{"id":1965,"date":"2017-03-21T00:51:51","date_gmt":"2017-03-21T00:51:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1965"},"modified":"2019-05-30T16:24:12","modified_gmt":"2019-05-30T16:24:12","slug":"the-positional-system-and-base-10","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/the-positional-system-and-base-10\/","title":{"raw":"The Positional System and Base 10","rendered":"The Positional System and Base 10"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Become familiar with the history of positional number systems<\/li>\r\n \t<li>Identify bases that have been used in number systems historically<\/li>\r\n \t<li>Convert numbers between bases<\/li>\r\n \t<li>Use two different methods for converting numbers between bases<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe Indians were not the first to use a positional system. The Babylonians (as we will see in Chapter 3) used a positional system with 60 as their base. However, there is not much evidence that the Babylonian system had much impact on later numeral systems, except with the Greeks. Also, the Chinese had a base-10 system, probably derived from the use of a counting board.[footnote]Ibid, page 230[\/footnote] Some believe that the positional system used in India was derived from the Chinese system.\r\n\r\nWherever it may have originated, it appears that around 600 CE, the Indians abandoned the use of symbols for numbers higher than nine and began to use our familiar system where the position of the symbol determines its overall value.[footnote]Ibid, page 231.[\/footnote]\u00a0Numerous documents from the seventh century demonstrate the use of this positional system.\r\n\r\nInterestingly, the earliest dated inscriptions using the system with a symbol for zero come from Cambodia. In 683, the 605th year of the Saka era is written with three digits and a dot in the middle. The 608th year uses three digits with a modern 0 in the middle.[footnote]Ibid, page 232.[\/footnote]\u00a0The dot as a symbol for zero also appears in a Chinese work (<em>Chiu<\/em><em>-chih li<\/em>). The author of this document gives a strikingly clear description of how the Indian system works:\r\n<blockquote>\r\n<div>Using the [Indian] numerals, multiplication and division are carried out. Each numeral is written in one stroke. When a number is counted to ten, it is advanced into the higher place. In each vacant place a dot is always put. Thus the numeral is always denoted in each place. Accordingly there can be no error in determining the place. With the numerals, calculations is easy.[footnote]Ibid, page 232.[\/footnote]<\/div><\/blockquote>\r\n<h3>Transmission to Europe<\/h3>\r\nIt is not completely known how the system got transmitted to Europe. Traders and travelers of the Mediterranean coast may have carried it there. It is found in a tenth-century Spanish manuscript and may have been introduced to Spain by the Arabs, who invaded the region in 711 CE and were there until 1492.\r\n\r\nIn many societies, a division formed between those who used numbers and calculation for practical, every day business and those who used them for ritualistic purposes or for state business.[footnote]McLeish, p. 18[\/footnote]\u00a0The former might often use older systems while the latter were inclined to use the newer, more elite written numbers. Competition between the two groups arose and continued for quite some time.\r\n\r\n[caption id=\"attachment_279\" align=\"alignright\" width=\"300\"]<img class=\"wp-image-279\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155211\/Margarita_philosophica.jpg\" alt=\"Margarita_philosophica\" width=\"300\" height=\"371\" \/> Figure 15.[\/caption]\r\n\r\nIn a fourteenth\u00a0century manuscript of Boethius\u2019 <em>The Consolations of Philosophy<\/em>, there appears a well-known drawing of two mathematicians. One is a merchant and is using an abacus (the \u201cabacist\u201d). The other is a Pythagorean philosopher (the \u201calgorist\u201d) using his \u201csacred\u201d numbers. They are in a competition that is being judged by the goddess of number. By 1500 CE, however, the newer symbols and system had won out and has persevered until today. The Seattle Times recently reported that the Hindu-Arabic numeral system has been included in the book <em>The Greatest Inventions of the Past 2000 Years<\/em>.