{"id":193,"date":"2016-01-25T22:05:38","date_gmt":"2016-01-25T22:05:38","guid":{"rendered":"https:\/\/courses.candelalearning.com\/math4libarts\/?post_type=chapter&#038;p=193"},"modified":"2020-09-14T18:46:21","modified_gmt":"2020-09-14T18:46:21","slug":"early-counting-systems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/chapter\/early-counting-systems\/","title":{"raw":"Early Counting Systems","rendered":"Early Counting Systems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine the number of objects being represented by pebbles placed on an Inca counting board.<\/li>\r\n \t<li>Determine the number represented by a quipu cord<\/li>\r\n \t<li>Identify uses other than counting for a quipu cord<\/li>\r\n \t<li>Become familiar with the evolution of the counting system we use every day<\/li>\r\n \t<li>Write numbers using Roman Numerals<\/li>\r\n \t<li>Convert between Hindu-Arabic and Roman Numerals<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs we begin our journey through the history of mathematics, one question to be asked is \u201cWhere do we start?\u201d Depending on how you view mathematics or numbers, you could choose any of a number of launching points from which to begin. Howard Eves suggests the following list of possibilities.[footnote]Eves, Howard; An Introduction to the History of Mathematics, p. 9.[\/footnote]\r\n\r\nWhere to start the study of the history of mathematics\u2026\r\n<ul>\r\n \t<li>At the first logical geometric \u201cproofs\u201d traditionally credited to Thales of Miletus (600 BCE).<\/li>\r\n \t<li>With the formulation of methods of measurement made by the Egyptians and Mesopotamians\/Babylonians.<\/li>\r\n \t<li>Where prehistoric peoples made efforts to organize the concepts of size, shape, and number.<\/li>\r\n \t<li>In pre-human times in the very simple number sense and pattern recognition that can be displayed by certain animals, birds, etc.<\/li>\r\n \t<li>Even before that in the amazing relationships of numbers and shapes found in plants.<\/li>\r\n \t<li>With the spiral nebulae, the natural course of planets, and other universe phenomena.<\/li>\r\n<\/ul>\r\nWe can choose no starting point at all and instead agree that mathematics has <em>always<\/em> existed and has simply been waiting in the wings for humans to discover. Each of these positions can be defended to some degree and which one you adopt (if any) largely depends on your philosophical ideas about mathematics and numbers.\r\n\r\nNevertheless, we need a starting point. Without passing judgment on the validity of any of these particular possibilities, we will choose as our starting point the emergence of the idea of number and the process of counting as our launching pad. This is done primarily as a practical matter given the nature of this course. In the following chapter, we will try to focus on two main ideas. The first will be an examination of basic number and counting systems and the symbols that we use for numbers. We will look at our own modern (Western) number system as well those of a couple of selected civilizations to see the differences and diversity that is possible when humans start counting. The second idea we will look at will be base systems. By comparing our own base-ten (decimal) system with other bases, we will quickly become aware that the system that we are so used to, when slightly changed, will challenge our notions about numbers and what symbols for those numbers actually mean.\r\n<h3>Recognition of More vs. Less<\/h3>\r\nThe idea of number and the process of counting goes back far beyond history began to be recorded. There is some archeological evidence that suggests that humans were counting as far back as 50,000 years ago.[footnote]Eves, p. 9.[\/footnote]\u00a0However, we do not really know how this process started or developed over time. The best we can do is to make a good guess as to how things progressed. It is probably not hard to believe that even the earliest humans had some sense of <em>more<\/em> and <em>less<\/em>. Even some small animals have been shown to have such a sense. For example, one naturalist tells of how he would secretly remove one egg each day from a plover\u2019s nest. The mother was diligent in laying an extra egg every day to make up for the missing egg. Some research has shown that hens can be trained to distinguish between even and odd numbers of pieces of food.[footnote]McLeish, John; The Story of Numbers\u2014How Mathematics Has Shaped Civilization, p. 7.[\/footnote]\u00a0With these sorts of findings in mind, it is not hard to conceive that early humans had (at least) a similar sense of more and less. However, our conjectures about how and when these ideas emerged among humans are simply that; educated guesses based on our own assumptions of what might or could have been.\r\n\r\n&nbsp;\r\n<h2>The Evolution of Counting and The Inca Counting System<\/h2>\r\n<h3>The Need for Simple Counting<\/h3>\r\nAs societies and humankind evolved, simply having a sense of more or less, even or odd, etc., would prove to be insufficient to meet the needs of everyday living. As tribes and groups formed, it became important to be able to know how many members were in the group, and perhaps how many were in the enemy\u2019s camp. Certainly it was important for them to know if the flock of sheep or other possessed animals were increasing or decreasing in size. \u201cJust how many of them do we have, anyway?\u201d is a question that we do not have a hard time imagining them asking themselves (or each other).\r\n\r\nIn order to count items such as animals, it is often conjectured that one of the earliest methods of doing so would be with \u201ctally sticks.\u201d These are objects used to track the numbers of items to be counted. With this method, each \u201cstick\u201d (or pebble, or whatever counting device being used) represents one animal or object. This method uses the idea of <strong>one to one correspondence<\/strong>. In a one to one correspondence, items that are being counted are uniquely linked with some counting tool.\r\n\r\n[caption id=\"attachment_280\" align=\"alignright\" width=\"200\"]<img class=\"wp-image-280\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155148\/Fig5_1_1.png\" alt=\"Fig5_1_1\" width=\"200\" height=\"239\" \/> Figure 1.[\/caption]\r\n\r\nIn the picture to the right, you see each stick corresponding to one horse. By examining the collection of sticks in hand one knows how many animals should be present. You can imagine the usefulness of such a system, at least for smaller numbers of items to keep track of. If a herder wanted to \u201ccount off\u201d his animals to make sure they were all present, he could mentally (or methodically) assign each stick to one animal and continue to do so until he was satisfied that all were accounted for.\r\n\r\nOf course, in our modern system, we have replaced the sticks with more abstract objects. In particular, the top stick is replaced with our symbol \u201c1,\u201d the second stick gets replaced by a \u201c2\u201d and the third stick is represented by the symbol \u201c3,\u201d but we are getting ahead of ourselves here. These modern symbols took many centuries to emerge.\r\n\r\nAnother possible way of employing the \u201ctally stick\u201d counting method is by making marks or cutting notches into pieces of wood, or even tying knots in string (as we shall see later). In 1937, Karl Absolom discovered a wolf bone that goes back possibly 30,000 years. It is believed to be a counting device.[footnote]Bunt, Lucas; Jones, Phillip; Bedient, Jack; The Historical Roots of Elementary Mathematics, p. 2.[\/footnote]\u00a0Another example of this kind of tool is the Ishango Bone, discovered in 1960 at Ishango, and shown below.[footnote]<a href=\"http:\/\/www.math.buffalo.edu\/mad\/Ancient-Africa\/mad_zaire-uganda.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.math.buffalo.edu\/mad\/Ancient-Africa\/mad_zaire-uganda.html<\/a>[\/footnote]\u00a0It is reported to be between six and nine thousand years old and shows what appear to be markings used to do counting of some sort.\r\n\r\nThe markings on rows (a) and (b) each add up to 60. Row (b) contains the prime numbers between 10 and 20. Row (c) seems to illustrate for the method of doubling and multiplication used by the Egyptians. It is believed that this may also represent a lunar phase counter.\r\n\r\n[caption id=\"attachment_281\" align=\"aligncenter\" width=\"517\"]<img class=\"wp-image-281 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155149\/Fig5_1_2.png\" alt=\"Fig5_1_2\" width=\"517\" height=\"293\" \/> Figure 2.[\/caption]\r\n<h3>Spoken Words<\/h3>\r\nAs methods for counting developed, and as language progressed as well, it is natural to expect that spoken words for numbers would appear. Unfortunately, the developments of these words, especially those corresponding to the numbers from one through ten, are not easy to trace. Past ten, however, we do see some patterns:\r\n<ul>\r\n \t<li>Eleven comes from \u201cein lifon,\u201d meaning \u201cone left over.