{"id":451,"date":"2017-10-28T22:23:09","date_gmt":"2017-10-28T22:23:09","guid":{"rendered":"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/?post_type=chapter&#038;p=451"},"modified":"2017-12-29T21:06:06","modified_gmt":"2017-12-29T21:06:06","slug":"number-systems","status":"publish","type":"chapter","link":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/chapter\/number-systems\/","title":{"raw":"Number Systems","rendered":"Number Systems"},"content":{"raw":"[latexpage]\r\n\r\nOur number system is a western adaptation of the Hindu-Arabic numeral system developed somewhere between the first and fourth centuries AD.\u00a0 However, numbers have been recorded with tally marks throughout history.\u00a0 The Ishango Bone[footnote]Image of Ishango bone by Ben2 (Own work), CC-BY-SA-3.0 or CC BY-SA 2.5-2.0-1.0], via Wikimedia Commons.[\/footnote] from Africa is about 25,000 years old. It's the lower leg bone from a baboon, and contains tally marks. We know the marks were used for counting because they appear in distinct groups.\r\n<p style=\"text-align: center\"><a title=\"By Ben2 (Own work) [GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html), CC-BY-SA-3.0 (http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/) or CC BY-SA 2.5-2.0-1.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/2.5-2.0-1.0)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AOs_d'Ishango_IRSNB.JPG\"><img width=\"256\" alt=\"Ishango bone\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/4\/42\/Os_d%27Ishango_IRSNB.JPG\/256px-Os_d%27Ishango_IRSNB.JPG\" class=\"aligncenter\" height=\"415\" \/><\/a><\/p>\r\nThis reindeer antler[footnote]Image of antler By Ryan Somma from Occoquan, USA [CC BY-SA 2.0], via Wikimedia Commons[\/footnote] from France is about 15,000 years old, and also shows clearly grouped tally marks.\r\n<p style=\"text-align: center\"><a title=\"By Ryan Somma from Occoquan, USA (Carved reindeer antler with tally marks) [CC BY-SA 2.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/2.0)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3ACarved_reindeer_antler_with_tally_marks_(4697848661).jpg\"><img width=\"512\" alt=\"Carved reindeer antler with tally marks (4697848661)\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/ce\/Carved_reindeer_antler_with_tally_marks_%284697848661%29.jpg\/512px-Carved_reindeer_antler_with_tally_marks_%284697848661%29.jpg\" class=\"aligncenter\" height=\"340\" \/><\/a><\/p>\r\n&nbsp;\r\n\r\nOf course, we still use tally marks today![footnote]Image of Hanakapiai beach warning sign by God of War at the English language Wikipedia [GFDL or CC-BY-SA-3.0], via Wikimedia Commons.[\/footnote]\r\n<p style=\"text-align: center\"><a title=\"God of War at the English language Wikipedia [GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html) or CC-BY-SA-3.0 (http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AHanakapiai_Beach_Warning_Sign_Only.jpg\"><img width=\"256\" alt=\"Hanakapiai Beach Warning Sign Only\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fa\/Hanakapiai_Beach_Warning_Sign_Only.jpg\/256px-Hanakapiai_Beach_Warning_Sign_Only.jpg\" \/><\/a><\/p>\r\nBase ten numbers (the ones you have probably been using your whole life), and base <em>b<\/em> numbers (the ones you've been learning about in this chapter) are both positional number systems.\r\n<div class=\"textbox key-takeaways\">\r\n<h3 itemprop=\"educationalUse\">Definition<\/h3>\r\nA <strong>positional number system<\/strong> is one way of writing numbers. It has unique symbols for 1 through <em>b<\/em>\u00a0\u2013 1, where <em>b<\/em> is the base of the system. Modern positional number systems also include a symbol for 0.\r\n\r\nThe <strong>positional value<\/strong> of each symbol depends on its position in the number:\r\n<ul>\r\n \t<li>The positional value of a symbol in the first position is just its face value.<\/li>\r\n \t<li>The positional value of a symbol in the second position is <em>b<\/em> times its value.<\/li>\r\n \t<li>The positional value of a symbol in the third position is $b^2$ times its value.<\/li>\r\n \t<li>And so on.<\/li>\r\n<\/ul>\r\nThe value of a number is the sum of the positional values of its digits.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3 itemprop=\"educationalUse\">Definition<\/h3>\r\nIn an <strong>additive number system,<\/strong> the value of a written number is the sum of the face values of the symbols that make up the number. The only symbol necessary for an additive number system is a symbol for 1, however many additive number systems contain other symbols.