{"id":4613,"date":"2020-04-21T00:19:08","date_gmt":"2020-04-21T00:19:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/the-mayan-numeral-system\/"},"modified":"2020-04-21T00:19:08","modified_gmt":"2020-04-21T00:19:08","slug":"the-mayan-numeral-system","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/the-mayan-numeral-system\/","title":{"raw":"The Mayan Numeral System","rendered":"The Mayan Numeral System"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Become familiar with the history of positional number systems<\/li>\n \t<li>Identify bases that have been used in number systems historically<\/li>\n<\/ul>\n<\/div>\n<h3>Background<\/h3>\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.\n\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]&nbsp;The 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.\n\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]&nbsp;like 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]&nbsp;Since 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.\n<div class=\"textbox examples\">\n<h3>Powers of numbers<\/h3>\nRecall that powers of numbers indicate how many times to multiply the base. In the Mayan base 20 number system, the smallest place value is [latex]20^{0} = 1[\/latex]. The next is [latex]20^{1}=20[\/latex]. The third is [latex]20^{2}=400[\/latex], and so on. See the table below for the base-ten values and Mayan names for each of the base 20 number positions.\n\n<\/div>\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>\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]\n\n<img class=\"aligncenter wp-image-304 size-full\" 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=\"A chart showing the Mayan numeral system. 0 is represented by an oval with two lines in it. 1 is represented by a dot. 2 is two dots. 3 is three dots. 4 is four dots. 5 is a line. 6 is a dot above a line. 7 is two dots above a line. 8 is three dots above a line. 9 is four dots above a line. 10 is two lines. 11 is a dot above two lines. 12 is two dots above two lines. 13 is three dots above two lines. 14 is four dots above two lines. 15 is three lines. 16 is a dot above three lines. 17 is two dots above three lines. 18 is three dots above three lines. 19 is four dots above three lines.\" width=\"438\" height=\"750\">\n\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]\n\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.\n\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.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nWhat is the value of this number, which is shown in vertical form?\n\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\">\n[reveal-answer q=\"170707\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"170707\"]\n\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.\n\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\">\n\nHence we can write this number in base-ten as:\n\n(3 \u00d7 20<sup>1<\/sup>) +&nbsp;(13 \u00d7 20<sup>0<\/sup>) =&nbsp;(3 \u00d7 20<sup>1<\/sup>) +&nbsp;(13 \u00d7 1) = 60 + 13 = 73\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nWhat is the value of the following Mayan number?\n\n<img class=\"alignnone wp-image-307 size-full\" 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=\"A depiction of a Mayan number. The bottom has two lines and a dot above them, the middle has a circle with two diagonal lines, and the top has three lines and three dots above them.\" width=\"57\" height=\"111\">\n[reveal-answer q=\"760307\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"760307\"]\n\nThis number has 11 in the ones place, zero in the 20s place, and 18 in the 20<sup>2&nbsp;<\/sup>= 400s place. Hence, the value of this number in base-ten is:\n\n18 \u00d7 400 + 0 \u00d7 20 + 11 \u00d7 1 = 7211.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nConvert the Mayan number below to base 10.\n\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\">\n[reveal-answer q=\"249582\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"249582\"]\n<div>\n\n1562\n\n<\/div>\n[\/hidden-answer]\n\n[ohm_question]6420[\/ohm_question]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nConvert the base 10 number 3575<sub>10<\/sub> to Mayan numerals.\n[reveal-answer q=\"549968\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"549968\"]\n\nThis problem is done in two stages:\n<p style=\"padding-left: 30px;\">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 style=\"padding-left: 90px;\">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 style=\"padding-left: 90px;\">3575 \u00f7 400 = 8.9375\n0.9375 \u00d7 20 = 18.75\n0.75 \u00d7 20 = 15.0<\/p>\n<p style=\"padding-left: 90px;\">This means that 3575<sub>10<\/sub> = 8,18,15<sub>20<\/sub><\/p>\n<p style=\"padding-left: 30px;\">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.<\/p>\n<p style=\"padding-left: 30px;\">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 style=\"padding-left: 90px;\"><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\"><\/p>\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.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Writing numbers with bases bigger than 10<\/h3>\nWhen the base of a number is larger than 10, separate each \u201cdigit\u201d with a comma to make the separation of digits clear.\n\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>.\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nConvert the base 10 number 10553<sub>10<\/sub> to Mayan numerals.\n[reveal-answer q=\"933764\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"933764\"]\n<div>\n\n[latex]10553_{10} = 1,6,7,13_{20}[\/latex]\n\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>\n[\/hidden-answer]\n\n[ohm_question]6423[\/ohm_question]\n\n<\/div>\n&nbsp;\n\nConvert the base 10 number 5617<sub>10<\/sub> to Mayan numerals.\n[reveal-answer q=\"551605\"]Show Solution[\/reveal-answer]\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&nbsp;places.\n\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>\n\n[\/hidden-answer]\n\n<\/div>\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.\n\nhttps:\/\/youtu.be\/gPUOrcilVS0\n\nThe next video shows more examples of converting base 10 numbers into Mayan numerals.\n\nhttps:\/\/youtu.be\/LrHNXoqQ_lI\n<h3>Adding Mayan Numbers<\/h3>\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.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nAdd, in Mayan, the numbers 37 and 29:\n\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]\n[reveal-answer q=\"193127\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"193127\"]\n\nFirst draw a box around each of the vertical places. This will help keep the place values from being mixed up.\n\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\">\n\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:\n\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\">\n\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:\n\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\">\n\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>.\n\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.\n\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\">\n\nNow there are only three dots in the next highest place, so draw them in the corresponding empty box.