\u201d<\/li>\r\n \t<li>Twelve comes from \u201ctwe lif,\u201d meaning \u201ctwo left over.\u201d<\/li>\r\n \t<li>Thirteen comes from \u201cThree and ten\u201d as do fourteen through nineteen.<\/li>\r\n \t<li>Twenty appears to come from \u201ctwe-tig\u201d which means \u201ctwo tens.\u201d<\/li>\r\n \t<li>Hundred probably comes from a term meaning \u201cten times.\u201d<\/li>\r\n<\/ul>\r\n<h3>Written Numbers<\/h3>\r\nWhen we speak of \u201cwritten\u201d numbers, we have to be careful because this could mean a variety of things. It is important to keep in mind that modern paper is only a little more than 100 years old, so \u201cwriting\u201d in times past often took on forms that might look quite unfamiliar to us today.\r\n\r\nAs we saw earlier, some might consider wooden sticks with notches carved in them as writing as these are means of recording information on a medium that can be \u201cread\u201d by others. Of course, the symbols used (simple notches) certainly did not leave a lot of flexibility for communicating a wide variety of ideas or information.\r\n\r\nOther mediums on which \u201cwriting\u201d may have taken place include carvings in stone or clay tablets, rag paper made by hand (twelfth century in Europe, but earlier in China), papyrus (invented by the Egyptians and used up until the Greeks), and parchments from animal skins. And these are just a few of the many possibilities.\r\n\r\nThese are just a few examples of early methods of counting and simple symbols for representing numbers. Extensive books, articles and research have been done on this topic and could provide enough information to fill this entire course if we allowed it to. The range and diversity of creative thought that has been used in the past to describe numbers and to count objects and people is staggering. Unfortunately, we don\u2019t have time to examine them all, but it is fun and interesting to look at one system in more detail to see just how ingenious people have been.\r\n<h2>The Number and Counting System of the Inca Civilization<\/h2>\r\n<h3>Background<\/h3>\r\nThere is generally a lack of books and research material concerning the historical foundations of the Americas. Most of the \u201cimportant\u201d information available concentrates on the eastern hemisphere, with Europe as the central focus. The reasons for this may be twofold: first, it is thought that there was a lack of specialized mathematics in the American regions; second, many of the secrets of ancient mathematics in the Americas have been closely guarded.[footnote]Diana, Lind Mae; The Peruvian Quipu in <em>Mathematics Teacher, <\/em>Issue 60 (Oct., 1967), p. 623\u201328.[\/footnote]\u00a0The Peruvian system does not seem to be an exception here. Two researchers, Leland Locke and Erland Nordenskiold, have carried out research that has attempted to discover what mathematical knowledge was known by the Incas and how they used the Peruvian quipu, a counting system using cords and knots, in their mathematics. These researchers have come to certain beliefs about the quipu that we will summarize here.\r\n<h3>Counting Boards<\/h3>\r\nIt should be noted that the Incas did not have a complicated system of computation. Where other peoples in the regions, such as the Mayans, were doing computations related to their rituals and calendars, the Incas seem to have been more concerned with the simpler task of record-keeping. To do this, they used what are called the \u201cquipu\u201d to record quantities of items. (We will describe them in more detail in a moment.) However, they first often needed to do computations whose results would be recorded on quipu. To do these computations, they would sometimes use a counting board constructed with a slab of stone. In the slab were cut rectangular and square compartments so that an octagonal (eight-sided) region was left in the middle. Two opposite corner rectangles were raised. Another two sections were mounted on the original surface of the slab so that there were actually three levels available. In the figure shown, the darkest shaded corner regions represent the highest, third level. The lighter shaded regions surrounding the corners are the second highest levels, while the clear white rectangles are the compartments cut into the stone slab.\r\n\r\n[caption id=\"attachment_282\" align=\"aligncenter\" width=\"476\"]<img class=\"wp-image-282 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155150\/Fig5_1_3.png\" alt=\"Fig5_1_3\" width=\"476\" height=\"311\" \/> Figure 3.[\/caption]\r\n\r\nPebbles were used to keep accounts and their positions within the various levels and compartments gave totals. For example, a pebble in a smaller (white) compartment represented one unit. Note that there are 12 such squares around the outer edge of the figure. If a pebble was put into one of the two (white) larger, rectangular compartments, its value was doubled. When a pebble was put in the octagonal region in the middle of the slab, its value was tripled. If a pebble was placed on the second (shaded) level, its value was multiplied by six. And finally, if a pebble was found on one of the two highest corner levels, its value was multiplied by twelve. Different objects could be counted at the same time by representing different objects by different colored pebbles.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose you have the following counting board with two different kind of pebbles places as illustrated. Let the solid black pebble represent a dog and the striped pebble represent a cat. How many dogs are being represented?\r\n\r\n<img class=\"wp-image-283 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155151\/Fig5_1_4.png\" alt=\"Fig5_1_4\" width=\"474\" height=\"311\" \/>\r\n[reveal-answer q=\"129586\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"129586\"]There are two black pebbles in the outer square regions\u2026these represent 2 dogs.\u00a0There are three black pebbles in the larger (white) rectangular compartments. These represent 6 dogs.\u00a0There is one black pebble in the middle region\u2026this represents 3 dogs.\u00a0There are three black pebbles on the second level\u2026these represent 18 dogs.\u00a0Finally, there is one black pebble on the highest corner level\u2026this represents 12 dogs. We then have a total of 2+6+3+18+12 = 41 dogs.[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nHow many cats are represented on this board?\r\n\r\n<img class=\"wp-image-283 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155151\/Fig5_1_4.png\" alt=\"Fig5_1_4\" width=\"474\" height=\"311\" \/>\r\n[reveal-answer q=\"881476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"881476\"]1+6\u00b43+3\u00b46+2\u00b412 = 61 cats[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch this short video lesson about Inca counting boards. You will find that this is a review of concepts presented here about counting boards.\r\n\r\nhttps:\/\/youtu.be\/fL1N_V89g78\r\n<h3>The Quipu<\/h3>\r\n[caption id=\"attachment_318\" align=\"alignright\" width=\"244\"]<img class=\"wp-image-318 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155153\/Fig5_1_5.jpg\" alt=\"Fig5_1_5\" width=\"244\" height=\"395\" \/> Figure 5.[\/caption]\r\n\r\nThis kind of board was good for doing quick computations, but it did not provide a good way to keep a permanent recording of quantities or computations. For this purpose, they used the quipu. The quipu is a collection of cords with knots in them. These cords and knots are carefully arranged so that the position and type of cord or knot gives specific information on how to decipher the cord.\r\n\r\nA quipu is made up of a main cord which has other cords (branches) tied to it. See pictures to the right.[footnote]Diana, Lind Mae; The Peruvian Quipu in <em>Mathematics Teacher, <\/em>Issue 60 (Oct., 1967), p. 623\u201328.[\/footnote]\r\n\r\nLocke called the branches H cords. They are attached to the main cord. B cords, in turn, were attached to the H cords. Most of these cords would have knots on them. Rarely are knots found on the main cord, however, and tend to be mainly on the H and B cords. A quipu might also have a \u201ctotalizer\u201d cord that summarizes all of the information on the cord group in one place.\r\n\r\nLocke points out that there are three types of knots, each representing a different value, depending on the kind of knot used and its position on the cord. The Incas, like us, had a decimal (base-ten) system, so each kind of knot had a specific decimal value. The Single knot, pictured in the middle of figure 6[footnote]<a href=\"http:\/\/wiscinfo.doit.wisc.edu\/chaysimire\/titulo2\/khipus\/what.htm\" target=\"_blank\" rel=\"noopener\">http:\/\/wiscinfo.doit.wisc.edu\/chaysimire\/titulo2\/khipus\/what.htm<\/a>[\/footnote]\u00a0was used to denote tens, hundreds, thousands, and ten thousands. They would be on the upper levels of the H cords. The figure-eight knot on the end was used to denote the integer \u201cone.