[\/latex] is less than [latex]5[\/latex], do not change the digit in the tens place.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215242\/CNX_BMath_Figure_01_01_034_img-03.png\" alt=\"The number 843 with the 3 underlined.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Replace all digits to the right of the tens place with zeros.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215243\/CNX_BMath_Figure_01_01_034_img-04.png\" alt=\"The number 840 with the 0 underlined.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rounding [latex]843[\/latex] to the nearest ten gives [latex]840[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2910[\/ohm2_question]<\/section>\r\n<\/li>\r\n<\/ul>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand where and how to round numbers<\/li>\n<\/ul>\n<\/section>\n<h2>Identify the Place Value of a Digit<\/h2>\n<p>Identifying the place value of a digit is a foundational step in mastering number rounding techniques. By using a place value chart, we can effectively categorize digits into groups of three, or periods, such as ones, thousands, millions, and beyond. This organization is not just for clarity\u2014it directly informs where to round numbers for simplification, as rounding typically occurs at these period boundaries. In written form, we use commas to separate these periods, providing visual cues for where rounding can take place in larger numbers.<\/p>\n<p>The chart below\u00a0shows how the number [latex]5,278,194[\/latex] is written in a place value chart.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 431px\" 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\/24215214\/CNX_BMath_Figure_01_01_011.png\" alt=\"A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'.\" width=\"431\" height=\"331\" \/><figcaption class=\"wp-caption-text\">Figure 1. A place value chart example with the number &#8220;5,278,194&#8221;<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<ul id=\"fs-id930962\">\n<li>The digit [latex]5[\/latex] is in the millions place. Its value is [latex]5,000,000[\/latex].<\/li>\n<li>The digit [latex]2[\/latex] is in the hundred thousands place. Its value is [latex]200,000[\/latex].<\/li>\n<li>The digit [latex]7[\/latex] is in the ten thousands place. Its value is [latex]70,000[\/latex].<\/li>\n<li>The digit [latex]8[\/latex] is in the thousands place. Its value is [latex]8,000[\/latex].<\/li>\n<li>The digit [latex]1[\/latex] is in the hundreds place. Its value is [latex]100[\/latex].<\/li>\n<li>The digit [latex]9[\/latex] is in the tens place. Its value is [latex]90[\/latex].<\/li>\n<li>The digit [latex]4[\/latex] is in the ones place. Its value is [latex]4[\/latex].<br \/>\n<section class=\"textbox example\">In the number [latex]63,407,218[\/latex]; find the place value of each of the following digits:<\/p>\n<ol id=\"fs-id2218367\" class=\"circled\">\n<li>[latex]7[\/latex]<\/li>\n<li>[latex]0[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]6[\/latex]<\/li>\n<li>[latex]3[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q17862\">Show Solution<\/button><\/p>\n<div id=\"q17862\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write the number in a place value chart, starting at the right.<\/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\/24215216\/CNX_BMath_Figure_01_01_012_img.png\" alt=\"A chart with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 7 values in the next row are blank. Starting with the ninth column, the values are '6', '3', '4', '0', '7', '2' , '1' and '8'.\" width=\"431\" height=\"297\" \/><\/div>\n<p>&nbsp;<\/p>\n<ul id=\"eip-id1168898159450\">\n<li>The [latex]7[\/latex] is in the thousands place.<\/li>\n<li>The [latex]0[\/latex] is in the ten thousands place.<\/li>\n<li>The [latex]1[\/latex] is in the tens place.<\/li>\n<li>The [latex]6[\/latex] is in the ten millions place.<\/li>\n<li>The [latex]3[\/latex] is in the millions place.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2907\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2907&theme=lumen&iframe_resize_id=ohm2907&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Rounding Numbers<\/h2>\n<p>Rounding numbers is a process used to simplify figures to make them easier to work with or understand. It involves adjusting a number to the nearest place value, which can be especially useful in estimating results or making numbers more readable. Here&#8217;s a step-by-step guide to help you round whole numbers accurately to any specific place value.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Round a Whole Number to a Specific Place Value<\/strong><\/p>\n<ol id=\"eip-id1168287215567\" class=\"stepwise\">\n<li>Locate the given place value. All digits to the left of that place value do not change.<\/li>\n<li>Underline the digit to the right of the given place value.<\/li>\n<li>Determine if this digit is greater than or equal to [latex]5[\/latex].\n<ul id=\"fs-id2486248\">\n<li>Yes\u2014add [latex]1[\/latex] to the digit in the given place value.<\/li>\n<li>No\u2014do not change the digit in the given place value.<\/li>\n<\/ul>\n<\/li>\n<li>Replace all digits to the right of the given place value with zeros.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Round [latex]843[\/latex] to the nearest ten.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q17812\">Show Solution<\/button><\/p>\n<div id=\"q17812\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-659\" class=\"unnumbered unstyled\" summary=\"This table is two columns. The first column states locate the tens place. Next to this is the number 843 with the label\">\n<tbody>\n<tr>\n<td>Locate the tens place.<\/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\/24215241\/CNX_BMath_Figure_01_01_034_img-01.png\" alt=\"The number 843 with the label &quot;tens place&quot; pointing to the 4.\" width=\"93\" height=\"78\" \/><\/td>\n<\/tr>\n<tr>\n<td>Underline the digit to the right of the tens place.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215242\/CNX_BMath_Figure_01_01_034_img-02.png\" alt=\"The number 843 with the 3 underlined.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Since [latex]3[\/latex] is less than [latex]5[\/latex], do not change the digit in the tens place.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215242\/CNX_BMath_Figure_01_01_034_img-03.png\" alt=\"The number 843 with the 3 underlined.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Replace all digits to the right of the tens place with zeros.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215243\/CNX_BMath_Figure_01_01_034_img-04.png\" alt=\"The number 840 with the 0 underlined.\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rounding [latex]843[\/latex] to the nearest ten gives [latex]840[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2910\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2910&theme=lumen&iframe_resize_id=ohm2910&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/li>\n<\/ul>\n","protected":false},"author":16,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"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":40,"module-header":"background_you_need","content_attributions":[{"type":"cc-attribution","description":"Prealgebra","author":"","organization":"","url":"","project":"","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\/204"}],"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":24,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/204\/revisions"}],"predecessor-version":[{"id":15554,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/204\/revisions\/15554"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/40"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/204\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=204"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=204"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=204"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=204"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}