{"id":1962,"date":"2017-03-21T00:48:21","date_gmt":"2017-03-21T00:48:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1962"},"modified":"2020-09-14T18:40:17","modified_gmt":"2020-09-14T18:40:17","slug":"introduction-positional-systems-and-bases","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/chapter\/introduction-positional-systems-and-bases\/","title":{"raw":"Positional Systems and Bases","rendered":"Positional Systems and Bases"},"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\nMore important than the form of the number symbols is the development of the place value system. Although it is in slight dispute, the earliest known document in which the Indian system displays a positional system dates back to 346 CE. However, some evidence suggests that they may have actually developed a positional system as far back as the first century CE.\r\n\r\nIn this lesson we will explore positional systems an their historical development. \u00a0We will also discuss some of the positional systems that have been used throughout history and the bases used for those systems. \u00a0Finally, we will learn how to convert numbers between bases and systems.\r\n<h2>The Positional System and Base 10<\/h2>\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<sup>1<\/sup><\/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 7<sup>0<\/sup><\/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_{10}[\/latex]\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\r\n<h2>Another Method For Converting From Base 10 to Other Bases<\/h2>\r\nAs you read the solution to this last example and attempted the \u201cTry It\u201d problems, you may have had to repeatedly stop and think about what was going on. The fact that you are probably struggling to follow the explanation and reproduce the process yourself is mostly due to the fact that the non-decimal systems are so unfamiliar to you. In fact, the only system that you are probably comfortable with is the decimal system.\r\n\r\nAs budding mathematicians, you should always be asking questions like \u201cHow could I simplify this process?\u201d In general, that is one of the main things that mathematicians do: they look for ways to take complicated situations and make them easier or more familiar. In this section we will attempt to do that.\r\n\r\nTo do so, we will start by looking at our own decimal system. What we do may seem obvious and maybe even intuitive but that\u2019s the point. We want to find a process that we readily recognize works and makes sense to us in a familiar system and then use it to extend our results to a different, unfamiliar system.\r\n\r\nLet\u2019s start with the decimal number, 4863<sub>10<\/sub>. We will convert this number to base 10. Yeah, I know it\u2019s already in base 10, but if you carefully follow what we\u2019re doing, you\u2019ll see it makes things work out very nicely with other bases later on. We first note that the highest power of 10 that will divide into 4863 at least once is 10<sup>3<\/sup> = 1000. <em>In general, this is the first step in our new process; we find the highest power that a given base that will divide at least once into our given number.<\/em>\r\n<p style=\"text-align: center;\">We now divide 1000 into 4863:<\/p>\r\n<p style=\"text-align: center;\">4863 \u00f7 1000 = 4.863<\/p>\r\nThis says that there are four thousands in 4863 (obviously). However, it also says that there are 0.863 thousands in 4863. This fractional part is our remainder and will be converted to lower powers of our base (10). If we take that decimal and multiply by 10 (since that\u2019s the base we\u2019re in) we get the following:\r\n<p style=\"text-align: center;\">0.863 \u00d7 10 = 8.63<\/p>\r\nWhy multiply by 10 at this point? We need to recognize here that 0.863 thousands is the same as 8.63 hundreds. Think about that until it sinks in.\r\n<p style=\"text-align: center;\">(0.863)(1000) = 863\r\n(8.63)(100) = 863<\/p>\r\nThese two statements are equivalent. So, what we are really doing here by multiplying by 10 is rephrasing or converting from one place (thousands) to the next place down (hundreds).\r\n<p style=\"text-align: center;\">0.863 \u00d7 10 \u21d2 8.63\r\n(Parts of Thousands) \u00d7 10 \u21d2 Hundreds<\/p>\r\nWhat we have now is 8 hundreds and a remainder of 0.63 hundreds, which is the same as 6.3 tens. We can do this again with the 0.63 that remains after this first step.\r\n<p style=\"text-align: center;\">0.63 \u00d7 10 \u21d2 6.3\r\nHundreds \u00d7 10 \u21d2 Tens<\/p>\r\nSo we have six tens and 0.3 tens, which is the same as 3 ones, our last place value.\r\n\r\nNow here\u2019s the punch line. Let\u2019s put all of the together in one place:\r\n\r\n<img class=\"aligncenter wp-image-300 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155212\/Fig5_1_20.png\" alt=\"4863 divided by 1000 = 4.863, 0.863x 10 = 8.63, 0.63x10 = 6.3, 0.3x10 = 3.0\" width=\"237\" height=\"200\" \/>\r\nConverting from Base 10 to Base <em>b<\/em>: Another methodNote that in each step, the remainder is carried down to the next step and multiplied by 10, the base. Also, at each step, the whole number part, which is circled, gives the digit that belongs in that particular place. What is amazing is that this works for any base! So, to convert from a base 10 number to some other base, <em>b<\/em>, we have the following steps we can follow:\r\n<div class=\"textbox\">\r\n<h3>Converting from Base 10 to Base <em>b<\/em>: Another method<\/h3>\r\n<ol>\r\n \t<li>Find the highest power of the base <em>b<\/em> that will divide into the given number at least once and then divide.<\/li>\r\n \t<li>Keep the whole number part, and multiply the fractional part by the base <em>b<\/em>.<\/li>\r\n \t<li>Repeat step two, keeping the whole number part (including 0), carrying the fractional part to the next step until only a whole number result is obtained.<\/li>\r\n \t<li>Collect all your whole number parts to get your number in base <em>b<\/em> notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\nWe will illustrate this procedure with some examples.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConvert the base 10 number, 348<sub>10<\/sub>, to base 5.\r\n[reveal-answer q=\"881622\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"881622\"]\r\n\r\nThis is actually a conversion that we have done in a previous example. The powers of five are:\r\n\r\n5<sup>0<\/sup> = 1\r\n5<sup>1<\/sup> = 5\r\n5<sup>2<\/sup> = 25\r\n5<sup>3<\/sup> = 125\r\n5<sup>4<\/sup> = 625\r\nEtc\u2026\r\n\r\nThe highest power of five that will go into 348 at least once is 5<sup>3<\/sup>.\r\n\r\nWe divide by 125 and then proceed.\r\n\r\n<img class=\"alignnone size-full wp-image-301\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155214\/Fig5_1_21.png\" alt=\"Fig5_1_21\" width=\"183\" height=\"284\" \/>\r\n\r\nBy keeping all the whole number parts, from top bottom, gives 2343 as our base 5 number. Thus, 2343<sub>5<\/sub>\u00a0= 348<sub>10<\/sub>.\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can compare our result with what we saw earlier, or simply check with our calculator, and find that these two numbers really are equivalent to each other.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConvert the base 10 number, 3007<sub>10<\/sub>, to base 5.\r\n[reveal-answer q=\"462788\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"462788\"]\r\n\r\nThe highest power of 5 that divides at least once into 3007 is 5<sup>4<\/sup> = 625. Thus, we have:\r\n\r\n3007 \u00f7 625 = \u2463.8112\r\n0.8112 \u00d7 5 = \u2463.056\r\n0.056 \u00d7 5 = \u24ea.28\r\n0.28 \u00d7 5 = \u24600.4\r\n0.4 \u00d7 5 = \u24610.0\r\n\r\nThis gives us that 3007<sub>10<\/sub> = 44012<sub>5<\/sub>. Notice that in the third line that multiplying by 5 gave us 0 for our whole number part. We don\u2019t discard that! The zero tells us that a zero in that place. That is, there are no 5<sup>2<\/sup>s in this number.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis last example shows the importance of using a calculator in certain situations and taking care to avoid clearing the calculator\u2019s memory or display until you get to the very end of the process.