\u201d Every other integer from 2 to 9 was represented with a long knot, shown on the left of the figure. (Sometimes long knots were used to represents tens and hundreds.) Note that the long knot has several turns in it\u2026the number of turns indicates which integer is being represented. The units (ones) were placed closest to the bottom of the cord, then tens right above them, then the hundreds, and so on.\r\n\r\n[caption id=\"attachment_286\" align=\"aligncenter\" width=\"506\"]<img class=\"size-full wp-image-286\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155154\/Fig5_1_7.png\" alt=\"Figure 6\" width=\"506\" height=\"209\" \/> Figure 6[\/caption]\r\n\r\nIn order to make reading these pictures easier, we will adopt a convention that is consistent. For the long knot with turns in it (representing the numbers 2 through 9), we will use the following notation:\r\n\r\nThe four horizontal bars represent four turns and the curved arc on the right links the four turns together. This would represent the number 4.\r\n\r\nWe will represent the single knot with a large dot ( \u00b7 ) and we will represent the figure eight knot with a sideways eight ( \u221e\u00a0).\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat number is represented on the cord shown in figure 7?\r\n\r\n<img class=\"wp-image-287\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155155\/Fig5_1_8.png\" alt=\"Figure 7.\" width=\"250\" height=\"200\" \/>\r\n[reveal-answer q=\"265241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265241\"]On the cord, we see a long knot with four turns in it\u2026this represents four in the ones place. Then 5 single knots appear in the tens position immediately above that, which represents 5 tens, or 50. Finally, 4 single knots are tied in the hundreds, representing four 4 hundreds, or 400. Thus, the total shown on this cord is 454.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWhat numbers are represented on each of the four cords hanging from the main cord?<strong>\u00a0<\/strong>\r\n\r\n<img class=\"wp-image-288 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155157\/Fig5_1_9.png\" alt=\"\" width=\"347\" height=\"281\" \/>\r\n[reveal-answer q=\"600164\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"600164\"]\r\n<div>\r\n\r\nFrom left to right:\r\n\r\nCord 1 = 2,162\r\n\r\nCord 2 = 301\r\n\r\nCord 3 = 0\r\n\r\nCord 4 = 2,070\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe colors of the cords had meaning and could distinguish one object from another. One color could represent llamas, while a different color might represent sheep, for example. When all the colors available were exhausted, they would have to be re-used. Because of this, the ability to read the quipu became a complicated task and specially trained individuals did this job. They were called Quipucamayoc, which means keeper of the quipus. They would build, guard, and decipher quipus.\r\n\r\n[caption id=\"attachment_289\" align=\"alignright\" width=\"350\"]<img class=\"wp-image-289\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155158\/Fig5_1_10.png\" alt=\"Fig5_1_10\" width=\"350\" height=\"283\" \/> Figure 9.[\/caption]\r\n\r\nAs you can see from this photograph of an actual quipu (figure 9), they could get quite complex.\r\n\r\nThere were various purposes for the quipu. Some believe that they were used to keep an account of their traditions and history, using knots to record history rather than some other formal system of writing. One writer has even suggested that the quipu replaced writing as it formed a role in the Incan postal system.[footnote]Diana, Lind Mae; The Peruvian Quipu in <em>Mathematics Teacher, <\/em>Issue 60 (Oct., 1967), p. 623\u201328.[\/footnote]\u00a0Another proposed use of the quipu is as a translation tool. After the conquest of the Incas by the Spaniards and subsequent \u201cconversion\u201d to Catholicism, an Inca supposedly could use the quipu to confess their sins to a priest. Yet another proposed use of the quipu was to record numbers related to magic and astronomy, although this is not a widely accepted interpretation.\r\n\r\nThe following video presents another introduction to the Inca's use of a quipu for record keeping.\r\n\r\nhttps:\/\/youtu.be\/EYq-VtyAd2s\r\n\r\nThe mysteries of the quipu have not been fully explored yet. Recently, Ascher and Ascher have published a book, <em>The Code of the Quipu: A Study in Media, Mathematics, and Culture<\/em><em>, which is \u201c<\/em>an extensive elaboration of the logical-numerical system of the quipu.\u201d[footnote]<a href=\"http:\/\/www.cs.uidaho.edu\/~casey931\/seminar\/quipu.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.cs.uidaho.edu\/~casey931\/seminar\/quipu.html<\/a>[\/footnote]\u00a0For more information on the quipu, you may want to check out <em><a href=\"https:\/\/www.maa.org\/press\/periodicals\/convergence\/mathematical-treasure-the-quipu\">Mathematical Treasure: The Quipu<\/a><\/em>.\r\n\r\nWe are so used to seeing the symbols 1, 2, 3, 4, etc. that it may be somewhat surprising to see such a creative and innovative way to compute and record numbers. Unfortunately, as we proceed through our mathematical education in grade and high school, we receive very little information about the wide range of number systems that have existed and which still exist all over the world. That\u2019s not to say our own system is not important or efficient. The fact that it has survived for hundreds of years and shows no sign of going away any time soon suggests that we may have finally found a system that works well and may not need further improvement, but only time will tell that whether or not that conjecture is valid or not. We now turn to a brief historical look at how our current system developed over history.\r\n<h2>The Hindu\u2014Arabic Number System and Roman Numerals<\/h2>\r\n<h3>The Evolution of a System<\/h3>\r\nOur own number system, composed of the ten symbols {0,1,2,3,4,5,6,7,8,9} is called the <em>Hindu<\/em><em>-Arabic system<\/em>. This is a base-ten (decimal) system since place values increase by powers of ten. Furthermore, this system is positional, which means that the position of a symbol has bearing on the value of that symbol within the number. For example, the position of the symbol 3 in the number 435,681 gives it a value much greater than the value of the symbol 8 in that same number. We\u2019ll explore base systems more thoroughly later. The development of these ten symbols and their use in a positional system comes to us primarily from India.[footnote]<a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html<\/a>[\/footnote]\r\n\r\n[caption id=\"attachment_278\" align=\"alignright\" width=\"200\"]<img class=\"wp-image-278\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155159\/Al_biruni_28-02-2010.jpg\" alt=\"Al-Biruni\" width=\"200\" height=\"263\" \/> Figure 10. Al-Biruni[\/caption]\r\n\r\nIt was not until the fifteenth\u00a0century that the symbols that we are familiar with today first took form in Europe. However, the history of these numbers and their development goes back hundreds of years. One important source of information on this topic is the writer al-Biruni, whose picture is shown in figure 10.[footnote]<a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/Mathematicians\/Al-Biruni.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/Mathematicians\/Al-Biruni.html<\/a>[\/footnote] Al-Biruni, who was born in modern day Uzbekistan, had visited India on several occasions and made comments on the Indian number system. When we look at the origins of the numbers that al-Biruni encountered, we have to go back to the third century BCE to explore their origins. It is then that the Brahmi numerals were being used.\r\n\r\nThe Brahmi numerals were more complicated than those used in our own modern system. They had separate symbols for the numbers 1 through 9, as well as distinct symbols for 10, 100, 1000,\u2026, also for 20, 30, 40,\u2026, and others for 200, 300, 400, \u2026, 900. The Brahmi symbols for 1, 2, and 3 are shown below.[footnote]<a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html<\/a>[\/footnote]\r\n\r\n<img class=\"alignnone size-full wp-image-290\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155200\/Fig5_1_11.png\" alt=\"Fig5_1_11\" width=\"220\" height=\"118\" \/>\r\n\r\nThese numerals were used all the way up to the fourth\u00a0century CE, with variations through time and geographic location. For example, in the first century CE, one particular set of Brahmi numerals took on the following form:[footnote]<a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html<\/a>[\/footnote]\r\n\r\n<img class=\"alignnone size-full wp-image-291\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155200\/Fig5_1_12.png\" alt=\"Fig5_1_12\" width=\"526\" height=\"119\" \/>\r\n\r\nFrom the fourth\u00a0century on, you can actually trace several different paths that the Brahmi numerals took to get to different points and incarnations. One of those paths led to our current numeral system, and went through what are called the Gupta numerals. The Gupta numerals were prominent during a time ruled by the Gupta dynasty and were spread throughout that empire as they conquered lands during the fourth\u00a0through sixth\u00a0centuries. They have the following form:[footnote]Ibid.