\r\n\r\n<\/div>\r\n<h1>History: Roman numerals<\/h1>\r\nThe ancient Romans used a version of an additive number systems. The Romans represented numbers this way:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center\"><strong>number<\/strong><\/td>\r\n<td style=\"text-align: center\"><strong>Roman Numeral<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">1<\/td>\r\n<td style=\"text-align: center\">I<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">5<\/td>\r\n<td style=\"text-align: center\">V<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">10<\/td>\r\n<td style=\"text-align: center\">X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">50<\/td>\r\n<td style=\"text-align: center\">L<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">100<\/td>\r\n<td style=\"text-align: center\">C<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">500<\/td>\r\n<td style=\"text-align: center\">D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">1,000<\/td>\r\n<td style=\"text-align: center\">M<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo the number 2013 would be represented as MMXIII. This is read as 2,000 (two M's), one ten (one X), and three ones (three I's).\r\n\r\nFor any additive number system very large numbers become impractical to write. To represent the number one million in Roman numerals it would take one thousand M\u2019s!\r\n\r\nHowever, the Roman numerals did have one efficiency advantage: The order of the symbols mattered. If a symbol to the left was smaller than the symbol to the right, it would be subtracted\u00a0instead of added. So for example nine is represented as IX rather than VIIII.\r\n<div class=\"textbox examples\">\r\n<h3 itemprop=\"educationalUse\">Think \/ Pair \/ Share<\/h3>\r\nIf you don't already know how to use Roman numerals, research it a little bit.\u00a0 Then answer these questions.\r\n<ul>\r\n \t<li>Write the numbers 1\u201320 in Roman numerals.<\/li>\r\n \t<li>What is the maximum number of symbols needed to write any number between 1 and 1,000 in Roman Numerals? Justify your answer.<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe earliest positional number systems are attributed to the Babylonians (base 60) and the Mayans (base 20). These positional systems were both developed before they had a symbol or a clear concept for zero. Instead of using 0, a blank space was used to indicate skipping a particular place value. This could lead to ambiguity.\r\n\r\nSuppose we didn't have a symbol for 0, and someone wrote the number\r\n\r\n\\[2 \\ \\ 3.\\]\r\n\r\nIt would be impossible to tell if they mean 23, 203, 2003, or maybe two separate numbers (two and three).\r\n<div class=\"textbox shaded\">\r\n\r\nLeonardo Pisano Bigollo, more commonly known as <strong>Fibonacci<\/strong>[footnote]Image of Fibonacci via Wikimedia Commons, in the public domain.[\/footnote], played a pivotal role in guiding Europe out of a long period in which the importance and development of math was in marked decline. He was born in Italy around 1170 CE to Guglielmo Bonacci, a successful merchant. Guglielmo brought his son with him to what is now Algeria, and Leonardo was educated in mathematics mathematics there.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"256\"]<a title=\"By The original uploader was Dr. Manuel at German Wikipedia [Public domain], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AFibonacci.jpeg\"><img width=\"256\" alt=\"Fibonacci\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/b5\/Fibonacci.jpeg\" height=\"326\" \/><\/a> Fibonacci[\/caption]At the time, Roman Numerals dominated Europe, and the official means of calculations was the abacus. Mu\u1e25ammad ibn M\u016bs\u0101 al-Khw\u0101rizm\u012b[footnote]Image of al-Khwarizmi statue by M. Tomczak [CC BY-SA 3.0], via Wikimedia Commons.[\/footnote] described the use of Hindu-Arabic system in his book <em>On the Calculation with Hindu Numerals<\/em> in 825 CE, but it was not well-known in Europe.[caption id=\"\" align=\"aligncenter\" width=\"256\"]<a title=\"By M. Tomczak [CC BY-SA 3.