[footnote]<a href=\"http:\/\/seattletimes.nwsource.com\/news\/health-science\/html98\/invs_20000201.html\">http:\/\/seattletimes.nwsource.com\/news\/health-science\/html98\/invs_20000201.html<\/a>, Seattle Times, Feb. 1, 2000[\/footnote]\r\n\r\nOne question to answer is <em>why<\/em> the Indians would develop such a positional notation. Unfortunately, an answer to that question is not currently known. Some suggest that the system has its origins with the Chinese counting boards. These boards were portable and it is thought that Chinese travelers who passed through India took their boards with them and ignited an idea in Indian mathematics.[footnote]Ibid, page 232.[\/footnote]\u00a0Others, such as G. G. Joseph propose that it is the Indian fascination with very large numbers that drove them to develop a system whereby these kinds of big numbers could easily be written down. In this theory, the system developed entirely within the Indian mathematical framework without considerable influence from other civilizations.\r\n<h2>The Development and Use of Different Number Bases<\/h2>\r\n<h3>Introduction and Basics<\/h3>\r\nDuring the previous discussions, we have been referring to positional base systems. In this section of the chapter, we will explore exactly what a base system is and what it means if a system is \u201cpositional.\u201d We will do so by first looking at our own familiar, base-ten system and then deepen our exploration by looking at other possible base systems. In the next part of this section, we will journey back to Mayan civilization and look at their unique base system, which is based on the number 20 rather than the number 10.\r\n\r\nA base system is a structure within which we count. The easiest way to describe a base system is to think about our own base-ten system. The base-ten system, which we call the \u201cdecimal\u201d system, requires a total of ten different symbols\/digits to write any number. They are, of course, 0, 1, 2, . . . , 9.\r\n\r\nThe decimal system is also an example of a <em>positional <\/em>base system, which simply means that the position of a digit gives its place value. Not all civilizations had a positional system even though they did have a base with which they worked.\r\n\r\nIn our base-ten system, a number like 5,783,216 has meaning to us because we are familiar with the system and its places. As we know, there are six ones, since there is a 6 in the ones place. Likewise, there are seven \u201chundred thousands,\u201d since the 7 resides in that place. Each digit has a value that is explicitly determined by its position within the number. We make a distinction between digit, which is just a symbol such as 5, and a number, which is made up of one or more digits. We can take this number and assign each of its digits a value. One way to do this is with a table, which follows:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>5,000,000<\/td>\r\n<td>= 5 \u00d7 1,000,000<\/td>\r\n<td>= 5 \u00d7 10<sup>6<\/sup><\/td>\r\n<td>Five million<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+700,000<\/td>\r\n<td>= 7 \u00d7 100,000<\/td>\r\n<td>= 7 \u00d7 10<sup>5<\/sup><\/td>\r\n<td>Seven hundred thousand<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+80,000<\/td>\r\n<td>= 8 \u00d7 10,000<\/td>\r\n<td>= 8 \u00d7 10<sup>4<\/sup><\/td>\r\n<td>Eighty thousand<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+3,000<\/td>\r\n<td>= 3 \u00d7 1000<\/td>\r\n<td>= 3 \u00d7 10<sup>3<\/sup><\/td>\r\n<td>Three thousand<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+200<\/td>\r\n<td>= 2 \u00d7 100<\/td>\r\n<td>= 2 \u00d7 10<sup>2<\/sup><\/td>\r\n<td>Two hundred<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+10<\/td>\r\n<td>= 1 \u00d7 10<\/td>\r\n<td>= 1 \u00d7 10<sup>1<\/sup><\/td>\r\n<td>Ten<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+6<\/td>\r\n<td>= 6 \u00d7 1<\/td>\r\n<td>= 6 \u00d7 10<sup>0<\/sup><\/td>\r\n<td>Six<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5,783,216<\/td>\r\n<td colspan=\"3\">Five million, seven hundred eighty-three thousand, two hundred sixteen<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the