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConvert the base 10 number, 63201<sub>10<\/sub>, to base 7.\r\n[reveal-answer q=\"186862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186862\"]\r\n\r\nThe powers of 7 are:\r\n\r\n7<sup>0<\/sup> = 1\r\n7<sup>1<\/sup> = 7\r\n7<sup>2<\/sup> = 49\r\n7<sup>3<\/sup> = 343\r\n7<sup>4<\/sup> = 2401\r\n7<sup>5<\/sup> = 16807\r\netc\u2026\r\n\r\nThe highest power of 7 that will divide at least once into 63201 is 7<sup>5<\/sup>. When we do the initial division on a calculator, we get the following:\r\n\r\n63201 \u00f7 7<sup>5<\/sup> = 3.760397453\r\n\r\nThe decimal part actually fills up the calculators display and we don\u2019t know if it terminates at some point or perhaps even repeats down the road. So if we clear our calculator at this point, we will introduce error that is likely to keep this process from ever ending. To avoid this problem, we leave the result in the calculator and simply subtract 3 from this to get the fractional part all by itself. <strong>Do not round off!<\/strong> Subtraction and then multiplication by seven gives:\r\n\r\n63201 \u00f7 7<sup>5<\/sup> = \u0083\u2462.760397453\r\n0.760397453 \u00d7 7 = \u0085\u2464.322782174\r\n0.322782174 \u00d7 7 =\u0082 \u2461.259475219\r\n0.259475219 \u00d7 7 =\u0081 \u2460.816326531\r\n0.816326531 \u00d7 7 = \u2464\u0085.714285714\r\n0.714285714 \u00d7 7 = \u0085\u2464.000000000\r\n\r\nYes, believe it or not, that last product is exactly 5, <em>as long as you don\u2019t clear anything out on your calculator<\/em>. This gives us our final result: 63201<sub>10<\/sub> = 352155<sub>7<\/sub>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf we round, even to two decimal places in each step, clearing our calculator out at each step along the way, we will get a series of numbers that do not terminate, but begin repeating themselves endlessly. (Try it!) We end up with something that doesn\u2019t make any sense, at least not in this context. So be careful to use your calculator cautiously on these conversion problems.\r\n\r\nAlso, remember that if your first division is by 7<sup>5<\/sup>, then you expect to have 6 digits in the final answer, corresponding to the places for 7<sup>5<\/sup>, 7<sup>4<\/sup>, and so on down to 7<sup>0<\/sup>. If you find yourself with more than 6 digits due to rounding errors, you know something went wrong.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConvert the base 10 number, 9352<sub>10<\/sub>, to base 5.\r\n[reveal-answer q=\"290694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"290694\"][latex]9352_{10} = 244402_{5}[\/latex][\/hidden-answer]\r\n\r\nConvert the base 10 number, 1500, to base 3.\r\n\r\nBe careful not to clear your calculator on this one. Also, if you\u2019re not careful in each step, you may not get all of the digits you\u2019re looking for, so move slowly and with caution.\r\n[reveal-answer q=\"148410\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"148410\"]\r\n\r\n[latex]1500_{10} = 2001120_{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=8681&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nThe following video shows how to use a calculator to convert numbers in base 10 into other bases.\r\n\r\nhttps:\/\/youtu.be\/YNPTYelCeIs\r\n<h2>The Mayan Numeral System<\/h2>\r\n<h3>Background<\/h3>\r\nAs you might imagine, the development of a base system is an important step in making the counting process more efficient. Our own base-ten system probably arose from the fact that we have 10 fingers (including thumbs) on two hands. This is a natural development. However, other civilizations have had a variety of bases other than ten. For example, the Natives of Queensland used a base-two system, counting as follows: \u201cone, two, two and one, two two\u2019s, much.\u201d Some Modern South American Tribes have a base-five system counting in this way: \u201cone, two, three, four, hand, hand and one, hand and two,\u201d and so on. The Babylonians used a base-sixty (sexigesimal) system. In this chapter, we wrap up with a specific example of a civilization that actually used a base system other than 10.\r\n\r\n<img class=\"alignright wp-image-302\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155215\/Fig5_1_22.png\" alt=\"Fig5_1_22\" width=\"350\" height=\"428\" \/>The Mayan civilization is generally dated from 1500 BCE to 1700 CE. The Yucatan Peninsula (see figure 16[footnote]http:\/\/www.gorp.com\/gorp\/location\/latamer\/map_maya.htm[\/footnote]) in Mexico was the scene for the development of one of the most advanced civilizations of the ancient world. The Mayans had a sophisticated ritual system that was overseen by a priestly class. This class of priests developed a philosophy with time as divine and eternal.[footnote]Bidwell, James; Mayan Arithmetic in <em>Mathematics Teacher<\/em>, Issue 74 (Nov., 1967), p. 762\u201368.[\/footnote]\u00a0The calendar, and calculations related to it, were thus very important to the ritual life of the priestly class, and hence the Mayan people. In fact, much of what we know about this culture comes from their calendar records and astronomy data. Another important source of information on the Mayans is the writings of Father Diego de Landa, who went to Mexico as a missionary in 1549.\r\n\r\n<img class=\"alignright size-full wp-image-303\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155218\/Fig5_1_23.png\" alt=\"Fig5_1_23\" width=\"162\" height=\"139\" \/>There were two numeral systems developed by the Mayans\u2014one for the common people and one for the priests. Not only did these two systems use different symbols, they also used different base systems. For the priests, the number system was governed by ritual. The days of the year were thought to be gods, so the formal symbols for the days were decorated heads,[footnote]<a href=\"http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html<\/a>[\/footnote]\u00a0like the sample to the left[footnote]<a href=\"http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html<\/a>[\/footnote]\u00a0Since the basic calendar was based on 360 days, the priestly numeral system used a mixed base system employing multiples of 20 and 360. This makes for a confusing system, the details of which we will skip.\r\n<div>\r\n<table style=\"width: 50%;\">\r\n<tbody>\r\n<tr>\r\n<td>Powers<\/td>\r\n<td>Base-Ten Value<\/td>\r\n<td>Place Name<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>7<\/sup><\/td>\r\n<td>12,800,000,000<\/td>\r\n<td>Hablat<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>6<\/sup><\/td>\r\n<td>64,000,000<\/td>\r\n<td>Alau<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>5<\/sup><\/td>\r\n<td>3,200,000<\/td>\r\n<td>Kinchil<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>4<\/sup><\/td>\r\n<td>160,000<\/td>\r\n<td>Cabal<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>3<\/sup><\/td>\r\n<td>8,000<\/td>\r\n<td>Pic<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>2<\/sup><\/td>\r\n<td>400<\/td>\r\n<td><em>Bak<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>1<\/sup><\/td>\r\n<td>20<\/td>\r\n<td>Kal<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<sup>0<\/sup><\/td>\r\n<td>1<\/td>\r\n<td>Hun<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<h3>The Mayan Number System<\/h3>\r\nInstead, we will focus on the numeration system of the \u201ccommon\u201d people, which used a more consistent base system. As we stated earlier, the Mayans used a base-20 system, called the \u201cvigesimal\u201d system. Like our system, it is positional, meaning that the position of a numeric symbol indicates its place value. In the following table you can see the place value in its vertical format.[footnote]Bidwell[\/footnote]\r\n\r\n<img class=\"size-full wp-image-304 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155218\/Fig5_1_24.png\" alt=\"Fig5_1_24\" width=\"438\" height=\"750\" \/>\r\n\r\nIn order to write numbers down, there were only three symbols needed in this system. A horizontal bar represented the quantity 5, a dot represented the quantity 1, and a special symbol (thought to be a shell) represented zero. The Mayan system may have been the first to make use of zero as a placeholder\/number. The first 20 numbers are shown in the table to the right.