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/3.0)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AKhwarizmi_Amirkabir_University_of_Technology.png\"><img width=\"256\" alt=\"Khwarizmi Amirkabir University of Technology\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/a6\/Khwarizmi_Amirkabir_University_of_Technology.png\" height=\"553\" \/><\/a> Statue of al-Khwarizmi at Amirkabir University of Technology[\/caption]\r\nFibonacci's book <em>Liber Abaci<\/em>\u00a0described the Hindu-Arabic system and its business applications for a European readership. His book was well-received throughout Europe, and it marked the beginning of a reawakening of European mathematics.<\/div>\r\n<h1>History: Hawaiian numbers<\/h1>\r\nThe Hindu-Arabic number system is now used nearly exclusively throughout the globe.\u00a0 But many cultures had their own number systems before contact and trade with other countries spread the work of\u00a0al-Khw\u0101rizm\u012b throughout the world.\r\n\r\nThere is evidence that pre-contact Hawaiians actually used two different number systems. \u00a0Depending on what they were counting, they might use base 4 instead (or a mixed base-10 and base-4 system). \u00a0One theory is that certain objects (fish, taro, etc.) were often put in bundles of 4, so were more natural to count by 4\u2019s than by 10\u2019s. \u00a0The number four also had spiritual significance in Hawaiian culture.\r\n<table style=\"width: 763px\">\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;width: 300.753px\"><img src=\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-199x300.png\" alt=\"\" width=\"199\" height=\"300\" class=\"alignnone size-medium wp-image-1758\" \/><\/td>\r\n<td style=\"text-align: center;width: 434.247px\">Humans have 5 fingers on each hand[footnote]Image of hand from https:\/\/pixabay.com, licensed under\u00a0<a href=\"https:\/\/pixabay.com\/en\/service\/terms\/#usage\">CC0 Creative Commons<\/a>.[\/footnote], making base ten a natural choice for counting. But there are 4 gaps between the fingers, meaning that a hand can carry four fish or taro plants or similar objects, making base four a natural choice for some cultures.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the mixed base system, instead of powers of 10, numbers are broken down into sums of numbers that look like 4 times a power of 10 (40, 400, 4000, etc.).\r\n<div>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>1<\/td>\r\n<td>\u2018ekahi<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>\u2018elua<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>\u2018ekolu<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>\u2018eh\u0101 (or kauna)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>\u2018elima<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>\u2018eono<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7<\/td>\r\n<td>\u2018ehiku<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8<\/td>\r\n<td>\u2018ewalu<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9<\/td>\r\n<td>\u2018eiwa<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>\u2018umi<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>11\u201319<\/td>\r\n<td>\u2018umi kumam\u0101 {kahi, lua, kolu, h\u0101, etc.}<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<\/td>\r\n<td>iwak\u0101lua<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>21\u201329<\/td>\r\n<td>Iwak\u0101lua kumam\u0101 {kahi, lua, kolu, h\u0101, etc.}<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>30<\/td>\r\n<td>kanakolu<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>31\u201339<\/td>\r\n<td>kanakolu kumam\u0101 {kahi, lua, etc.}<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>40<\/td>\r\n<td>kanah\u0101<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>400<\/td>\r\n<td>lau<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4,000<\/td>\r\n<td>mano<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>40,000<\/td>\r\n<td>kini<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>400,000<\/td>\r\n<td>lehu<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nHere are a few examples (refer to the table above for the Hawaiian names of the numbers):\r\n<div class=\"textbox key-takeaways\">\r\n<h3 itemprop=\"educationalUse\">Example<\/h3>\r\n\u2018ekolu kini, \u2018ewalu lau me \u2018ekahi\r\n\r\ntranslates to three 40,000\u2019s, eight 400\u2019s, and one;\r\n\r\n3 \u22c5 40000 + 8 \u22c5 400 + 1 = 123201\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3 itemprop=\"educationalUse\">Example<\/h3>\r\n5207 = 1 \u22c5 4000 + 3 \u22c5 400 + 7\r\n\r\nwould be \u2018ekahi mano, \u2018ekolu lau me \u2018ehiku\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>On Your Own<\/h3>\r\nWork on the following exercises on your own or with a partner.