[\/footnote]\r\n\r\n<img class=\"alignnone size-full wp-image-292\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155202\/Fig5_1_13.png\" alt=\"Fig5_1_13\" width=\"524\" height=\"118\" \/>\r\n\r\nHow the numbers got to their Gupta form is open to considerable debate. Many possible hypotheses have been offered, most of which boil down to two basic types.[footnote]Ibid.[\/footnote] The first type of hypothesis states that the numerals came from the initial letters of the names of the numbers. This is not uncommon . . . the Greek numerals developed in this manner. The second type of hypothesis states that they were derived from some earlier number system. However, there are other hypotheses that are offered, one of which is by the researcher Ifrah. His theory is that there were originally nine numerals, each represented by a corresponding number of vertical lines. One possibility is this:[footnote]Ibid.[\/footnote]\r\n\r\n<img class=\"alignnone size-full wp-image-293\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155203\/Fig5_1_14.png\" alt=\"Fig5_1_14\" width=\"522\" height=\"164\" \/>\r\n\r\nBecause these symbols would have taken a lot of time to write, they eventually evolved into cursive symbols that could be written more quickly. If we compare these to the Gupta numerals above, we can try to see how that evolutionary process might have taken place, but our imagination would be just about all we would have to depend upon since we do not know exactly how the process unfolded.<strong>\u00a0<\/strong>\r\n\r\nThe Gupta numerals eventually evolved into another form of numerals called the Nagari numerals, and these continued to evolve until the eleventh\u00a0century, at which time they looked like this:[footnote]Ibid.[\/footnote]\r\n\r\n<img class=\"alignnone size-full wp-image-294\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155204\/Fig5_1_15.png\" alt=\"Fig5_1_15\" width=\"579\" height=\"116\" \/>\r\n\r\nNote that by this time, the symbol for 0 has appeared! The Mayans in the Americas had a symbol for zero long before this, however, as we shall see later in the chapter.\r\n\r\nThese numerals were adopted by the Arabs, most likely in the eighth century during Islamic incursions into the northern part of India.[footnote]Katz, page 230[\/footnote]\u00a0It is believed that the Arabs were instrumental in spreading them to other parts of the world, including Spain (see below).\r\n\r\nOther examples of variations up to the eleventh century include:[footnote]Burton, David M., <em>History of Mathematics, An Introduction<\/em>, p. 254\u2013255[\/footnote]\r\n\r\n[caption id=\"attachment_296\" align=\"alignnone\" width=\"417\"]<img class=\"wp-image-296 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155205\/Fig5_1_16.png\" alt=\"Fig5_1_16\" width=\"417\" height=\"60\" \/> Figure 11. Devangari, eighth century[\/caption]\r\n\r\n[caption id=\"attachment_297\" align=\"alignnone\" width=\"452\"]<img class=\"wp-image-297 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155207\/Fig5_1_17.png\" alt=\"Fig5_1_17\" width=\"452\" height=\"55\" \/> Figure 12. West Arab Gobar, tenth century[\/caption]\r\n\r\n[caption id=\"attachment_298\" align=\"alignnone\" width=\"433\"]<img class=\"wp-image-298 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155208\/Fig5_1_18.png\" alt=\"Fig5_1_18\" width=\"433\" height=\"56\" \/> Figure 13. Spain, 976 BCE[\/caption]\r\n\r\nFinally, figure 14[footnote]Katz, page 231.[\/footnote]\u00a0shows various forms of these numerals as they developed and eventually converged to the fifteenth\u00a0century in Europe.\r\n\r\n[caption id=\"attachment_299\" align=\"aligncenter\" width=\"655\"]<img class=\"wp-image-299 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155209\/Fig5_1_19.png\" alt=\"Fig5_1_19\" width=\"655\" height=\"556\" \/> Figure 14.[\/caption]\r\n<h2>Roman Numerals<\/h2>\r\nThe numeric system represented by <b>Roman numerals<\/b> originated in ancient Rome (<b>753 BC\u2013476 AD)\u00a0<\/b>and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages (generally comprising the 14th and 15th centuries (c. 1301\u20131500)). Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols:\r\n<table style=\"width: 50%;\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Symbol<\/strong><\/td>\r\n<td>I<\/td>\r\n<td>V<\/td>\r\n<td>X<\/td>\r\n<td>L<\/td>\r\n<td>C<\/td>\r\n<td>D<\/td>\r\n<td>M<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Value<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>5<\/td>\r\n<td>10<\/td>\r\n<td>50<\/td>\r\n<td>100<\/td>\r\n<td>500<\/td>\r\n<td>1,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Hindu-Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some minor applications to this day.\r\n\r\nThe numbers 1 to 10 are usually expressed in Roman numerals as follows:\r\n<dl>\r\n \t<dd><b><span class=\"times-serif\" title=\"Roman numeral\">I, II, III, IV, V, VI, VII, VIII, IX, X<\/span><\/b>.<\/dd>\r\n<\/dl>\r\nNumbers are formed by combining symbols and adding the values, so <span class=\"times-serif\" title=\"Roman numeral\">II<\/span> is two (two ones) and <span class=\"times-serif\" title=\"Roman numeral\">XIII<\/span> is thirteen (a ten and three ones). Because each numeral has a fixed value rather than representing multiples of ten, one hundred and so on, according to <i>position<\/i>, there is no need for \"place keeping\" zeros, as in numbers like 207 or 1066; those numbers are written as <span class=\"times-serif\" title=\"Roman numeral\">CCVII<\/span> (two hundreds, a five and two ones) and <span class=\"times-serif\" title=\"Roman numeral\">MLXVI<\/span> (a thousand, a fifty, a ten, a five and a one).\r\n\r\nSymbols are placed from left to right in order of value, starting with the largest. However, in a few specific cases,\u00a0to avoid four characters being repeated in succession (such as <span class=\"times-serif\" title=\"Roman numeral\">IIII<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">XXXX<\/span>), subtractive notation is used: as in this table:\r\n<table style=\"width: 50%;\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Number<\/strong><\/td>\r\n<td>4<\/td>\r\n<td>9<\/td>\r\n<td>40<\/td>\r\n<td>90<\/td>\r\n<td>400<\/td>\r\n<td>900<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Roman Numeral<\/strong><\/td>\r\n<td>IV<\/td>\r\n<td>IX<\/td>\r\n<td>XL<\/td>\r\n<td>XC<\/td>\r\n<td>CD<\/td>\r\n<td>CM<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn summary:\r\n<ul>\r\n \t<li><span class=\"times-serif\" title=\"Roman numeral\">I<\/span> placed before <span class=\"times-serif\" title=\"Roman numeral\">V<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">X<\/span> indicates one less, so four is <span class=\"times-serif\" title=\"Roman numeral\">IV<\/span> (one less than five) and nine is <span class=\"times-serif\" title=\"Roman numeral\">IX<\/span> (one less than ten)<\/li>\r\n \t<li><span class=\"times-serif\" title=\"Roman numeral\">X<\/span> placed before <span class=\"times-serif\" title=\"Roman numeral\">L<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">C<\/span> indicates ten less, so forty is <span class=\"times-serif\" title=\"Roman numeral\">XL<\/span> (ten less than fifty) and ninety is <span class=\"times-serif\" title=\"Roman numeral\">XC<\/span> (ten less than a hundred)<\/li>\r\n \t<li><span class=\"times-serif\" title=\"Roman numeral\">C<\/span> placed before <span class=\"times-serif\" title=\"Roman numeral\">D<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">M<\/span> indicates a hundred less, so four hundred is <span class=\"times-serif\" title=\"Roman numeral\">CD<\/span> (a hundred less than five hundred) and nine hundred is <span class=\"times-serif\" title=\"Roman numeral\">CM<\/span> (a hundred less than a thousand)<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the Hindu-Arabic numeral for\u00a0<span class=\"times-serif\" title=\"Roman numeral\">MCMIV.