[footnote]<a href=\"http:\/\/www.vpds.wsu.edu\/fair_95\/gym\/UM001.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.vpds.wsu.edu\/fair_95\/gym\/UM001.html<\/a>[\/footnote]\r\n\r\nUnlike our system, where the ones place starts on the right and then moves to the left, the Mayan systems places the ones on the <strong>bottom<\/strong> of a vertical orientation and moves up as the place value increases.\r\n\r\nWhen numbers are written in vertical form, there should never be more than four dots in a single place. When writing Mayan numbers, every group of five dots becomes one bar. Also, there should never be more than three bars in a single place\u2026four bars would be converted to one dot in the next place up. It\u2019s the same as 10 getting converted to a 1 in the next place up when we carry during addition.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat is the value of this number, which is shown in vertical form?\r\n\r\n<img class=\"alignnone size-full wp-image-305\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155220\/Fig5_1_25.png\" alt=\"Fig5_1_25\" width=\"48\" height=\"83\" \/>\r\n[reveal-answer q=\"170707\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"170707\"]\r\n\r\nStarting from the bottom, we have the ones place. There are two bars and three dots in this place. Since each bar is worth 5, we have 13 ones when we count the three dots in the ones place. Looking to the place value above it (the twenties places), we see there are three dots so we have three twenties.\r\n\r\n<img class=\"alignnone size-full wp-image-306\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155220\/Fig5_1_26.png\" alt=\"Fig5_1_26\" width=\"206\" height=\"82\" \/>\r\n\r\nHence we can write this number in base-ten as:\r\n\r\n(3 \u00d7 20<sup>1<\/sup>) +\u00a0(13 \u00d7 20<sup>0<\/sup>) =\u00a0(3 \u00d7 20<sup>1<\/sup>) +\u00a0(13 \u00d7 1) = 60 + 13 = 73\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat is the value of the following Mayan number?\r\n\r\n<img class=\"alignnone size-full wp-image-307\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155221\/Fig5_1_27.png\" alt=\"Fig5_1_27\" width=\"57\" height=\"111\" \/>\r\n[reveal-answer q=\"760307\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"760307\"]\r\n\r\nThis number has 11 in the ones place, zero in the 20s place, and 18 in the 20<sup>2\u00a0<\/sup>= 400s place. Hence, the value of this number in base-ten is:\r\n\r\n18 \u00d7 400 + 0 \u00d7 20 + 11 \u00d7 1 = 7211.\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 the Mayan number below to base 10.\r\n\r\n<img class=\"alignnone size-full wp-image-308\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155221\/Fig5_1_28.png\" alt=\"Fig5_1_28\" width=\"60\" height=\"115\" \/>\r\n[reveal-answer q=\"249582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"249582\"]\r\n<div>\r\n\r\n1562\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6420&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConvert the base 10 number 3575<sub>10<\/sub> to Mayan numerals.\r\n[reveal-answer q=\"549968\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"549968\"]\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nThis problem is done in two stages. First we need to convert to a base 20 number. We will do so using the method provided in the last section of the text. The second step is to convert that number to Mayan symbols.\r\n\r\nThe highest power of 20 that will divide into 3575 is 20<sup>2<\/sup> = 400, so we start by dividing that and then proceed from there:\r\n\r\n3575 \u00f7 400 = 8.9375\r\n0.9375 \u00d7 20 = 18.75\r\n0.75 \u00d7 20 = 15.0\r\n\r\nThis means that 3575<sub>10<\/sub> = 8,18,15<sub>20<\/sub>\r\n\r\nThe second step is to convert this to Mayan notation. This number indicates that we have 15 in the ones position. That\u2019s three bars at the bottom of the number. We also have 18 in the 20s place, so that\u2019s three bars and three dots in the second position. Finally, we have 8 in the 400s place, so that\u2019s one bar and three dots on the top. We get the following:\r\n\r\n<img class=\"alignnone size-full wp-image-309\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155221\/Fig5_1_29.png\" alt=\"Fig5_1_29\" width=\"66\" height=\"93\" \/>\r\n\r\n<\/div>\r\nNote that in the previous example a new notation was used when we wrote 8,18,15<sub>20<\/sub>. The commas between the three numbers 8, 18, and 15 are now separating place values for us so that we can keep them separate from each other. This use of the comma is slightly different than how they\u2019re used in the decimal system. When we write a number in base 10, such as 7,567,323, the commas are used primarily as an aide to read the number easily but they do not separate single place values from each other. We will need this notation whenever the base we use is larger than 10.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Writing numbers with bases bigger than 10<\/h3>\r\nWhen the base of a number is larger than 10, separate each \u201cdigit\u201d with a comma to make the separation of digits clear.\r\n\r\nFor example, in base 20, to write the number corresponding to 17 \u00d7 20<sup>2<\/sup> + 6 \u00d7 20<sup>1<\/sup> + 13 \u00d7 20<sup>0<\/sup>, we\u2019d write 17,6,13<sub>20<\/sub>.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConvert the base 10 number 10553<sub>10<\/sub> to Mayan numerals.\r\n[reveal-answer q=\"933764\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"933764\"]\r\n<div>\r\n\r\n[latex]10553_{10} = 1,6,7,13_{20}[\/latex]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192231\/mayan-numeral.png\"><img class=\"alignnone size-full wp-image-2009\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192231\/mayan-numeral.png\" alt=\"\" width=\"36\" height=\"87\" \/><\/a>\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6423&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nConvert the base 10 number 5617<sub>10<\/sub> to Mayan numerals.\r\n[reveal-answer q=\"551605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"551605\"][latex]5617_{10} = 14,0,17_{20}[\/latex]. Note that there is a zero in the 20\u2019s place, so you\u2019ll need to use the appropriate zero symbol in between the ones and 400\u2019s\u00a0places.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192356\/mayan-2.png\"><img class=\"alignnone size-full wp-image-2010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192356\/mayan-2.png\" alt=\"\" width=\"36\" height=\"57\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nIn the following video we present more examples of how to write numbers using Mayan numerals as well as converting numerals written in Mayan for into base 10 form.\r\n\r\nhttps:\/\/youtu.be\/gPUOrcilVS0\r\n\r\nThe next video shows more examples of converting base 10 numbers into Mayan numerals.\r\n\r\nhttps:\/\/youtu.be\/LrHNXoqQ_lI\r\n<h3>Adding Mayan Numbers<\/h3>\r\nWhen adding Mayan numbers together, we\u2019ll adopt a scheme that the Mayans probably did not use but which will make life a little easier for us.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd, in Mayan, the numbers 37 and 29:\r\n\r\n<img class=\"alignnone size-full wp-image-310\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155222\/Fig5_1_30.png\" alt=\"Fig5_1_30\" width=\"146\" height=\"83\" \/>[footnote]<a href=\"http:\/\/forum.swarthmore.edu\/k12\/mayan.math\/mayan2.html\">http:\/\/forum.swarthmore.edu\/k12\/mayan.math\/mayan2.html<\/a>[\/footnote]\r\n[reveal-answer q=\"193127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"193127\"]\r\n\r\nFirst draw a box around each of the vertical places. This will help keep the place values from being mixed up.