\r\n\r\n1. Translate this Hawaiian number to English and then write it in base ten.\r\n<p style=\"text-align: center\">\u2018ekahi kanah\u0101 me kanakolu kumam\u0101iwa<\/p>\r\n2.\u00a0 Translate this base\u2010ten number to Hawaiian.\r\n<p style=\"text-align: center\">1,573<\/p>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3 itemprop=\"educationalUse\">Think \/ Pair \/ Share<\/h3>\r\nHow is learning about different number systems (including representing numbers in different bases) valuable to you as a future teacher?\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p>Our number system is a western adaptation of the Hindu-Arabic numeral system developed somewhere between the first and fourth centuries AD.\u00a0 However, numbers have been recorded with tally marks throughout history.\u00a0 The Ishango Bone<a class=\"footnote\" title=\"Image of Ishango bone by Ben2 (Own work), CC-BY-SA-3.0 or CC BY-SA 2.5-2.0-1.0], via Wikimedia Commons.\" id=\"return-footnote-451-1\" href=\"#footnote-451-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> from Africa is about 25,000 years old. It&#8217;s the lower leg bone from a baboon, and contains tally marks. We know the marks were used for counting because they appear in distinct groups.<\/p>\n<p style=\"text-align: center\"><a title=\"By Ben2 (Own work) [GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html), CC-BY-SA-3.0 (http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/) or CC BY-SA 2.5-2.0-1.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/2.5-2.0-1.0)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AOs_d'Ishango_IRSNB.JPG\"><img loading=\"lazy\" decoding=\"async\" width=\"256\" alt=\"Ishango bone\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/4\/42\/Os_d%27Ishango_IRSNB.JPG\/256px-Os_d%27Ishango_IRSNB.JPG\" class=\"aligncenter\" height=\"415\" \/><\/a><\/p>\n<p>This reindeer antler<a class=\"footnote\" title=\"Image of antler By Ryan Somma from Occoquan, USA [CC BY-SA 2.0], via Wikimedia Commons\" id=\"return-footnote-451-2\" href=\"#footnote-451-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> from France is about 15,000 years old, and also shows clearly grouped tally marks.<\/p>\n<p style=\"text-align: center\"><a title=\"By Ryan Somma from Occoquan, USA (Carved reindeer antler with tally marks) [CC BY-SA 2.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/2.0)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3ACarved_reindeer_antler_with_tally_marks_(4697848661).jpg\"><img loading=\"lazy\" decoding=\"async\" width=\"512\" alt=\"Carved reindeer antler with tally marks (4697848661)\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/ce\/Carved_reindeer_antler_with_tally_marks_%284697848661%29.jpg\/512px-Carved_reindeer_antler_with_tally_marks_%284697848661%29.jpg\" class=\"aligncenter\" height=\"340\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>Of course, we still use tally marks today!<a class=\"footnote\" title=\"Image of Hanakapiai beach warning sign by God of War at the English language Wikipedia [GFDL or CC-BY-SA-3.0], via Wikimedia Commons.\" id=\"return-footnote-451-3\" href=\"#footnote-451-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a><\/p>\n<p style=\"text-align: center\"><a title=\"God of War at the English language Wikipedia [GFDL (http:\/\/www.gnu.org\/copyleft\/fdl.html) or CC-BY-SA-3.0 (http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AHanakapiai_Beach_Warning_Sign_Only.jpg\"><img decoding=\"async\" width=\"256\" alt=\"Hanakapiai Beach Warning Sign Only\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fa\/Hanakapiai_Beach_Warning_Sign_Only.jpg\/256px-Hanakapiai_Beach_Warning_Sign_Only.jpg\" \/><\/a><\/p>\n<p>Base ten numbers (the ones you have probably been using your whole life), and base <em>b<\/em> numbers (the ones you&#8217;ve been learning about in this chapter) are both positional number systems.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3 itemprop=\"educationalUse\">Definition<\/h3>\n<p>A <strong>positional number system<\/strong> is one way of writing numbers. It has unique symbols for 1 through <em>b<\/em>\u00a0\u2013 1, where <em>b<\/em> is the base of the system. Modern positional number systems also include a symbol for 0.<\/p>\n<p>The <strong>positional value<\/strong> of each symbol depends on its position in the number:<\/p>\n<ul>\n<li>The positional value of a symbol in the first position is just its face value.