\r\n[reveal-answer q=\"559978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"559978\"]One thousand nine hundred four, 1904 (M is a thousand, CM is nine hundred and IV is four)[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=86577&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h3><span id=\"Modern_use\" class=\"mw-headline\">Modern use<\/span><\/h3>\r\nBy the 11th century, Hindu\u2013Arabic numerals had been introduced into Europe from al-Andalus, by way of Arab traders and arithmetic treatises. Roman numerals, however, proved very persistent, remaining in common use in the West well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Replacement by their more convenient \"Arabic\" equivalents was quite gradual, and Roman numerals are still used today in certain contexts. A few examples of their current use are:\r\n<div class=\"thumb tright\">\r\n<div class=\"thumbinner\">\r\n\r\n<a class=\"image\" href=\"https:\/\/en.wikipedia.org\/wiki\/File:Carlos_IV_Coin.jpg\"><img class=\"thumbimage\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8b\/Carlos_IV_Coin.jpg\/360px-Carlos_IV_Coin.jpg\" srcset=\"\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8b\/Carlos_IV_Coin.jpg\/540px-Carlos_IV_Coin.jpg 1.5x, \/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8b\/Carlos_IV_Coin.jpg\/720px-Carlos_IV_Coin.jpg 2x\" alt=\"\" width=\"360\" height=\"188\" data-file-width=\"1022\" data-file-height=\"535\" \/><\/a>\r\n<div class=\"thumbcaption\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"thumb tright\">\r\n<div class=\"thumbinner\">\r\n<div class=\"thumbcaption\">Spanish Real using \"IIII\" instead of IV<\/div>\r\n<\/div>\r\n<\/div>\r\n<ul>\r\n \t<li>Names of monarchs and popes, e.g. Elizabeth II of the United Kingdom, Pope Benedict XVI. These are referred to as regnal numbers; e.g. <span class=\"times-serif\" title=\"Roman numeral\">II<\/span> is pronounced \"the second\". This tradition began in Europe sporadically in the Middle Ages, gaining widespread use in England only during the reign of Henry VIII. Previously, the monarch was not known by numeral but by an epithet such as Edward the Confessor. Some monarchs (e.g. Charles IV of Spain and Louis XIV of France) seem to have preferred the use of <span class=\"times-serif\" title=\"Roman numeral\">IIII<\/span> instead of <span class=\"times-serif\" title=\"Roman numeral\">IV<\/span> on their coinage (see illustration).<\/li>\r\n \t<li>Generational suffixes, particularly in the US, for people sharing the same name across generations, for example William Howard Taft IV.<\/li>\r\n \t<li>In the French Republican Calendar, initiated during the French Revolution, years were numbered by Roman numerals - from the year I (1792) when this calendar was introduced to the year XIV (1805) when it was abandoned.<\/li>\r\n \t<li>The year of production of films, television shows and other works of art within the work itself. It has been suggested \u2013 by BBC News, perhaps facetiously \u2013 that this was originally done \"in an attempt to disguise the age of films or television programmes.\"<sup id=\"cite_ref-23\" class=\"reference\">[23]<\/sup> Outside reference to the work will use regular Hindu\u2013Arabic numerals.<\/li>\r\n \t<li>Hour marks on timepieces. In this context, 4 is usually written <span class=\"times-serif\" title=\"Roman numeral\">IIII<\/span>.<\/li>\r\n \t<li>The year of construction on building faces and cornerstones.<\/li>\r\n \t<li>Page numbering of prefaces and introductions of books, and sometimes of annexes, too.<\/li>\r\n \t<li>Book volume and chapter numbers, as well as the several acts within a play (e.g. Act iii, Scene 2).<\/li>\r\n \t<li>Sequels of some movies, video games, and other works (as in <i>Rocky II<\/i>).<\/li>\r\n \t<li>Outlines that use numbers to show hierarchical relationships.<\/li>\r\n \t<li>Occurrences of a recurring grand event, for instance:\r\n<ul>\r\n \t<li>The Summer and Winter Olympic Games (e.g. the <span class=\"times-serif\" title=\"Roman numeral\">XXI<\/span> Olympic Winter Games; the Games of the <span class=\"times-serif\" title=\"Roman numeral\">XXX<\/span> Olympiad)<\/li>\r\n \t<li>The Super Bowl, the annual championship game of the National Football League (e.g. Super Bowl <span class=\"times-serif\" title=\"Roman numeral\">XXXVII<\/span>; Super Bowl 50 is a one-time exception<sup id=\"cite_ref-24\" class=\"reference\">[24]<\/sup>)<\/li>\r\n \t<li>WrestleMania, the annual professional wrestling event for the WWE (e.g. WrestleMania <span class=\"times-serif\" title=\"Roman numeral\">XXX<\/span>). This usage has also been inconsistent.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine the number of objects being represented by pebbles placed on an Inca counting board.<\/li>\n<li>Determine the number represented by a quipu cord<\/li>\n<li>Identify uses other than counting for a quipu cord<\/li>\n<li>Become familiar with the evolution of the counting system we use every day<\/li>\n<li>Write numbers using Roman Numerals<\/li>\n<li>Convert between Hindu-Arabic and Roman Numerals<\/li>\n<\/ul>\n<\/div>\n<p>As we begin our journey through the history of mathematics, one question to be asked is \u201cWhere do we start?\u201d Depending on how you view mathematics or numbers, you could choose any of a number of launching points from which to begin. Howard Eves suggests the following list of possibilities.<a class=\"footnote\" title=\"Eves, Howard; An Introduction to the History of Mathematics, p. 9.\" id=\"return-footnote-193-1\" href=\"#footnote-193-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p>Where to start the study of the history of mathematics\u2026<\/p>\n<ul>\n<li>At the first logical geometric \u201cproofs\u201d traditionally credited to Thales of Miletus (600 BCE).<\/li>\n<li>With the formulation of methods of measurement made by the Egyptians and Mesopotamians\/Babylonians.<\/li>\n<li>Where prehistoric peoples made efforts to organize the concepts of size, shape, and number.<\/li>\n<li>In pre-human times in the very simple number sense and pattern recognition that can be displayed by certain animals, birds, etc.<\/li>\n<li>Even before that in the amazing relationships of numbers and shapes found in plants.<\/li>\n<li>With the spiral nebulae, the natural course of planets, and other universe phenomena.<\/li>\n<\/ul>\n<p>We can choose no starting point at all and instead agree that mathematics has <em>always<\/em> existed and has simply been waiting in the wings for humans to discover. Each of these positions can be defended to some degree and which one you adopt (if any) largely depends on your philosophical ideas about mathematics and numbers.<\/p>\n<p>Nevertheless, we need a starting point. Without passing judgment on the validity of any of these particular possibilities, we will choose as our starting point the emergence of the idea of number and the process of counting as our launching pad. This is done primarily as a practical matter given the nature of this course. In the following chapter, we will try to focus on two main ideas. The first will be an examination of basic number and counting systems and the symbols that we use for numbers. We will look at our own modern (Western) number system as well those of a couple of selected civilizations to see the differences and diversity that is possible when humans start counting. The second idea we will look at will be base systems. By comparing our own base-ten (decimal) system with other bases, we will quickly become aware that the system that we are so used to, when slightly changed, will challenge our notions about numbers and what symbols for those numbers actually mean.<\/p>\n<h3>Recognition of More vs. Less<\/h3>\n<p>The idea of number and the process of counting goes back far beyond history began to be recorded. There is some archeological evidence that suggests that humans were counting as far back as 50,000 years ago.<a class=\"footnote\" title=\"Eves, p. 9.\" id=\"return-footnote-193-2\" href=\"#footnote-193-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>\u00a0However, we do not really know how this process started or developed over time. The best we can do is to make a good guess as to how things progressed. It is probably not hard to believe that even the earliest humans had some sense of <em>more<\/em> and <em>less<\/em>. Even some small animals have been shown to have such a sense. For example, one naturalist tells of how he would secretly remove one egg each day from a plover\u2019s nest. The mother was diligent in laying an extra egg every day to make up for the missing egg. Some research has shown that hens can be trained to distinguish between even and odd numbers of pieces of food.<a class=\"footnote\" title=\"McLeish, John; The Story of Numbers\u2014How Mathematics Has Shaped Civilization, p. 7.\" id=\"return-footnote-193-3\" href=\"#footnote-193-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>\u00a0With these sorts of findings in mind, it is not hard to conceive that early humans had (at least) a similar sense of more and less. However, our conjectures about how and when these ideas emerged among humans are simply that; educated guesses based on our own assumptions of what might or could have been.<\/p>\n<p>&nbsp;<\/p>\n<h2>The Evolution of Counting and The Inca Counting System<\/h2>\n<h3>The Need for Simple Counting<\/h3>\n<p>As societies and humankind evolved, simply having a sense of more or less, even or odd, etc., would prove to be insufficient to meet the needs of everyday living. As tribes and groups formed, it became important to be able to know how many members were in the group, and perhaps how many were in the enemy\u2019s camp. Certainly it was important for them to know if the flock of sheep or other possessed animals were increasing or decreasing in size. \u201cJust how many of them do we have, anyway?\u201d is a question that we do not have a hard time imagining them asking themselves (or each other).<\/p>\n<p>In order to count items such as animals, it is often conjectured that one of the earliest methods of doing so would be with \u201ctally sticks.