\r\n\r\n<img class=\"alignnone size-full wp-image-311\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155222\/Fig5_1_31.png\" alt=\"Fig5_1_31\" width=\"169\" height=\"154\" \/>\r\n\r\nNext, put all of the symbols from both numbers into a single set of places (boxes), and to the right of this new number draw a set of empty boxes where you will place the final sum:\r\n\r\n<img class=\"alignnone size-full wp-image-312\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155223\/Fig5_1_32.png\" alt=\"Fig5_1_32\" width=\"167\" height=\"150\" \/>\r\n\r\nYou are now ready to start carrying. Begin with the place that has the lowest value, just as you do with Arabic numbers. Start at the bottom place, where each dot is worth 1. There are six dots, but a maximum of four are allowed in any one place; once you get to five dots, you must convert to a bar. Since five dots make one bar, we draw a bar through five of the dots, leaving us with one dot which is under the four-dot limit. Put this dot into the bottom place of the empty set of boxes you just drew:\r\n\r\n<img class=\"alignnone size-full wp-image-313\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155224\/Fig5_1_33.png\" alt=\"Fig5_1_33\" width=\"168\" height=\"150\" \/>\r\n\r\nNow look at the bars in the bottom place. There are five, and the maximum number the place can hold is three. <em>Four bars are equal to one dot in the next highest place<\/em>.\r\n\r\nWhenever we have four bars in a single place we will automatically convert that to a <em>dot <\/em>in the next place up. We draw a circle around four of the bars and an arrow up to the dots' section of the higher place. At the end of that arrow, draw a new dot. That dot represents 20 just the same as the other dots in that place. Not counting the circled bars in the bottom place, there is one bar left. One bar is under the three-bar limit; put it under the dot in the set of empty places to the right.\r\n\r\n<img class=\"alignnone size-full wp-image-314\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155224\/Fig5_1_34.png\" alt=\"Fig5_1_34\" width=\"178\" height=\"151\" \/>\r\n\r\nNow there are only three dots in the next highest place, so draw them in the corresponding empty box.\r\n\r\n<img class=\"alignnone size-full wp-image-315\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155227\/Fig5_1_35.png\" alt=\"Fig5_1_35\" width=\"166\" height=\"150\" \/>\r\n\r\nWe can see here that we have 3 twenties (60), and 6 ones, for a total of 66. We check and note that 37 + 29 = 66, so we have done this addition correctly. Is it easier to just do it in base-ten? Probably, but that\u2019s only because it\u2019s more familiar to you. Your task here is to try to learn a new base system and how addition can be done in slightly different ways than what you have seen in the past. Note, however, that the concept of carrying is still used, just as it is in our own addition algorithm.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nTry adding 174 and 78 in Mayan by first converting to Mayan numbers and then working entirely within that system. Do not add in base-ten (decimal) until the very end when you <em>check<\/em> your work.\r\n[reveal-answer q=\"374361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"374361\"]A sample solution is shown.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192716\/mayan-3.png\"><img class=\"alignnone size-medium wp-image-2011\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192716\/mayan-3-300x98.png\" alt=\"\" width=\"300\" height=\"98\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last video we show more examples of adding Mayan numerals.\r\n\r\nhttps:\/\/youtu.be\/NpH5oMCrubM\r\nIn this module, we have briefly sketched the development of numbers and our counting system, with the emphasis on the \u201cbrief\u201d part. There are numerous sources of information and research that fill many volumes of books on this topic. Unfortunately, we cannot begin to come close to covering all of the information that is out there.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/28182439\/204px-Numeral_Systems_of_the_World.svg_.png\"><img class=\" wp-image-2292 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/28182439\/204px-Numeral_Systems_of_the_World.svg_.png\" alt=\"\" width=\"302\" height=\"215\" \/><\/a>\r\n\r\nWe have only scratched the surface of the wealth of research and information that exists on the development of numbers and counting throughout human history. What is important to note is that the system that we use every day is a product of thousands of years of progress and development. It represents contributions by many civilizations and cultures. It does not come down to us from the sky, a gift from the gods. It is not the creation of a textbook publisher. It is indeed as human as we are, as is the rest of mathematics. Behind every symbol, formula and rule there is a human face to be found, or at least sought.\r\n\r\nFurthermore, we hope that you now have a basic appreciation for just how interesting and diverse number systems can get. Also, we\u2019re pretty sure that you have also begun to recognize that we take our own number system for granted so much that when we try to adapt to other systems or bases, we find ourselves truly having to concentrate and think about what is going on.","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>More important than the form of the number symbols is the development of the place value system. Although it is in slight dispute, the earliest known document in which the Indian system displays a positional system dates back to 346 CE. However, some evidence suggests that they may have actually developed a positional system as far back as the first century CE.<\/p>\n<p>In this lesson we will explore positional systems an their historical development. \u00a0We will also discuss some of the positional systems that have been used throughout history and the bases used for those systems. \u00a0Finally, we will learn how to convert numbers between bases and systems.<\/p>\n<h2>The Positional System and Base 10<\/h2>\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-1962-1\" href=\"#footnote-1962-1\" 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-1962-2\" href=\"#footnote-1962-2\" 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-1962-3\" href=\"#footnote-1962-3\" 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-1962-4\" href=\"#footnote-1962-4\" 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-1962-5\" href=\"#footnote-1962-5\" 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-1962-6\" href=\"#footnote-1962-6\" 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-1962-7\" href=\"#footnote-1962-7\" 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<sup>1<\/sup><\/td>\n<td>= 3\u00a0\u00d7 7<\/td>\n<td>= 21<\/td>\n<\/tr>\n<tr>\n<td>+<\/td>\n<td>= 4\u00a0\u00d7 7<sup>0<\/sup><\/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_{10}[\/latex]<\/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<h2>Another Method For Converting From Base 10 to Other Bases<\/h2>\n<p>As you read the solution to this last example and attempted the \u201cTry It\u201d problems, you may have had to repeatedly stop and think about what was going on. The fact that you are probably struggling to follow the explanation and reproduce the process yourself is mostly due to the fact that the non-decimal systems are so unfamiliar to you. In fact, the only system that you are probably comfortable with is the decimal system.<\/p>\n<p>As budding mathematicians, you should always be asking questions like \u201cHow could I simplify this process?\u201d In general, that is one of the main things that mathematicians do: they look for ways to take complicated situations and make them easier or more familiar. In this section we will attempt to do that.<\/p>\n<p>To do so, we will start by looking at our own decimal system. What we do may seem obvious and maybe even intuitive but that\u2019s the point. We want to find a process that we readily recognize works and makes sense to us in a familiar system and then use it to extend our results to a different, unfamiliar system.<\/p>\n<p>Let\u2019s start with the decimal number, 4863<sub>10<\/sub>. We will convert this number to base 10. Yeah, I know it\u2019s already in base 10, but if you carefully follow what we\u2019re doing, you\u2019ll see it makes things work out very nicely with other bases later on. We first note that the highest power of 10 that will divide into 4863 at least once is 10<sup>3<\/sup> = 1000. <em>In general, this is the first step in our new process; we find the highest power that a given base that will divide at least once into our given number.