<\/li>\n<li>The positional value of a symbol in the second position is <em>b<\/em> times its value.<\/li>\n<li>The positional value of a symbol in the third position is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-55408115d44d3e7d9ff4d46eee600892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: 0px;\" \/> times its value.<\/li>\n<li>And so on.<\/li>\n<\/ul>\n<p>The value of a number is the sum of the positional values of its digits.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3 itemprop=\"educationalUse\">Definition<\/h3>\n<p>In an <strong>additive number system,<\/strong> the value of a written number is the sum of the face values of the symbols that make up the number. The only symbol necessary for an additive number system is a symbol for 1, however many additive number systems contain other symbols.<\/p>\n<\/div>\n<h1>History: Roman numerals<\/h1>\n<p>The ancient Romans used a version of an additive number systems. The Romans represented numbers this way:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center\"><strong>number<\/strong><\/td>\n<td style=\"text-align: center\"><strong>Roman Numeral<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">1<\/td>\n<td style=\"text-align: center\">I<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">5<\/td>\n<td style=\"text-align: center\">V<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">10<\/td>\n<td style=\"text-align: center\">X<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">50<\/td>\n<td style=\"text-align: center\">L<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">100<\/td>\n<td style=\"text-align: center\">C<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">500<\/td>\n<td style=\"text-align: center\">D<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">1,000<\/td>\n<td style=\"text-align: center\">M<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So the number 2013 would be represented as MMXIII. This is read as 2,000 (two M&#8217;s), one ten (one X), and three ones (three I&#8217;s).<\/p>\n<p>For any additive number system very large numbers become impractical to write. To represent the number one million in Roman numerals it would take one thousand M\u2019s!<\/p>\n<p>However, the Roman numerals did have one efficiency advantage: The order of the symbols mattered. If a symbol to the left was smaller than the symbol to the right, it would be subtracted\u00a0instead of added. So for example nine is represented as IX rather than VIIII.<\/p>\n<div class=\"textbox examples\">\n<h3 itemprop=\"educationalUse\">Think \/ Pair \/ Share<\/h3>\n<p>If you don&#8217;t already know how to use Roman numerals, research it a little bit.\u00a0 Then answer these questions.<\/p>\n<ul>\n<li>Write the numbers 1\u201320 in Roman numerals.<\/li>\n<li>What is the maximum number of symbols needed to write any number between 1 and 1,000 in Roman Numerals? Justify your answer.<\/li>\n<\/ul>\n<\/div>\n<p>The earliest positional number systems are attributed to the Babylonians (base 60) and the Mayans (base 20). These positional systems were both developed before they had a symbol or a clear concept for zero. Instead of using 0, a blank space was used to indicate skipping a particular place value. This could lead to ambiguity.<\/p>\n<p>Suppose we didn&#8217;t have a symbol for 0, and someone wrote the number<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 12px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-d9274e864a34a4d10a9e06dd15fa02d8_l3.png\" height=\"12\" width=\"33\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#091;&#50;&#32;&#92;&#32;&#92;&#32;&#51;&#46;&#92;&#093;\" title=\"Rendered by QuickLaTeX.com\" \/><\/p>\n<p>It would be impossible to tell if they mean 23, 203, 2003, or maybe two separate numbers (two and three).<\/p>\n<div class=\"textbox shaded\">\n<p>Leonardo Pisano Bigollo, more commonly known as <strong>Fibonacci<\/strong><a class=\"footnote\" title=\"Image of Fibonacci via Wikimedia Commons, in the public domain.\" id=\"return-footnote-451-4\" href=\"#footnote-451-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a>, played a pivotal role in guiding Europe out of a long period in which the importance and development of math was in marked decline. He was born in Italy around 1170 CE to Guglielmo Bonacci, a successful merchant. Guglielmo brought his son with him to what is now Algeria, and Leonardo was educated in mathematics mathematics there.