third column in the table we can see that each place is simply a multiple of ten. Of course, this makes sense given that our base is ten. The digits that are multiplying each place simply tell us how many of that place we have. We are restricted to having at most 9 in any one place before we have to \u201ccarry\u201d over to the next place. We cannot, for example, have 11 in the hundreds place. Instead, we would carry 1 to the thousands place and retain 1 in the hundreds place. This comes as no surprise to us since we readily see that 11 hundreds is the same as one thousand, one hundred. Carrying is a pretty typical occurrence in a base system.\r\n\r\nHowever, base-ten is not the only option we have. Practically any positive integer greater than or equal to 2 can be used as a base for a number system. Such systems can work just like the decimal system except the number of symbols will be different and each position will depend on the base itself.\r\n<h3>Other Bases<\/h3>\r\nFor example, let\u2019s suppose we adopt a base-five system. The only modern digits we would need for this system are 0,1,2,3 and 4. What are the place values in such a system? To answer that, we start with the ones place, as most base systems do. However, if we were to count in this system, we could only get to four (4) before we had to jump up to the next place. Our base is 5, after all! What is that next place that we would jump to? It would not be tens, since we are no longer in base-ten. We\u2019re in a different numerical world. As the base-ten system progresses from 10<sup>0<\/sup> to 10<sup>1<\/sup>, so the base-five system moves from 5<sup>0<\/sup> to 5<sup>1<\/sup> = 5. Thus, we move from the ones to the fives.\r\n\r\nAfter the fives, we would move to the 5<sup>2<\/sup> place, or the twenty fives. Note that in base-ten, we would have gone from the tens to the hundreds, which is, of course, 10<sup>2<\/sup>.\r\n\r\nLet\u2019s take an example and build a table. Consider the number 30412 in base five. We will write this as 30412<sub>5<\/sub>, where the subscript 5 is not part of the number but indicates the base we\u2019re using. First off, note that this is NOT the number \u201cthirty thousand, four hundred twelve.\u201d We must be careful not to impose the base-ten system on this number. Here\u2019s what our table might look like. We will use it to convert this number to our more familiar base-ten system.\r\n\r\n&nbsp;\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>Base 5<\/td>\r\n<td>This column coverts to base-ten<\/td>\r\n<td>In Base-Ten<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>3 \u00d7 5<sup>4<\/sup><\/td>\r\n<td>= 3 \u00d7 625<\/td>\r\n<td>= 1875<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+<\/td>\r\n<td>0 \u00d7 5<sup>3<\/sup><\/td>\r\n<td>= 0 \u00d7 125<\/td>\r\n<td>= 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+<\/td>\r\n<td>4 \u00d7 5<sup>2<\/sup><\/td>\r\n<td>= 4 \u00d7 25<\/td>\r\n<td>= 100<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+<\/td>\r\n<td>1 \u00d7 5<sup>1<\/sup><\/td>\r\n<td>= 1 \u00d7 5<\/td>\r\n<td>= 5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+<\/td>\r\n<td>2 \u00d7 5<sup>0<\/sup><\/td>\r\n<td>= 2 \u00d7 1<\/td>\r\n<td>= 2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>Total<\/td>\r\n<td>1982<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs you can see, the number 30412<sub>5<\/sub> is equivalent to 1,982 in base-ten. We will say 30412<sub>5<\/sub> = 1982<sub>10<\/sub>. All of this may seem strange to you, but that\u2019s only because you are so used to the only system that you\u2019ve ever seen.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConvert 6234<sub>7<\/sub> to a base 10 number.