<\/em><\/p>\n<p style=\"text-align: center;\">We now divide 1000 into 4863:<\/p>\n<p style=\"text-align: center;\">4863 \u00f7 1000 = 4.863<\/p>\n<p>This says that there are four thousands in 4863 (obviously). However, it also says that there are 0.863 thousands in 4863. This fractional part is our remainder and will be converted to lower powers of our base (10). If we take that decimal and multiply by 10 (since that\u2019s the base we\u2019re in) we get the following:<\/p>\n<p style=\"text-align: center;\">0.863 \u00d7 10 = 8.63<\/p>\n<p>Why multiply by 10 at this point? We need to recognize here that 0.863 thousands is the same as 8.63 hundreds. Think about that until it sinks in.<\/p>\n<p style=\"text-align: center;\">(0.863)(1000) = 863<br \/>\n(8.63)(100) = 863<\/p>\n<p>These two statements are equivalent. So, what we are really doing here by multiplying by 10 is rephrasing or converting from one place (thousands) to the next place down (hundreds).<\/p>\n<p style=\"text-align: center;\">0.863 \u00d7 10 \u21d2 8.63<br \/>\n(Parts of Thousands) \u00d7 10 \u21d2 Hundreds<\/p>\n<p>What we have now is 8 hundreds and a remainder of 0.63 hundreds, which is the same as 6.3 tens. We can do this again with the 0.63 that remains after this first step.<\/p>\n<p style=\"text-align: center;\">0.63 \u00d7 10 \u21d2 6.3<br \/>\nHundreds \u00d7 10 \u21d2 Tens<\/p>\n<p>So we have six tens and 0.3 tens, which is the same as 3 ones, our last place value.<\/p>\n<p>Now here\u2019s the punch line. Let\u2019s put all of the together in one place:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-300 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155212\/Fig5_1_20.png\" alt=\"4863 divided by 1000 = 4.863, 0.863x 10 = 8.63, 0.63x10 = 6.3, 0.3x10 = 3.0\" width=\"237\" height=\"200\" \/><br \/>\nConverting from Base 10 to Base <em>b<\/em>: Another methodNote that in each step, the remainder is carried down to the next step and multiplied by 10, the base. Also, at each step, the whole number part, which is circled, gives the digit that belongs in that particular place. What is amazing is that this works for any base! So, to convert from a base 10 number to some other base, <em>b<\/em>, we have the following steps we can follow:<\/p>\n<div class=\"textbox\">\n<h3>Converting from Base 10 to Base <em>b<\/em>: Another method<\/h3>\n<ol>\n<li>Find the highest power of the base <em>b<\/em> that will divide into the given number at least once and then divide.<\/li>\n<li>Keep the whole number part, and multiply the fractional part by the base <em>b<\/em>.<\/li>\n<li>Repeat step two, keeping the whole number part (including 0), carrying the fractional part to the next step until only a whole number result is obtained.<\/li>\n<li>Collect all your whole number parts to get your number in base <em>b<\/em> notation.<\/li>\n<\/ol>\n<\/div>\n<p>We will illustrate this procedure with some examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert the base 10 number, 348<sub>10<\/sub>, to base 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q881622\">Show Solution<\/span><\/p>\n<div id=\"q881622\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is actually a conversion that we have done in a previous example. The powers of five are:<\/p>\n<p>5<sup>0<\/sup> = 1<br \/>\n5<sup>1<\/sup> = 5<br \/>\n5<sup>2<\/sup> = 25<br \/>\n5<sup>3<\/sup> = 125<br \/>\n5<sup>4<\/sup> = 625<br \/>\nEtc\u2026<\/p>\n<p>The highest power of five that will go into 348 at least once is 5<sup>3<\/sup>.<\/p>\n<p>We divide by 125 and then proceed.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-301\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155214\/Fig5_1_21.png\" alt=\"Fig5_1_21\" width=\"183\" height=\"284\" \/><\/p>\n<p>By keeping all the whole number parts, from top bottom, gives 2343 as our base 5 number. Thus, 2343<sub>5<\/sub>\u00a0= 348<sub>10<\/sub>.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can compare our result with what we saw earlier, or simply check with our calculator, and find that these two numbers really are equivalent to each other.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert the base 10 number, 3007<sub>10<\/sub>, to base 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q462788\">Show Solution<\/span><\/p>\n<div id=\"q462788\" class=\"hidden-answer\" style=\"display: none\">\n<p>The highest power of 5 that divides at least once into 3007 is 5<sup>4<\/sup> = 625. Thus, we have:<\/p>\n<p>3007 \u00f7 625 = \u2463.8112<br \/>\n0.8112 \u00d7 5 = \u2463.056<br \/>\n0.056 \u00d7 5 = \u24ea.28<br \/>\n0.28 \u00d7 5 = \u24600.4<br \/>\n0.4 \u00d7 5 = \u24610.0<\/p>\n<p>This gives us that 3007<sub>10<\/sub> = 44012<sub>5<\/sub>. Notice that in the third line that multiplying by 5 gave us 0 for our whole number part. We don\u2019t discard that! The zero tells us that a zero in that place. That is, there are no 5<sup>2<\/sup>s in this number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This last example shows the importance of using a calculator in certain situations and taking care to avoid clearing the calculator\u2019s memory or display until you get to the very end of the process.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert the base 10 number, 63201<sub>10<\/sub>, to base 7.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186862\">Show Solution<\/span><\/p>\n<div id=\"q186862\" class=\"hidden-answer\" style=\"display: none\">\n<p>The powers of 7 are:<\/p>\n<p>7<sup>0<\/sup> = 1<br \/>\n7<sup>1<\/sup> = 7<br \/>\n7<sup>2<\/sup> = 49<br \/>\n7<sup>3<\/sup> = 343<br \/>\n7<sup>4<\/sup> = 2401<br \/>\n7<sup>5<\/sup> = 16807<br \/>\netc\u2026<\/p>\n<p>The highest power of 7 that will divide at least once into 63201 is 7<sup>5<\/sup>. When we do the initial division on a calculator, we get the following:<\/p>\n<p>63201 \u00f7 7<sup>5<\/sup> = 3.760397453<\/p>\n<p>The decimal part actually fills up the calculators display and we don\u2019t know if it terminates at some point or perhaps even repeats down the road. So if we clear our calculator at this point, we will introduce error that is likely to keep this process from ever ending. To avoid this problem, we leave the result in the calculator and simply subtract 3 from this to get the fractional part all by itself. <strong>Do not round off!<\/strong> Subtraction and then multiplication by seven gives:<\/p>\n<p>63201 \u00f7 7<sup>5<\/sup> = \u0083\u2462.760397453<br \/>\n0.760397453 \u00d7 7 = \u0085\u2464.322782174<br \/>\n0.322782174 \u00d7 7 =\u0082 \u2461.259475219<br \/>\n0.259475219 \u00d7 7 =\u0081 \u2460.816326531<br \/>\n0.816326531 \u00d7 7 = \u2464\u0085.714285714<br \/>\n0.714285714 \u00d7 7 = \u0085\u2464.000000000<\/p>\n<p>Yes, believe it or not, that last product is exactly 5, <em>as long as you don\u2019t clear anything out on your calculator<\/em>. This gives us our final result: 63201<sub>10<\/sub> = 352155<sub>7<\/sub>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>If we round, even to two decimal places in each step, clearing our calculator out at each step along the way, we will get a series of numbers that do not terminate, but begin repeating themselves endlessly. (Try it!) We end up with something that doesn\u2019t make any sense, at least not in this context. So be careful to use your calculator cautiously on these conversion problems.<\/p>\n<p>Also, remember that if your first division is by 7<sup>5<\/sup>, then you expect to have 6 digits in the final answer, corresponding to the places for 7<sup>5<\/sup>, 7<sup>4<\/sup>, and so on down to 7<sup>0<\/sup>. If you find yourself with more than 6 digits due to rounding errors, you know something went wrong.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert the base 10 number, 9352<sub>10<\/sub>, to base 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q290694\">Show Solution<\/span><\/p>\n<div id=\"q290694\" class=\"hidden-answer\" style=\"display: none\">[latex]9352_{10} = 244402_{5}[\/latex]<\/div>\n<\/div>\n<p>Convert the base 10 number, 1500, to base 3.