\u201d These are objects used to track the numbers of items to be counted. With this method, each \u201cstick\u201d (or pebble, or whatever counting device being used) represents one animal or object. This method uses the idea of <strong>one to one correspondence<\/strong>. In a one to one correspondence, items that are being counted are uniquely linked with some counting tool.<\/p>\n<div id=\"attachment_280\" style=\"width: 210px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-280\" class=\"wp-image-280\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155148\/Fig5_1_1.png\" alt=\"Fig5_1_1\" width=\"200\" height=\"239\" \/><\/p>\n<p id=\"caption-attachment-280\" class=\"wp-caption-text\">Figure 1.<\/p>\n<\/div>\n<p>In the picture to the right, you see each stick corresponding to one horse. By examining the collection of sticks in hand one knows how many animals should be present. You can imagine the usefulness of such a system, at least for smaller numbers of items to keep track of. If a herder wanted to \u201ccount off\u201d his animals to make sure they were all present, he could mentally (or methodically) assign each stick to one animal and continue to do so until he was satisfied that all were accounted for.<\/p>\n<p>Of course, in our modern system, we have replaced the sticks with more abstract objects. In particular, the top stick is replaced with our symbol \u201c1,\u201d the second stick gets replaced by a \u201c2\u201d and the third stick is represented by the symbol \u201c3,\u201d but we are getting ahead of ourselves here. These modern symbols took many centuries to emerge.<\/p>\n<p>Another possible way of employing the \u201ctally stick\u201d counting method is by making marks or cutting notches into pieces of wood, or even tying knots in string (as we shall see later). In 1937, Karl Absolom discovered a wolf bone that goes back possibly 30,000 years. It is believed to be a counting device.<a class=\"footnote\" title=\"Bunt, Lucas; Jones, Phillip; Bedient, Jack; The Historical Roots of Elementary Mathematics, p. 2.\" id=\"return-footnote-193-4\" href=\"#footnote-193-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a>\u00a0Another example of this kind of tool is the Ishango Bone, discovered in 1960 at Ishango, and shown below.<a class=\"footnote\" title=\"http:\/\/www.math.buffalo.edu\/mad\/Ancient-Africa\/mad_zaire-uganda.html\" id=\"return-footnote-193-5\" href=\"#footnote-193-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a>\u00a0It is reported to be between six and nine thousand years old and shows what appear to be markings used to do counting of some sort.<\/p>\n<p>The markings on rows (a) and (b) each add up to 60. Row (b) contains the prime numbers between 10 and 20. Row (c) seems to illustrate for the method of doubling and multiplication used by the Egyptians. It is believed that this may also represent a lunar phase counter.<\/p>\n<div id=\"attachment_281\" style=\"width: 527px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-281\" class=\"wp-image-281 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155149\/Fig5_1_2.png\" alt=\"Fig5_1_2\" width=\"517\" height=\"293\" \/><\/p>\n<p id=\"caption-attachment-281\" class=\"wp-caption-text\">Figure 2.<\/p>\n<\/div>\n<h3>Spoken Words<\/h3>\n<p>As methods for counting developed, and as language progressed as well, it is natural to expect that spoken words for numbers would appear. Unfortunately, the developments of these words, especially those corresponding to the numbers from one through ten, are not easy to trace. Past ten, however, we do see some patterns:<\/p>\n<ul>\n<li>Eleven comes from \u201cein lifon,\u201d meaning \u201cone left over.\u201d<\/li>\n<li>Twelve comes from \u201ctwe lif,\u201d meaning \u201ctwo left over.\u201d<\/li>\n<li>Thirteen comes from \u201cThree and ten\u201d as do fourteen through nineteen.<\/li>\n<li>Twenty appears to come from \u201ctwe-tig\u201d which means \u201ctwo tens.\u201d<\/li>\n<li>Hundred probably comes from a term meaning \u201cten times.\u201d<\/li>\n<\/ul>\n<h3>Written Numbers<\/h3>\n<p>When we speak of \u201cwritten\u201d numbers, we have to be careful because this could mean a variety of things. It is important to keep in mind that modern paper is only a little more than 100 years old, so \u201cwriting\u201d in times past often took on forms that might look quite unfamiliar to us today.<\/p>\n<p>As we saw earlier, some might consider wooden sticks with notches carved in them as writing as these are means of recording information on a medium that can be \u201cread\u201d by others. Of course, the symbols used (simple notches) certainly did not leave a lot of flexibility for communicating a wide variety of ideas or information.<\/p>\n<p>Other mediums on which \u201cwriting\u201d may have taken place include carvings in stone or clay tablets, rag paper made by hand (twelfth century in Europe, but earlier in China), papyrus (invented by the Egyptians and used up until the Greeks), and parchments from animal skins. And these are just a few of the many possibilities.<\/p>\n<p>These are just a few examples of early methods of counting and simple symbols for representing numbers. Extensive books, articles and research have been done on this topic and could provide enough information to fill this entire course if we allowed it to. The range and diversity of creative thought that has been used in the past to describe numbers and to count objects and people is staggering. Unfortunately, we don\u2019t have time to examine them all, but it is fun and interesting to look at one system in more detail to see just how ingenious people have been.<\/p>\n<h2>The Number and Counting System of the Inca Civilization<\/h2>\n<h3>Background<\/h3>\n<p>There is generally a lack of books and research material concerning the historical foundations of the Americas. Most of the \u201cimportant\u201d information available concentrates on the eastern hemisphere, with Europe as the central focus. The reasons for this may be twofold: first, it is thought that there was a lack of specialized mathematics in the American regions; second, many of the secrets of ancient mathematics in the Americas have been closely guarded.<a class=\"footnote\" title=\"Diana, Lind Mae; The Peruvian Quipu in Mathematics Teacher, Issue 60 (Oct., 1967), p. 623\u201328.\" id=\"return-footnote-193-6\" href=\"#footnote-193-6\" aria-label=\"Footnote 6\"><sup class=\"footnote\">[6]<\/sup><\/a>\u00a0The Peruvian system does not seem to be an exception here. Two researchers, Leland Locke and Erland Nordenskiold, have carried out research that has attempted to discover what mathematical knowledge was known by the Incas and how they used the Peruvian quipu, a counting system using cords and knots, in their mathematics. These researchers have come to certain beliefs about the quipu that we will summarize here.<\/p>\n<h3>Counting Boards<\/h3>\n<p>It should be noted that the Incas did not have a complicated system of computation. Where other peoples in the regions, such as the Mayans, were doing computations related to their rituals and calendars, the Incas seem to have been more concerned with the simpler task of record-keeping. To do this, they used what are called the \u201cquipu\u201d to record quantities of items. (We will describe them in more detail in a moment.) However, they first often needed to do computations whose results would be recorded on quipu. To do these computations, they would sometimes use a counting board constructed with a slab of stone. In the slab were cut rectangular and square compartments so that an octagonal (eight-sided) region was left in the middle. Two opposite corner rectangles were raised. Another two sections were mounted on the original surface of the slab so that there were actually three levels available. In the figure shown, the darkest shaded corner regions represent the highest, third level. The lighter shaded regions surrounding the corners are the second highest levels, while the clear white rectangles are the compartments cut into the stone slab.<\/p>\n<div id=\"attachment_282\" style=\"width: 486px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-282\" class=\"wp-image-282 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155150\/Fig5_1_3.png\" alt=\"Fig5_1_3\" width=\"476\" height=\"311\" \/><\/p>\n<p id=\"caption-attachment-282\" class=\"wp-caption-text\">Figure 3.<\/p>\n<\/div>\n<p>Pebbles were used to keep accounts and their positions within the various levels and compartments gave totals. For example, a pebble in a smaller (white) compartment represented one unit. Note that there are 12 such squares around the outer edge of the figure. If a pebble was put into one of the two (white) larger, rectangular compartments, its value was doubled. When a pebble was put in the octagonal region in the middle of the slab, its value was tripled. If a pebble was placed on the second (shaded) level, its value was multiplied by six. And finally, if a pebble was found on one of the two highest corner levels, its value was multiplied by twelve. Different objects could be counted at the same time by representing different objects by different colored pebbles.