<\/p>\n<figure style=\"width: 256px\" class=\"wp-caption aligncenter\"><a title=\"By The original uploader was Dr. Manuel at German Wikipedia [Public domain], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AFibonacci.jpeg\"><img loading=\"lazy\" decoding=\"async\" width=\"256\" alt=\"Fibonacci\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/b5\/Fibonacci.jpeg\" height=\"326\" \/><\/a><figcaption class=\"wp-caption-text\">Fibonacci<\/figcaption><\/figure>\n<p>At the time, Roman Numerals dominated Europe, and the official means of calculations was the abacus. Mu\u1e25ammad ibn M\u016bs\u0101 al-Khw\u0101rizm\u012b<a class=\"footnote\" title=\"Image of al-Khwarizmi statue by M. Tomczak [CC BY-SA 3.0], via Wikimedia Commons.\" id=\"return-footnote-451-5\" href=\"#footnote-451-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a> described the use of Hindu-Arabic system in his book <em>On the Calculation with Hindu Numerals<\/em> in 825 CE, but it was not well-known in Europe.<\/p>\n<figure style=\"width: 256px\" class=\"wp-caption aligncenter\"><a title=\"By M. Tomczak [CC BY-SA 3.0 (https:\/\/creativecommons.org\/licenses\/by-sa\/3.0)], via Wikimedia Commons\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AKhwarizmi_Amirkabir_University_of_Technology.png\"><img loading=\"lazy\" decoding=\"async\" width=\"256\" alt=\"Khwarizmi Amirkabir University of Technology\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/a6\/Khwarizmi_Amirkabir_University_of_Technology.png\" height=\"553\" \/><\/a><figcaption class=\"wp-caption-text\">Statue of al-Khwarizmi at Amirkabir University of Technology<\/figcaption><\/figure>\n<p>Fibonacci&#8217;s book <em>Liber Abaci<\/em>\u00a0described the Hindu-Arabic system and its business applications for a European readership. His book was well-received throughout Europe, and it marked the beginning of a reawakening of European mathematics.<\/div>\n<h1>History: Hawaiian numbers<\/h1>\n<p>The Hindu-Arabic number system is now used nearly exclusively throughout the globe.\u00a0 But many cultures had their own number systems before contact and trade with other countries spread the work of\u00a0al-Khw\u0101rizm\u012b throughout the world.<\/p>\n<p>There is evidence that pre-contact Hawaiians actually used two different number systems. \u00a0Depending on what they were counting, they might use base 4 instead (or a mixed base-10 and base-4 system). \u00a0One theory is that certain objects (fish, taro, etc.) were often put in bundles of 4, so were more natural to count by 4\u2019s than by 10\u2019s. \u00a0The number four also had spiritual significance in Hawaiian culture.<\/p>\n<table style=\"width: 763px\">\n<tbody>\n<tr>\n<td style=\"text-align: center;width: 300.753px\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-199x300.png\" alt=\"\" width=\"199\" height=\"300\" class=\"alignnone size-medium wp-image-1758\" srcset=\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-199x300.png 199w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-768x1155.png 768w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-681x1024.png 681w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-65x98.png 65w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-225x338.png 225w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280-350x526.png 350w, https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hand-294018_1280.png 851w\" sizes=\"auto, (max-width: 199px) 100vw, 199px\" \/><\/td>\n<td style=\"text-align: center;width: 434.247px\">Humans have 5 fingers on each hand<a class=\"footnote\" title=\"Image of hand from https:\/\/pixabay.com, licensed under\u00a0CC0 Creative Commons.\" id=\"return-footnote-451-6\" href=\"#footnote-451-6\" aria-label=\"Footnote 6\"><sup class=\"footnote\">[6]<\/sup><\/a>, making base ten a natural choice for counting. But there are 4 gaps between the fingers, meaning that a hand can carry four fish or taro plants or similar objects, making base four a natural choice for some cultures.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the mixed base system, instead of powers of 10, numbers are broken down into sums of numbers that look like 4 times a power of 10 (40, 400, 4000, etc.).