\r\n[reveal-answer q=\"482364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"482364\"]We first note that we are given a base-7 number that we are to convert. Thus, our places will start at the ones (7<sup>0<\/sup>), and then move up to the 7s, 49s (7<sup>2<\/sup>), etc. Here\u2019s the breakdown:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>Base 7<\/td>\r\n<td>Convert<\/td>\r\n<td>Base 10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>= 6\u00a0\u00d7\u00a07<sup>3<\/sup><\/td>\r\n<td>= 6\u00a0\u00d7 343<\/td>\r\n<td>= 2058<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+<\/td>\r\n<td>= 2\u00a0\u00d7\u00a07<sup>2<\/sup><\/td>\r\n<td>= 2\u00a0\u00d7 49<\/td>\r\n<td>= 98<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+<\/td>\r\n<td>= 3\u00a0\u00d7\u00a07<\/td>\r\n<td>= 3\u00a0\u00d7 7<\/td>\r\n<td>= 21<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>+<\/td>\r\n<td>= 4\u00a0\u00d7 1<\/td>\r\n<td>= 4\u00a0\u00d7 1<\/td>\r\n<td>= 4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>Total<\/td>\r\n<td>2181<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nThus 6234<sub>7<\/sub> = 2181<sub>10<\/sub>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConvert 41065<sub>7<\/sub> to a base 10 number.\r\n[reveal-answer q=\"896067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"896067\"][latex]41065_{7} = 9994[\/latex]_{10}\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=8680&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nWatch this video to see more examples of converting numbers in bases other than 10 into a base 10 number.\r\n\r\nhttps:\/\/youtu.be\/TjvexIVV_gI","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Become familiar with the history of positional number systems<\/li>\n<li>Identify bases that have been used in number systems historically<\/li>\n<li>Convert numbers between bases<\/li>\n<li>Use two different methods for converting numbers between bases<\/li>\n<\/ul>\n<\/div>\n<p>The Indians were not the first to use a positional system. The Babylonians (as we will see in Chapter 3) used a positional system with 60 as their base. However, there is not much evidence that the Babylonian system had much impact on later numeral systems, except with the Greeks. Also, the Chinese had a base-10 system, probably derived from the use of a counting board.<a class=\"footnote\" title=\"Ibid, page 230\" id=\"return-footnote-1965-1\" href=\"#footnote-1965-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> Some believe that the positional system used in India was derived from the Chinese system.<\/p>\n<p>Wherever it may have originated, it appears that around 600 CE, the Indians abandoned the use of symbols for numbers higher than nine and began to use our familiar system where the position of the symbol determines its overall value.<a class=\"footnote\" title=\"Ibid, page 231.\" id=\"return-footnote-1965-2\" href=\"#footnote-1965-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>\u00a0Numerous documents from the seventh century demonstrate the use of this positional system.<\/p>\n<p>Interestingly, the earliest dated inscriptions using the system with a symbol for zero come from Cambodia. In 683, the 605th year of the Saka era is written with three digits and a dot in the middle. The 608th year uses three digits with a modern 0 in the middle.<a class=\"footnote\" title=\"Ibid, page 232.\" id=\"return-footnote-1965-3\" href=\"#footnote-1965-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>\u00a0The dot as a symbol for zero also appears in a Chinese work (<em>Chiu<\/em><em>-chih li<\/em>). The author of this document gives a strikingly clear description of how the Indian system works:<\/p>\n<blockquote>\n<div>Using the [Indian] numerals, multiplication and division are carried out. Each numeral is written in one stroke. When a number is counted to ten, it is advanced into the higher place. In each vacant place a dot is always put. Thus the numeral is always denoted in each place. Accordingly there can be no error in determining the place. With the numerals, calculations is easy.<a class=\"footnote\" title=\"Ibid, page 232.\" id=\"return-footnote-1965-4\" href=\"#footnote-1965-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a><\/div>\n<\/blockquote>\n<h3>Transmission to Europe<\/h3>\n<p>It is not completely known how the system got transmitted to Europe. Traders and travelers of the Mediterranean coast may have carried it there. It is found in a tenth-century Spanish manuscript and may have been introduced to Spain by the Arabs, who invaded the region in 711 CE and were there until 1492.<\/p>\n<p>In many societies, a division formed between those who used numbers and calculation for practical, every day business and those who used them for ritualistic purposes or for state business.