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose you have the following counting board with two different kind of pebbles places as illustrated. Let the solid black pebble represent a dog and the striped pebble represent a cat. How many dogs are being represented?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-283 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155151\/Fig5_1_4.png\" alt=\"Fig5_1_4\" width=\"474\" height=\"311\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q129586\">Show Solution<\/span><\/p>\n<div id=\"q129586\" class=\"hidden-answer\" style=\"display: none\">There are two black pebbles in the outer square regions\u2026these represent 2 dogs.\u00a0There are three black pebbles in the larger (white) rectangular compartments. These represent 6 dogs.\u00a0There is one black pebble in the middle region\u2026this represents 3 dogs.\u00a0There are three black pebbles on the second level\u2026these represent 18 dogs.\u00a0Finally, there is one black pebble on the highest corner level\u2026this represents 12 dogs. We then have a total of 2+6+3+18+12 = 41 dogs.<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>How many cats are represented on this board?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-283 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155151\/Fig5_1_4.png\" alt=\"Fig5_1_4\" width=\"474\" height=\"311\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q881476\">Show Solution<\/span><\/p>\n<div id=\"q881476\" class=\"hidden-answer\" style=\"display: none\">1+6\u00b43+3\u00b46+2\u00b412 = 61 cats<\/div>\n<\/div>\n<\/div>\n<p>Watch this short video lesson about Inca counting boards. You will find that this is a review of concepts presented here about counting boards.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Inca Counting Boards\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fL1N_V89g78?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>The Quipu<\/h3>\n<div id=\"attachment_318\" style=\"width: 254px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-318\" class=\"wp-image-318 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155153\/Fig5_1_5.jpg\" alt=\"Fig5_1_5\" width=\"244\" height=\"395\" \/><\/p>\n<p id=\"caption-attachment-318\" class=\"wp-caption-text\">Figure 5.<\/p>\n<\/div>\n<p>This kind of board was good for doing quick computations, but it did not provide a good way to keep a permanent recording of quantities or computations. For this purpose, they used the quipu. The quipu is a collection of cords with knots in them. These cords and knots are carefully arranged so that the position and type of cord or knot gives specific information on how to decipher the cord.<\/p>\n<p>A quipu is made up of a main cord which has other cords (branches) tied to it. See pictures to the right.<a class=\"footnote\" title=\"Diana, Lind Mae; The Peruvian Quipu in Mathematics Teacher, Issue 60 (Oct., 1967), p. 623\u201328.\" id=\"return-footnote-193-7\" href=\"#footnote-193-7\" aria-label=\"Footnote 7\"><sup class=\"footnote\">[7]<\/sup><\/a><\/p>\n<p>Locke called the branches H cords. They are attached to the main cord. B cords, in turn, were attached to the H cords. Most of these cords would have knots on them. Rarely are knots found on the main cord, however, and tend to be mainly on the H and B cords. A quipu might also have a \u201ctotalizer\u201d cord that summarizes all of the information on the cord group in one place.<\/p>\n<p>Locke points out that there are three types of knots, each representing a different value, depending on the kind of knot used and its position on the cord. The Incas, like us, had a decimal (base-ten) system, so each kind of knot had a specific decimal value. The Single knot, pictured in the middle of figure 6<a class=\"footnote\" title=\"http:\/\/wiscinfo.doit.wisc.edu\/chaysimire\/titulo2\/khipus\/what.htm\" id=\"return-footnote-193-8\" href=\"#footnote-193-8\" aria-label=\"Footnote 8\"><sup class=\"footnote\">[8]<\/sup><\/a>\u00a0was used to denote tens, hundreds, thousands, and ten thousands. They would be on the upper levels of the H cords. The figure-eight knot on the end was used to denote the integer \u201cone.\u201d Every other integer from 2 to 9 was represented with a long knot, shown on the left of the figure. (Sometimes long knots were used to represents tens and hundreds.) Note that the long knot has several turns in it\u2026the number of turns indicates which integer is being represented. The units (ones) were placed closest to the bottom of the cord, then tens right above them, then the hundreds, and so on.<\/p>\n<div id=\"attachment_286\" style=\"width: 516px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-286\" class=\"size-full wp-image-286\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155154\/Fig5_1_7.png\" alt=\"Figure 6\" width=\"506\" height=\"209\" \/><\/p>\n<p id=\"caption-attachment-286\" class=\"wp-caption-text\">Figure 6<\/p>\n<\/div>\n<p>In order to make reading these pictures easier, we will adopt a convention that is consistent. For the long knot with turns in it (representing the numbers 2 through 9), we will use the following notation:<\/p>\n<p>The four horizontal bars represent four turns and the curved arc on the right links the four turns together. This would represent the number 4.<\/p>\n<p>We will represent the single knot with a large dot ( \u00b7 ) and we will represent the figure eight knot with a sideways eight ( \u221e\u00a0).<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What number is represented on the cord shown in figure 7?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-287\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155155\/Fig5_1_8.png\" alt=\"Figure 7.\" width=\"250\" height=\"200\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265241\">Show Solution<\/span><\/p>\n<div id=\"q265241\" class=\"hidden-answer\" style=\"display: none\">On the cord, we see a long knot with four turns in it\u2026this represents four in the ones place. Then 5 single knots appear in the tens position immediately above that, which represents 5 tens, or 50. Finally, 4 single knots are tied in the hundreds, representing four 4 hundreds, or 400. Thus, the total shown on this cord is 454.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>What numbers are represented on each of the four cords hanging from the main cord?<strong>\u00a0<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-288 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155157\/Fig5_1_9.png\" alt=\"\" width=\"347\" height=\"281\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q600164\">Show Solution<\/span><\/p>\n<div id=\"q600164\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>From left to right:<\/p>\n<p>Cord 1 = 2,162<\/p>\n<p>Cord 2 = 301<\/p>\n<p>Cord 3 = 0<\/p>\n<p>Cord 4 = 2,070<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The colors of the cords had meaning and could distinguish one object from another. One color could represent llamas, while a different color might represent sheep, for example. When all the colors available were exhausted, they would have to be re-used. Because of this, the ability to read the quipu became a complicated task and specially trained individuals did this job. They were called Quipucamayoc, which means keeper of the quipus. They would build, guard, and decipher quipus.<\/p>\n<div id=\"attachment_289\" style=\"width: 360px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-289\" class=\"wp-image-289\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155158\/Fig5_1_10.png\" alt=\"Fig5_1_10\" width=\"350\" height=\"283\" \/><\/p>\n<p id=\"caption-attachment-289\" class=\"wp-caption-text\">Figure 9.<\/p>\n<\/div>\n<p>As you can see from this photograph of an actual quipu (figure 9), they could get quite complex.<\/p>\n<p>There were various purposes for the quipu. Some believe that they were used to keep an account of their traditions and history, using knots to record history rather than some other formal system of writing. One writer has even suggested that the quipu replaced writing as it formed a role in the Incan postal system.<a class=\"footnote\" title=\"Diana, Lind Mae; The Peruvian Quipu in Mathematics Teacher, Issue 60 (Oct., 1967), p. 623\u201328.\" id=\"return-footnote-193-9\" href=\"#footnote-193-9\" aria-label=\"Footnote 9\"><sup class=\"footnote\">[9]<\/sup><\/a>\u00a0Another proposed use of the quipu is as a translation tool. After the conquest of the Incas by the Spaniards and subsequent \u201cconversion\u201d to Catholicism, an Inca supposedly could use the quipu to confess their sins to a priest. Yet another proposed use of the quipu was to record numbers related to magic and astronomy, although this is not a widely accepted interpretation.<\/p>\n<p>The following video presents another introduction to the Inca&#8217;s use of a quipu for record keeping.