\n\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\">\n\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.\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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.\n[reveal-answer q=\"374361\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"374361\"]A sample solution is shown.\n\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>\n\n[\/hidden-answer]\n\n<\/div>\nIn the last video we show more examples of adding Mayan numerals.\n\nhttps:\/\/youtu.be\/NpH5oMCrubM\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.\n\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>\n\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.\n\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.\n","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<\/ul>\n<\/div>\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-4613-1\" href=\"#footnote-4613-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/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-4613-2\" href=\"#footnote-4613-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>&nbsp;The 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-4613-3\" href=\"#footnote-4613-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>&nbsp;like the sample to the left<a class=\"footnote\" title=\"http:\/\/www.ukans.edu\/~lctls\/Mayan\/numbers.html\" id=\"return-footnote-4613-4\" href=\"#footnote-4613-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a>&nbsp;Since 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 class=\"textbox examples\">\n<h3>Powers of numbers<\/h3>\n<p>Recall that powers of numbers indicate how many times to multiply the base. In the Mayan base 20 number system, the smallest place value is [latex]20^{0} = 1[\/latex]. The next is [latex]20^{1}=20[\/latex]. The third is [latex]20^{2}=400[\/latex], and so on. See the table below for the base-ten values and Mayan names for each of the base 20 number positions.<\/p>\n<\/div>\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-4613-5\" href=\"#footnote-4613-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-304 size-full\" 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=\"A chart showing the Mayan numeral system. 0 is represented by an oval with two lines in it. 1 is represented by a dot. 2 is two dots. 3 is three dots. 4 is four dots. 5 is a line. 6 is a dot above a line. 7 is two dots above a line. 8 is three dots above a line. 9 is four dots above a line. 10 is two lines. 11 is a dot above two lines. 12 is two dots above two lines. 13 is three dots above two lines. 14 is four dots above two lines. 15 is three lines. 16 is a dot above three lines. 17 is two dots above three lines. 18 is three dots above three lines. 19 is four dots above three lines.\" 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-4613-6\" href=\"#footnote-4613-6\" aria-label=\"Footnote 6\"><sup class=\"footnote\">[6]<\/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>) +&nbsp;(13 \u00d7 20<sup>0<\/sup>) =&nbsp;(3 \u00d7 20<sup>1<\/sup>) +&nbsp;(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 wp-image-307 size-full\" 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=\"A depiction of a Mayan number. The bottom has two lines and a dot above them, the middle has a circle with two diagonal lines, and the top has three lines and three dots above them.\" 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&nbsp;<\/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=\"ohm6420\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6420&theme=oea&iframe_resize_id=ohm6420&show_question_numbers\" width=\"100%\" height=\"150\"><\/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<p>This problem is done in two stages:<\/p>\n<p style=\"padding-left: 30px;\">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 style=\"padding-left: 90px;\">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 style=\"padding-left: 90px;\">3575 \u00f7 400 = 8.9375<br \/>\n0.9375 \u00d7 20 = 18.75<br \/>\n0.75 \u00d7 20 = 15.0<\/p>\n<p style=\"padding-left: 90px;\">This means that 3575<sub>10<\/sub> = 8,18,15<sub>20<\/sub><\/p>\n<p style=\"padding-left: 30px;\">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.<\/p>\n<p style=\"padding-left: 30px;\">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 style=\"padding-left: 90px;\"><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<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=\"ohm6423\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6423&theme=oea&iframe_resize_id=ohm6423&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\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&nbsp;places.<\/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<\/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-1\" 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-2\" 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-4613-7\" href=\"#footnote-4613-7\" aria-label=\"Footnote 7\"><sup class=\"footnote\">[7]<\/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-3\" 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-4613\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided 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>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>Math in Society. <strong>Authored by<\/strong>: Lippman, David. <strong>Provided by<\/strong>: http:\/\/www.opentextbookstore.com\/mathinsociety\/. <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>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>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-4613-1\">http:\/\/www.gorp.com\/gorp\/location\/latamer\/map_maya.htm <a href=\"#return-footnote-4613-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-4613-2\">Bidwell, James; Mayan Arithmetic in <em>Mathematics Teacher<\/em>, Issue 74 (Nov., 1967), p. 762\u201368. <a href=\"#return-footnote-4613-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-4613-3\"><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-4613-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-4613-4\"><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-4613-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-4613-5\">Bidwell <a href=\"#return-footnote-4613-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><li id=\"footnote-4613-6\"><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-4613-6\" class=\"return-footnote\" aria-label=\"Return to footnote 6\">&crarr;<\/a><\/li><li id=\"footnote-4613-7\"><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-4613-7\" class=\"return-footnote\" aria-label=\"Return to footnote 7\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":17533,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 6420, 6423\",\"author\":\"Morales,Lawrence\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Lippman, David\",\"organization\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"The Mayan Number System: Writing Mayan Number in Base 10\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/gPUOrcilVS0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"The Mayan Number System: Writing Mayan Number in Base 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