<a class=\"footnote\" title=\"McLeish, p. 18\" id=\"return-footnote-1965-5\" href=\"#footnote-1965-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a>\u00a0The former might often use older systems while the latter were inclined to use the newer, more elite written numbers. Competition between the two groups arose and continued for quite some time.<\/p>\n<div id=\"attachment_279\" style=\"width: 310px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-279\" class=\"wp-image-279\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155211\/Margarita_philosophica.jpg\" alt=\"Margarita_philosophica\" width=\"300\" height=\"371\" \/><\/p>\n<p id=\"caption-attachment-279\" class=\"wp-caption-text\">Figure 15.<\/p>\n<\/div>\n<p>In a fourteenth\u00a0century manuscript of Boethius\u2019 <em>The Consolations of Philosophy<\/em>, there appears a well-known drawing of two mathematicians. One is a merchant and is using an abacus (the \u201cabacist\u201d). The other is a Pythagorean philosopher (the \u201calgorist\u201d) using his \u201csacred\u201d numbers. They are in a competition that is being judged by the goddess of number. By 1500 CE, however, the newer symbols and system had won out and has persevered until today. The Seattle Times recently reported that the Hindu-Arabic numeral system has been included in the book <em>The Greatest Inventions of the Past 2000 Years<\/em>.<a class=\"footnote\" title=\"http:\/\/seattletimes.nwsource.com\/news\/health-science\/html98\/invs_20000201.html, Seattle Times, Feb. 1, 2000\" id=\"return-footnote-1965-6\" href=\"#footnote-1965-6\" aria-label=\"Footnote 6\"><sup class=\"footnote\">[6]<\/sup><\/a><\/p>\n<p>One question to answer is <em>why<\/em> the Indians would develop such a positional notation. Unfortunately, an answer to that question is not currently known. Some suggest that the system has its origins with the Chinese counting boards. These boards were portable and it is thought that Chinese travelers who passed through India took their boards with them and ignited an idea in Indian mathematics.<a class=\"footnote\" title=\"Ibid, page 232.\" id=\"return-footnote-1965-7\" href=\"#footnote-1965-7\" aria-label=\"Footnote 7\"><sup class=\"footnote\">[7]<\/sup><\/a>\u00a0Others, such as G. G. Joseph propose that it is the Indian fascination with very large numbers that drove them to develop a system whereby these kinds of big numbers could easily be written down. In this theory, the system developed entirely within the Indian mathematical framework without considerable influence from other civilizations.<\/p>\n<h2>The Development and Use of Different Number Bases<\/h2>\n<h3>Introduction and Basics<\/h3>\n<p>During the previous discussions, we have been referring to positional base systems. In this section of the chapter, we will explore exactly what a base system is and what it means if a system is \u201cpositional.\u201d We will do so by first looking at our own familiar, base-ten system and then deepen our exploration by looking at other possible base systems. In the next part of this section, we will journey back to Mayan civilization and look at their unique base system, which is based on the number 20 rather than the number 10.<\/p>\n<p>A base system is a structure within which we count. The easiest way to describe a base system is to think about our own base-ten system. The base-ten system, which we call the \u201cdecimal\u201d system, requires a total of ten different symbols\/digits to write any number. They are, of course, 0, 1, 2, . . . , 9.<\/p>\n<p>The decimal system is also an example of a <em>positional <\/em>base system, which simply means that the position of a digit gives its place value. Not all civilizations had a positional system even though they did have a base with which they worked.