<\/p>\n<p>Be careful not to clear your calculator on this one. Also, if you\u2019re not careful in each step, you may not get all of the digits you\u2019re looking for, so move slowly and with caution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q148410\">Show Solution<\/span><\/p>\n<div id=\"q148410\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1500_{10} = 2001120_{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=8681&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows how to use a calculator to convert numbers in base 10 into other bases.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Convert Numbers in Base Ten to Different Bases:  Calculator Method\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/YNPTYelCeIs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Mayan Numeral System<\/h2>\n<h3>Background<\/h3>\n<p>As you might imagine, the development of a base system is an important step in making the counting process more efficient. Our own base-ten system probably arose from the fact that we have 10 fingers (including thumbs) on two hands. This is a natural development. However, other civilizations have had a variety of bases other than ten. For example, the Natives of Queensland used a base-two system, counting as follows: \u201cone, two, two and one, two two\u2019s, much.\u201d Some Modern South American Tribes have a base-five system counting in this way: \u201cone, two, three, four, hand, hand and one, hand and two,\u201d and so on. The Babylonians used a base-sixty (sexigesimal) system. In this chapter, we wrap up with a specific example of a civilization that actually used a base system other than 10.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-302\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155215\/Fig5_1_22.png\" alt=\"Fig5_1_22\" width=\"350\" height=\"428\" \/>The Mayan civilization is generally dated from 1500 BCE to 1700 CE. The Yucatan Peninsula (see figure 16<a class=\"footnote\" title=\"http:\/\/www.gorp.com\/gorp\/location\/latamer\/map_maya.htm\" id=\"return-footnote-1962-8\" href=\"#footnote-1962-8\" aria-label=\"Footnote 8\"><sup class=\"footnote\">[8]<\/sup><\/a>) in Mexico was the scene for the development of one of the most advanced civilizations of the ancient world. The Mayans had a sophisticated ritual system that was overseen by a priestly class. This class of priests developed a philosophy with time as divine and eternal.<a class=\"footnote\" title=\"Bidwell, James; Mayan Arithmetic in Mathematics Teacher, Issue 74 (Nov., 1967), p. 762\u201368.\" id=\"return-footnote-1962-9\" href=\"#footnote-1962-9\" aria-label=\"Footnote 9\"><sup class=\"footnote\">[9]<\/sup><\/a>\u00a0The calendar, and calculations related to it, were thus very important to the ritual life of the priestly class, and hence the Mayan people. In fact, much of what we know about this culture comes from their calendar records and astronomy data. Another important source of information on the Mayans is the writings of Father Diego de Landa, who went to Mexico as a missionary in 1549.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright size-full wp-image-303\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155218\/Fig5_1_23.png\" alt=\"Fig5_1_23\" width=\"162\" height=\"139\" \/>There were two numeral systems developed by the Mayans\u2014one for the common people and one for the priests. Not only did these two systems use different symbols, they also used different base systems. For the priests, the number system was governed by ritual. The days of the year were thought to be gods, so the formal symbols for the days were decorated heads,<a class=\"footnote\" title=\"http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html\" id=\"return-footnote-1962-10\" href=\"#footnote-1962-10\" aria-label=\"Footnote 10\"><sup class=\"footnote\">[10]<\/sup><\/a>\u00a0like the sample to the left<a class=\"footnote\" title=\"http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html\" id=\"return-footnote-1962-11\" href=\"#footnote-1962-11\" aria-label=\"Footnote 11\"><sup class=\"footnote\">[11]<\/sup><\/a>\u00a0Since the basic calendar was based on 360 days, the priestly numeral system used a mixed base system employing multiples of 20 and 360. This makes for a confusing system, the details of which we will skip.<\/p>\n<div>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<td>Powers<\/td>\n<td>Base-Ten Value<\/td>\n<td>Place Name<\/td>\n<\/tr>\n<tr>\n<td>20<sup>7<\/sup><\/td>\n<td>12,800,000,000<\/td>\n<td>Hablat<\/td>\n<\/tr>\n<tr>\n<td>20<sup>6<\/sup><\/td>\n<td>64,000,000<\/td>\n<td>Alau<\/td>\n<\/tr>\n<tr>\n<td>20<sup>5<\/sup><\/td>\n<td>3,200,000<\/td>\n<td>Kinchil<\/td>\n<\/tr>\n<tr>\n<td>20<sup>4<\/sup><\/td>\n<td>160,000<\/td>\n<td>Cabal<\/td>\n<\/tr>\n<tr>\n<td>20<sup>3<\/sup><\/td>\n<td>8,000<\/td>\n<td>Pic<\/td>\n<\/tr>\n<tr>\n<td>20<sup>2<\/sup><\/td>\n<td>400<\/td>\n<td><em>Bak<\/em><\/td>\n<\/tr>\n<tr>\n<td>20<sup>1<\/sup><\/td>\n<td>20<\/td>\n<td>Kal<\/td>\n<\/tr>\n<tr>\n<td>20<sup>0<\/sup><\/td>\n<td>1<\/td>\n<td>Hun<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3>The Mayan Number System<\/h3>\n<p>Instead, we will focus on the numeration system of the \u201ccommon\u201d people, which used a more consistent base system. As we stated earlier, the Mayans used a base-20 system, called the \u201cvigesimal\u201d system. Like our system, it is positional, meaning that the position of a numeric symbol indicates its place value. In the following table you can see the place value in its vertical format.<a class=\"footnote\" title=\"Bidwell\" id=\"return-footnote-1962-12\" href=\"#footnote-1962-12\" aria-label=\"Footnote 12\"><sup class=\"footnote\">[12]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-304 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155218\/Fig5_1_24.png\" alt=\"Fig5_1_24\" width=\"438\" height=\"750\" \/><\/p>\n<p>In order to write numbers down, there were only three symbols needed in this system. A horizontal bar represented the quantity 5, a dot represented the quantity 1, and a special symbol (thought to be a shell) represented zero. The Mayan system may have been the first to make use of zero as a placeholder\/number. The first 20 numbers are shown in the table to the right.<a class=\"footnote\" title=\"http:\/\/www.vpds.wsu.edu\/fair_95\/gym\/UM001.html\" id=\"return-footnote-1962-13\" href=\"#footnote-1962-13\" aria-label=\"Footnote 13\"><sup class=\"footnote\">[13]<\/sup><\/a><\/p>\n<p>Unlike our system, where the ones place starts on the right and then moves to the left, the Mayan systems places the ones on the <strong>bottom<\/strong> of a vertical orientation and moves up as the place value increases.<\/p>\n<p>When numbers are written in vertical form, there should never be more than four dots in a single place. When writing Mayan numbers, every group of five dots becomes one bar. Also, there should never be more than three bars in a single place\u2026four bars would be converted to one dot in the next place up. It\u2019s the same as 10 getting converted to a 1 in the next place up when we carry during addition.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What is the value of this number, which is shown in vertical form?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-305\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155220\/Fig5_1_25.png\" alt=\"Fig5_1_25\" width=\"48\" height=\"83\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q170707\">Show Solution<\/span><\/p>\n<div id=\"q170707\" class=\"hidden-answer\" style=\"display: none\">\n<p>Starting from the bottom, we have the ones place. There are two bars and three dots in this place. Since each bar is worth 5, we have 13 ones when we count the three dots in the ones place. Looking to the place value above it (the twenties places), we see there are three dots so we have three twenties.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-306\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155220\/Fig5_1_26.png\" alt=\"Fig5_1_26\" width=\"206\" height=\"82\" \/><\/p>\n<p>Hence we can write this number in base-ten as:<\/p>\n<p>(3 \u00d7 20<sup>1<\/sup>) +\u00a0(13 \u00d7 20<sup>0<\/sup>) =\u00a0(3 \u00d7 20<sup>1<\/sup>) +\u00a0(13 \u00d7 1) = 60 + 13 = 73<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What is the value of the following Mayan number?