<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>\u2018ekahi<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>\u2018elua<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>\u2018ekolu<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>\u2018eh\u0101 (or kauna)<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>\u2018elima<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>\u2018eono<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>\u2018ehiku<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>\u2018ewalu<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>\u2018eiwa<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>\u2018umi<\/td>\n<\/tr>\n<tr>\n<td>11\u201319<\/td>\n<td>\u2018umi kumam\u0101 {kahi, lua, kolu, h\u0101, etc.}<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>iwak\u0101lua<\/td>\n<\/tr>\n<tr>\n<td>21\u201329<\/td>\n<td>Iwak\u0101lua kumam\u0101 {kahi, lua, kolu, h\u0101, etc.}<\/td>\n<\/tr>\n<tr>\n<td>30<\/td>\n<td>kanakolu<\/td>\n<\/tr>\n<tr>\n<td>31\u201339<\/td>\n<td>kanakolu kumam\u0101 {kahi, lua, etc.}<\/td>\n<\/tr>\n<tr>\n<td>40<\/td>\n<td>kanah\u0101<\/td>\n<\/tr>\n<tr>\n<td>400<\/td>\n<td>lau<\/td>\n<\/tr>\n<tr>\n<td>4,000<\/td>\n<td>mano<\/td>\n<\/tr>\n<tr>\n<td>40,000<\/td>\n<td>kini<\/td>\n<\/tr>\n<tr>\n<td>400,000<\/td>\n<td>lehu<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Here are a few examples (refer to the table above for the Hawaiian names of the numbers):<\/p>\n<div class=\"textbox key-takeaways\">\n<h3 itemprop=\"educationalUse\">Example<\/h3>\n<p>\u2018ekolu kini, \u2018ewalu lau me \u2018ekahi<\/p>\n<p>translates to three 40,000\u2019s, eight 400\u2019s, and one;<\/p>\n<p>3 \u22c5 40000 + 8 \u22c5 400 + 1 = 123201<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3 itemprop=\"educationalUse\">Example<\/h3>\n<p>5207 = 1 \u22c5 4000 + 3 \u22c5 400 + 7<\/p>\n<p>would be \u2018ekahi mano, \u2018ekolu lau me \u2018ehiku<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>On Your Own<\/h3>\n<p>Work on the following exercises on your own or with a partner.<\/p>\n<p>1. Translate this Hawaiian number to English and then write it in base ten.<\/p>\n<p style=\"text-align: center\">\u2018ekahi kanah\u0101 me kanakolu kumam\u0101iwa<\/p>\n<p>2.\u00a0 Translate this base\u2010ten number to Hawaiian.<\/p>\n<p style=\"text-align: center\">1,573<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3 itemprop=\"educationalUse\">Think \/ Pair \/ Share<\/h3>\n<p>How is learning about different number systems (including representing numbers in different bases) valuable to you as a future teacher?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-451-1\">Image of Ishango bone by Ben2 (Own work), CC-BY-SA-3.0 or CC BY-SA 2.5-2.0-1.0], via Wikimedia Commons. <a href=\"#return-footnote-451-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-451-2\">Image of antler By Ryan Somma from Occoquan, USA [CC BY-SA 2.0], via Wikimedia Commons <a href=\"#return-footnote-451-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-451-3\">Image of Hanakapiai beach warning sign by God of War at the English language Wikipedia [GFDL or CC-BY-SA-3.0], via Wikimedia Commons. <a href=\"#return-footnote-451-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-451-4\">Image of Fibonacci via Wikimedia Commons, in the public domain. <a href=\"#return-footnote-451-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-451-5\">Image of al-Khwarizmi statue by M. Tomczak [CC BY-SA 3.0], via Wikimedia Commons. <a href=\"#return-footnote-451-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><li id=\"footnote-451-6\">Image of hand from https:\/\/pixabay.com, licensed under\u00a0<a href=\"https:\/\/pixabay.com\/en\/service\/terms\/#usage\">CC0 Creative Commons<\/a>. <a href=\"#return-footnote-451-6\" class=\"return-footnote\" aria-label=\"Return to footnote 6\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-451","chapter","type-chapter","status-publish","hentry"],"part":134,"_links":{"self":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/451","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":39,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/451\/revisions"}],"predecessor-version":[{"id":2292,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/451\/revisions\/2292"}],"part":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/parts\/134"}],"metadata":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/451\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/media?parent=451"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapter-type?post=451"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/contributor?post=451"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/license?post=451"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}