<\/p>\n<p>In our base-ten system, a number like 5,783,216 has meaning to us because we are familiar with the system and its places. As we know, there are six ones, since there is a 6 in the ones place. Likewise, there are seven \u201chundred thousands,\u201d since the 7 resides in that place. Each digit has a value that is explicitly determined by its position within the number. We make a distinction between digit, which is just a symbol such as 5, and a number, which is made up of one or more digits. We can take this number and assign each of its digits a value. One way to do this is with a table, which follows:<\/p>\n<table>\n<tbody>\n<tr>\n<td>5,000,000<\/td>\n<td>= 5 \u00d7 1,000,000<\/td>\n<td>= 5 \u00d7 10<sup>6<\/sup><\/td>\n<td>Five million<\/td>\n<\/tr>\n<tr>\n<td>+700,000<\/td>\n<td>= 7 \u00d7 100,000<\/td>\n<td>= 7 \u00d7 10<sup>5<\/sup><\/td>\n<td>Seven hundred thousand<\/td>\n<\/tr>\n<tr>\n<td>+80,000<\/td>\n<td>= 8 \u00d7 10,000<\/td>\n<td>= 8 \u00d7 10<sup>4<\/sup><\/td>\n<td>Eighty thousand<\/td>\n<\/tr>\n<tr>\n<td>+3,000<\/td>\n<td>= 3 \u00d7 1000<\/td>\n<td>= 3 \u00d7 10<sup>3<\/sup><\/td>\n<td>Three thousand<\/td>\n<\/tr>\n<tr>\n<td>+200<\/td>\n<td>= 2 \u00d7 100<\/td>\n<td>= 2 \u00d7 10<sup>2<\/sup><\/td>\n<td>Two hundred<\/td>\n<\/tr>\n<tr>\n<td>+10<\/td>\n<td>= 1 \u00d7 10<\/td>\n<td>= 1 \u00d7 10<sup>1<\/sup><\/td>\n<td>Ten<\/td>\n<\/tr>\n<tr>\n<td>+6<\/td>\n<td>= 6 \u00d7 1<\/td>\n<td>= 6 \u00d7 10<sup>0<\/sup><\/td>\n<td>Six<\/td>\n<\/tr>\n<tr>\n<td>5,783,216<\/td>\n<td colspan=\"3\">Five million, seven hundred eighty-three thousand, two hundred sixteen<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the third column in the table we can see that each place is simply a multiple of ten. Of course, this makes sense given that our base is ten. The digits that are multiplying each place simply tell us how many of that place we have. We are restricted to having at most 9 in any one place before we have to \u201ccarry\u201d over to the next place. We cannot, for example, have 11 in the hundreds place. Instead, we would carry 1 to the thousands place and retain 1 in the hundreds place. This comes as no surprise to us since we readily see that 11 hundreds is the same as one thousand, one hundred. Carrying is a pretty typical occurrence in a base system.<\/p>\n<p>However, base-ten is not the only option we have. Practically any positive integer greater than or equal to 2 can be used as a base for a number system. Such systems can work just like the decimal system except the number of symbols will be different and each position will depend on the base itself.<\/p>\n<h3>Other Bases<\/h3>\n<p>For example, let\u2019s suppose we adopt a base-five system. The only modern digits we would need for this system are 0,1,2,3 and 4. What are the place values in such a system? To answer that, we start with the ones place, as most base systems do. However, if we were to count in this system, we could only get to four (4) before we had to jump up to the next place. Our base is 5, after all! What is that next place that we would jump to? It would not be tens, since we are no longer in base-ten. We\u2019re in a different numerical world. As the base-ten system progresses from 10<sup>0<\/sup> to 10<sup>1<\/sup>, so the base-five system moves from 5<sup>0<\/sup> to 5<sup>1<\/sup> = 5. Thus, we move from the ones to the fives.<\/p>\n<p>After the fives, we would move to the 5<sup>2<\/sup> place, or the twenty fives. Note that in base-ten, we would have gone from the tens to the hundreds, which is, of course, 10<sup>2<\/sup>.<\/p>\n<p>Let\u2019s take an example and build a table. Consider the number 30412 in base five. We will write this as 30412<sub>5<\/sub>, where the subscript 5 is not part of the number but indicates the base we\u2019re using. First off, note that this is NOT the number \u201cthirty thousand, four hundred twelve.\u201d We must be careful not to impose the base-ten system on this number. Here\u2019s what our table might look like. We will use it to convert this number to our more familiar base-ten system.