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-307\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155221\/Fig5_1_27.png\" alt=\"Fig5_1_27\" width=\"57\" height=\"111\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q760307\">Show Solution<\/span><\/p>\n<div id=\"q760307\" class=\"hidden-answer\" style=\"display: none\">\n<p>This number has 11 in the ones place, zero in the 20s place, and 18 in the 20<sup>2\u00a0<\/sup>= 400s place. Hence, the value of this number in base-ten is:<\/p>\n<p>18 \u00d7 400 + 0 \u00d7 20 + 11 \u00d7 1 = 7211.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert the Mayan number below to base 10.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-308\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155221\/Fig5_1_28.png\" alt=\"Fig5_1_28\" width=\"60\" height=\"115\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q249582\">Show Solution<\/span><\/p>\n<div id=\"q249582\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>1562<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6420&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert the base 10 number 3575<sub>10<\/sub> to Mayan numerals.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q549968\">Show Solution<\/span><\/p>\n<div id=\"q549968\" class=\"hidden-answer\" style=\"display: none\">\n<div class=\"bcc-box bcc-success\">\n<p>This problem is done in two stages. First we need to convert to a base 20 number. We will do so using the method provided in the last section of the text. The second step is to convert that number to Mayan symbols.<\/p>\n<p>The highest power of 20 that will divide into 3575 is 20<sup>2<\/sup> = 400, so we start by dividing that and then proceed from there:<\/p>\n<p>3575 \u00f7 400 = 8.9375<br \/>\n0.9375 \u00d7 20 = 18.75<br \/>\n0.75 \u00d7 20 = 15.0<\/p>\n<p>This means that 3575<sub>10<\/sub> = 8,18,15<sub>20<\/sub><\/p>\n<p>The second step is to convert this to Mayan notation. This number indicates that we have 15 in the ones position. That\u2019s three bars at the bottom of the number. We also have 18 in the 20s place, so that\u2019s three bars and three dots in the second position. Finally, we have 8 in the 400s place, so that\u2019s one bar and three dots on the top. We get the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-309\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155221\/Fig5_1_29.png\" alt=\"Fig5_1_29\" width=\"66\" height=\"93\" \/><\/p>\n<\/div>\n<p>Note that in the previous example a new notation was used when we wrote 8,18,15<sub>20<\/sub>. The commas between the three numbers 8, 18, and 15 are now separating place values for us so that we can keep them separate from each other. This use of the comma is slightly different than how they\u2019re used in the decimal system. When we write a number in base 10, such as 7,567,323, the commas are used primarily as an aide to read the number easily but they do not separate single place values from each other. We will need this notation whenever the base we use is larger than 10.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Writing numbers with bases bigger than 10<\/h3>\n<p>When the base of a number is larger than 10, separate each \u201cdigit\u201d with a comma to make the separation of digits clear.<\/p>\n<p>For example, in base 20, to write the number corresponding to 17 \u00d7 20<sup>2<\/sup> + 6 \u00d7 20<sup>1<\/sup> + 13 \u00d7 20<sup>0<\/sup>, we\u2019d write 17,6,13<sub>20<\/sub>.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert the base 10 number 10553<sub>10<\/sub> to Mayan numerals.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933764\">Show Solution<\/span><\/p>\n<div id=\"q933764\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>[latex]10553_{10} = 1,6,7,13_{20}[\/latex]<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192231\/mayan-numeral.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2009\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192231\/mayan-numeral.png\" alt=\"\" width=\"36\" height=\"87\" \/><\/a>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6423&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>Convert the base 10 number 5617<sub>10<\/sub> to Mayan numerals.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q551605\">Show Solution<\/span><\/p>\n<div id=\"q551605\" class=\"hidden-answer\" style=\"display: none\">[latex]5617_{10} = 14,0,17_{20}[\/latex]. Note that there is a zero in the 20\u2019s place, so you\u2019ll need to use the appropriate zero symbol in between the ones and 400\u2019s\u00a0places.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192356\/mayan-2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192356\/mayan-2.png\" alt=\"\" width=\"36\" height=\"57\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In the following video we present more examples of how to write numbers using Mayan numerals as well as converting numerals written in Mayan for into base 10 form.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"The Mayan Number System: Writing Mayan Number in Base 10\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/gPUOrcilVS0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The next video shows more examples of converting base 10 numbers into Mayan numerals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"The Mayan Number System: Writing Base 10 Numbers as Mayan Number (base 20)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LrHNXoqQ_lI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Adding Mayan Numbers<\/h3>\n<p>When adding Mayan numbers together, we\u2019ll adopt a scheme that the Mayans probably did not use but which will make life a little easier for us.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add, in Mayan, the numbers 37 and 29:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-310\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155222\/Fig5_1_30.png\" alt=\"Fig5_1_30\" width=\"146\" height=\"83\" \/><a class=\"footnote\" title=\"http:\/\/forum.swarthmore.edu\/k12\/mayan.math\/mayan2.html\" id=\"return-footnote-1962-14\" href=\"#footnote-1962-14\" aria-label=\"Footnote 14\"><sup class=\"footnote\">[14]<\/sup><\/a><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q193127\">Show Solution<\/span><\/p>\n<div id=\"q193127\" class=\"hidden-answer\" style=\"display: none\">\n<p>First draw a box around each of the vertical places. This will help keep the place values from being mixed up.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-311\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155222\/Fig5_1_31.png\" alt=\"Fig5_1_31\" width=\"169\" height=\"154\" \/><\/p>\n<p>Next, put all of the symbols from both numbers into a single set of places (boxes), and to the right of this new number draw a set of empty boxes where you will place the final sum:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-312\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155223\/Fig5_1_32.png\" alt=\"Fig5_1_32\" width=\"167\" height=\"150\" \/><\/p>\n<p>You are now ready to start carrying. Begin with the place that has the lowest value, just as you do with Arabic numbers. Start at the bottom place, where each dot is worth 1. There are six dots, but a maximum of four are allowed in any one place; once you get to five dots, you must convert to a bar. Since five dots make one bar, we draw a bar through five of the dots, leaving us with one dot which is under the four-dot limit. Put this dot into the bottom place of the empty set of boxes you just drew:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-313\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155224\/Fig5_1_33.png\" alt=\"Fig5_1_33\" width=\"168\" height=\"150\" \/><\/p>\n<p>Now look at the bars in the bottom place. There are five, and the maximum number the place can hold is three. <em>Four bars are equal to one dot in the next highest place<\/em>.