<\/p>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>Base 5<\/td>\n<td>This column coverts to base-ten<\/td>\n<td>In Base-Ten<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>3 \u00d7 5<sup>4<\/sup><\/td>\n<td>= 3 \u00d7 625<\/td>\n<td>= 1875<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>0 \u00d7 5<sup>3<\/sup><\/td>\n<td>= 0 \u00d7 125<\/td>\n<td>= 0<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>4 \u00d7 5<sup>2<\/sup><\/td>\n<td>= 4 \u00d7 25<\/td>\n<td>= 100<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>1 \u00d7 5<sup>1<\/sup><\/td>\n<td>= 1 \u00d7 5<\/td>\n<td>= 5<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>2 \u00d7 5<sup>0<\/sup><\/td>\n<td>= 2 \u00d7 1<\/td>\n<td>= 2<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>Total<\/td>\n<td>1982<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As you can see, the number 30412<sub>5<\/sub> is equivalent to 1,982 in base-ten. We will say 30412<sub>5<\/sub> = 1982<sub>10<\/sub>. All of this may seem strange to you, but that\u2019s only because you are so used to the only system that you\u2019ve ever seen.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert 6234<sub>7<\/sub> to a base 10 number.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q482364\">Show Solution<\/span><\/p>\n<div id=\"q482364\" class=\"hidden-answer\" style=\"display: none\">We first note that we are given a base-7 number that we are to convert. Thus, our places will start at the ones (7<sup>0<\/sup>), and then move up to the 7s, 49s (7<sup>2<\/sup>), etc. Here\u2019s the breakdown:<\/p>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>Base 7<\/td>\n<td>Convert<\/td>\n<td>Base 10<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>= 6\u00a0\u00d7\u00a07<sup>3<\/sup><\/td>\n<td>= 6\u00a0\u00d7 343<\/td>\n<td>= 2058<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>= 2\u00a0\u00d7\u00a07<sup>2<\/sup><\/td>\n<td>= 2\u00a0\u00d7 49<\/td>\n<td>= 98<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>= 3\u00a0\u00d7\u00a07<\/td>\n<td>= 3\u00a0\u00d7 7<\/td>\n<td>= 21<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>= 4\u00a0\u00d7 1<\/td>\n<td>= 4\u00a0\u00d7 1<\/td>\n<td>= 4<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>Total<\/td>\n<td>2181<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Thus 6234<sub>7<\/sub> = 2181<sub>10<\/sub><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert 41065<sub>7<\/sub> to a base 10 number.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q896067\">Show Solution<\/span><\/p>\n<div id=\"q896067\" class=\"hidden-answer\" style=\"display: none\">[latex]41065_{7} = 9994[\/latex]_{10}<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=8680&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of converting numbers in bases other than 10 into a base 10 number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Convert Numbers in Base Ten to Different Bases:  Remainder Method\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TjvexIVV_gI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1965\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Math in Society. <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Convert Numbers in Base Ten to Different Bases:  Remainder Method External link. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/TjvexIVV_gI\">https:\/\/youtu.be\/TjvexIVV_gI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 8680. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1965-1\">Ibid, page 230 <a href=\"#return-footnote-1965-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1965-2\">Ibid, page 231. <a href=\"#return-footnote-1965-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-1965-3\">Ibid, page 232. <a href=\"#return-footnote-1965-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-1965-4\">Ibid, page 232. <a href=\"#return-footnote-1965-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-1965-5\">McLeish, p. 18 <a href=\"#return-footnote-1965-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><li id=\"footnote-1965-6\"><a href=\"http:\/\/seattletimes.nwsource.com\/news\/health-science\/html98\/invs_20000201.html\">http:\/\/seattletimes.nwsource.com\/news\/health-science\/html98\/invs_20000201.html<\/a>, Seattle Times, Feb. 1, 2000 <a href=\"#return-footnote-1965-6\" class=\"return-footnote\" aria-label=\"Return to footnote 6\">&crarr;<\/a><\/li><li id=\"footnote-1965-7\">Ibid, page 232. <a href=\"#return-footnote-1965-7\" class=\"return-footnote\" aria-label=\"Return to footnote 7\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Convert Numbers in Base Ten to Different Bases:  Remainder Method External link\",\"author\":\"James Sousa 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