<\/p>\n<p>Whenever we have four bars in a single place we will automatically convert that to a <em>dot <\/em>in the next place up. We draw a circle around four of the bars and an arrow up to the dots&#8217; section of the higher place. At the end of that arrow, draw a new dot. That dot represents 20 just the same as the other dots in that place. Not counting the circled bars in the bottom place, there is one bar left. One bar is under the three-bar limit; put it under the dot in the set of empty places to the right.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-314\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155224\/Fig5_1_34.png\" alt=\"Fig5_1_34\" width=\"178\" height=\"151\" \/><\/p>\n<p>Now there are only three dots in the next highest place, so draw them in the corresponding empty box.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-315\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155227\/Fig5_1_35.png\" alt=\"Fig5_1_35\" width=\"166\" height=\"150\" \/><\/p>\n<p>We can see here that we have 3 twenties (60), and 6 ones, for a total of 66. We check and note that 37 + 29 = 66, so we have done this addition correctly. Is it easier to just do it in base-ten? Probably, but that\u2019s only because it\u2019s more familiar to you. Your task here is to try to learn a new base system and how addition can be done in slightly different ways than what you have seen in the past. Note, however, that the concept of carrying is still used, just as it is in our own addition algorithm.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Try adding 174 and 78 in Mayan by first converting to Mayan numbers and then working entirely within that system. Do not add in base-ten (decimal) until the very end when you <em>check<\/em> your work.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q374361\">Show Solution<\/span><\/p>\n<div id=\"q374361\" class=\"hidden-answer\" style=\"display: none\">A sample solution is shown.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192716\/mayan-3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2011\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/21192716\/mayan-3-300x98.png\" alt=\"\" width=\"300\" height=\"98\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last video we show more examples of adding Mayan numerals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"The Mayan Number System: Addition of Mayan Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/NpH5oMCrubM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nIn this module, we have briefly sketched the development of numbers and our counting system, with the emphasis on the \u201cbrief\u201d part. There are numerous sources of information and research that fill many volumes of books on this topic. Unfortunately, we cannot begin to come close to covering all of the information that is out there.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/28182439\/204px-Numeral_Systems_of_the_World.svg_.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2292 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/03\/28182439\/204px-Numeral_Systems_of_the_World.svg_.png\" alt=\"\" width=\"302\" height=\"215\" \/><\/a><\/p>\n<p>We have only scratched the surface of the wealth of research and information that exists on the development of numbers and counting throughout human history. What is important to note is that the system that we use every day is a product of thousands of years of progress and development. It represents contributions by many civilizations and cultures. It does not come down to us from the sky, a gift from the gods. It is not the creation of a textbook publisher. It is indeed as human as we are, as is the rest of mathematics. Behind every symbol, formula and rule there is a human face to be found, or at least sought.<\/p>\n<p>Furthermore, we hope that you now have a basic appreciation for just how interesting and diverse number systems can get. Also, we\u2019re pretty sure that you have also begun to recognize that we take our own number system for granted so much that when we try to adapt to other systems or bases, we find ourselves truly having to concentrate and think about what is going on.<\/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-1962\">\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>Introduction: Positional Systems and Bases. <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, 8681. <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><li> Convert Numbers in Base Ten to Different Bases: Calculator Method. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/YNPTYelCeIs\">https:\/\/youtu.be\/YNPTYelCeIs<\/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 6420, 6423. <strong>Authored by<\/strong>: Morales, Lawrence. <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><li>The Mayan Number System: Writing Mayan Number in Base 10. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/gPUOrcilVS0\">https:\/\/youtu.be\/gPUOrcilVS0<\/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>The Mayan Number System: Writing Mayan Number in Base 10. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LrHNXoqQ_lI\">https:\/\/youtu.be\/LrHNXoqQ_lI<\/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>The Mayan Number System: Addition of Mayan Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/NpH5oMCrubM\">https:\/\/youtu.be\/NpH5oMCrubM<\/a>. <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>\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-1962-1\">Ibid, page 230 <a href=\"#return-footnote-1962-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-1962-2\">Ibid, page 231. <a href=\"#return-footnote-1962-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-1962-3\">Ibid, page 232. <a href=\"#return-footnote-1962-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-1962-4\">Ibid, page 232. <a href=\"#return-footnote-1962-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-1962-5\">McLeish, p. 18 <a href=\"#return-footnote-1962-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><li id=\"footnote-1962-6\"><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-1962-6\" class=\"return-footnote\" aria-label=\"Return to footnote 6\">&crarr;<\/a><\/li><li id=\"footnote-1962-7\">Ibid, page 232. <a href=\"#return-footnote-1962-7\" class=\"return-footnote\" aria-label=\"Return to footnote 7\">&crarr;<\/a><\/li><li id=\"footnote-1962-8\">http:\/\/www.gorp.com\/gorp\/location\/latamer\/map_maya.htm <a href=\"#return-footnote-1962-8\" class=\"return-footnote\" aria-label=\"Return to footnote 8\">&crarr;<\/a><\/li><li id=\"footnote-1962-9\">Bidwell, James; Mayan Arithmetic in <em>Mathematics Teacher<\/em>, Issue 74 (Nov., 1967), p. 762\u201368. <a href=\"#return-footnote-1962-9\" class=\"return-footnote\" aria-label=\"Return to footnote 9\">&crarr;<\/a><\/li><li id=\"footnote-1962-10\"><a href=\"http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html<\/a> <a href=\"#return-footnote-1962-10\" class=\"return-footnote\" aria-label=\"Return to footnote 10\">&crarr;<\/a><\/li><li id=\"footnote-1962-11\"><a href=\"http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html<\/a> <a href=\"#return-footnote-1962-11\" class=\"return-footnote\" aria-label=\"Return to footnote 11\">&crarr;<\/a><\/li><li id=\"footnote-1962-12\">Bidwell <a href=\"#return-footnote-1962-12\" class=\"return-footnote\" aria-label=\"Return to footnote 12\">&crarr;<\/a><\/li><li id=\"footnote-1962-13\"><a href=\"http:\/\/www.vpds.wsu.edu\/fair_95\/gym\/UM001.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.vpds.wsu.edu\/fair_95\/gym\/UM001.html<\/a> <a href=\"#return-footnote-1962-13\" class=\"return-footnote\" aria-label=\"Return to footnote 13\">&crarr;<\/a><\/li><li id=\"footnote-1962-14\"><a href=\"http:\/\/forum.swarthmore.edu\/k12\/mayan.math\/mayan2.html\">http:\/\/forum.swarthmore.edu\/k12\/mayan.math\/mayan2.html<\/a> <a href=\"#return-footnote-1962-14\" class=\"return-footnote\" aria-label=\"Return to footnote 14\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Introduction: Positional Systems and Bases\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Lippman, 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