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"The Inca Quipu\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EYq-VtyAd2s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The mysteries of the quipu have not been fully explored yet. Recently, Ascher and Ascher have published a book, <em>The Code of the Quipu: A Study in Media, Mathematics, and Culture<\/em><em>, which is \u201c<\/em>an extensive elaboration of the logical-numerical system of the quipu.\u201d<a class=\"footnote\" title=\"http:\/\/www.cs.uidaho.edu\/~casey931\/seminar\/quipu.html\" id=\"return-footnote-193-10\" href=\"#footnote-193-10\" aria-label=\"Footnote 10\"><sup class=\"footnote\">[10]<\/sup><\/a>\u00a0For more information on the quipu, you may want to check out <em><a href=\"https:\/\/www.maa.org\/press\/periodicals\/convergence\/mathematical-treasure-the-quipu\">Mathematical Treasure: The Quipu<\/a><\/em>.<\/p>\n<p>We are so used to seeing the symbols 1, 2, 3, 4, etc. that it may be somewhat surprising to see such a creative and innovative way to compute and record numbers. Unfortunately, as we proceed through our mathematical education in grade and high school, we receive very little information about the wide range of number systems that have existed and which still exist all over the world. That\u2019s not to say our own system is not important or efficient. The fact that it has survived for hundreds of years and shows no sign of going away any time soon suggests that we may have finally found a system that works well and may not need further improvement, but only time will tell that whether or not that conjecture is valid or not. We now turn to a brief historical look at how our current system developed over history.<\/p>\n<h2>The Hindu\u2014Arabic Number System and Roman Numerals<\/h2>\n<h3>The Evolution of a System<\/h3>\n<p>Our own number system, composed of the ten symbols {0,1,2,3,4,5,6,7,8,9} is called the <em>Hindu<\/em><em>-Arabic system<\/em>. This is a base-ten (decimal) system since place values increase by powers of ten. Furthermore, this system is positional, which means that the position of a symbol has bearing on the value of that symbol within the number. For example, the position of the symbol 3 in the number 435,681 gives it a value much greater than the value of the symbol 8 in that same number. We\u2019ll explore base systems more thoroughly later. The development of these ten symbols and their use in a positional system comes to us primarily from India.<a class=\"footnote\" title=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" id=\"return-footnote-193-11\" href=\"#footnote-193-11\" aria-label=\"Footnote 11\"><sup class=\"footnote\">[11]<\/sup><\/a><\/p>\n<div id=\"attachment_278\" style=\"width: 210px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-278\" class=\"wp-image-278\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155159\/Al_biruni_28-02-2010.jpg\" alt=\"Al-Biruni\" width=\"200\" height=\"263\" \/><\/p>\n<p id=\"caption-attachment-278\" class=\"wp-caption-text\">Figure 10. Al-Biruni<\/p>\n<\/div>\n<p>It was not until the fifteenth\u00a0century that the symbols that we are familiar with today first took form in Europe. However, the history of these numbers and their development goes back hundreds of years. One important source of information on this topic is the writer al-Biruni, whose picture is shown in figure 10.<a class=\"footnote\" title=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/Mathematicians\/Al-Biruni.html\" id=\"return-footnote-193-12\" href=\"#footnote-193-12\" aria-label=\"Footnote 12\"><sup class=\"footnote\">[12]<\/sup><\/a> Al-Biruni, who was born in modern day Uzbekistan, had visited India on several occasions and made comments on the Indian number system. When we look at the origins of the numbers that al-Biruni encountered, we have to go back to the third century BCE to explore their origins. It is then that the Brahmi numerals were being used.<\/p>\n<p>The Brahmi numerals were more complicated than those used in our own modern system. They had separate symbols for the numbers 1 through 9, as well as distinct symbols for 10, 100, 1000,\u2026, also for 20, 30, 40,\u2026, and others for 200, 300, 400, \u2026, 900. The Brahmi symbols for 1, 2, and 3 are shown below.<a class=\"footnote\" title=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" id=\"return-footnote-193-13\" href=\"#footnote-193-13\" aria-label=\"Footnote 13\"><sup class=\"footnote\">[13]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-290\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155200\/Fig5_1_11.png\" alt=\"Fig5_1_11\" width=\"220\" height=\"118\" \/><\/p>\n<p>These numerals were used all the way up to the fourth\u00a0century CE, with variations through time and geographic location. For example, in the first century CE, one particular set of Brahmi numerals took on the following form:<a class=\"footnote\" title=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" id=\"return-footnote-193-14\" href=\"#footnote-193-14\" aria-label=\"Footnote 14\"><sup class=\"footnote\">[14]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-291\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155200\/Fig5_1_12.png\" alt=\"Fig5_1_12\" width=\"526\" height=\"119\" \/><\/p>\n<p>From the fourth\u00a0century on, you can actually trace several different paths that the Brahmi numerals took to get to different points and incarnations. One of those paths led to our current numeral system, and went through what are called the Gupta numerals. The Gupta numerals were prominent during a time ruled by the Gupta dynasty and were spread throughout that empire as they conquered lands during the fourth\u00a0through sixth\u00a0centuries. They have the following form:<a class=\"footnote\" title=\"Ibid.\" id=\"return-footnote-193-15\" href=\"#footnote-193-15\" aria-label=\"Footnote 15\"><sup class=\"footnote\">[15]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-292\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155202\/Fig5_1_13.png\" alt=\"Fig5_1_13\" width=\"524\" height=\"118\" \/><\/p>\n<p>How the numbers got to their Gupta form is open to considerable debate. Many possible hypotheses have been offered, most of which boil down to two basic types.<a class=\"footnote\" title=\"Ibid.\" id=\"return-footnote-193-16\" href=\"#footnote-193-16\" aria-label=\"Footnote 16\"><sup class=\"footnote\">[16]<\/sup><\/a> The first type of hypothesis states that the numerals came from the initial letters of the names of the numbers. This is not uncommon . . . the Greek numerals developed in this manner. The second type of hypothesis states that they were derived from some earlier number system. However, there are other hypotheses that are offered, one of which is by the researcher Ifrah. His theory is that there were originally nine numerals, each represented by a corresponding number of vertical lines. One possibility is this:<a class=\"footnote\" title=\"Ibid.\" id=\"return-footnote-193-17\" href=\"#footnote-193-17\" aria-label=\"Footnote 17\"><sup class=\"footnote\">[17]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-293\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155203\/Fig5_1_14.png\" alt=\"Fig5_1_14\" width=\"522\" height=\"164\" \/><\/p>\n<p>Because these symbols would have taken a lot of time to write, they eventually evolved into cursive symbols that could be written more quickly. If we compare these to the Gupta numerals above, we can try to see how that evolutionary process might have taken place, but our imagination would be just about all we would have to depend upon since we do not know exactly how the process unfolded.<strong>\u00a0<\/strong><\/p>\n<p>The Gupta numerals eventually evolved into another form of numerals called the Nagari numerals, and these continued to evolve until the eleventh\u00a0century, at which time they looked like this:<a class=\"footnote\" title=\"Ibid.\" id=\"return-footnote-193-18\" href=\"#footnote-193-18\" aria-label=\"Footnote 18\"><sup class=\"footnote\">[18]<\/sup><\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-294\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155204\/Fig5_1_15.png\" alt=\"Fig5_1_15\" width=\"579\" height=\"116\" \/><\/p>\n<p>Note that by this time, the symbol for 0 has appeared! The Mayans in the Americas had a symbol for zero long before this, however, as we shall see later in the chapter.<\/p>\n<p>These numerals were adopted by the Arabs, most likely in the eighth century during Islamic incursions into the northern part of India.<a class=\"footnote\" title=\"Katz, page 230\" id=\"return-footnote-193-19\" href=\"#footnote-193-19\" aria-label=\"Footnote 19\"><sup class=\"footnote\">[19]<\/sup><\/a>\u00a0It is believed that the Arabs were instrumental in spreading them to other parts of the world, including Spain (see below).<\/p>\n<p>Other examples of variations up to the eleventh century include:<a class=\"footnote\" title=\"Burton, David M., History of Mathematics, An Introduction, p. 254\u2013255\" id=\"return-footnote-193-20\" href=\"#footnote-193-20\" aria-label=\"Footnote 20\"><sup class=\"footnote\">[20]<\/sup><\/a><\/p>\n<div id=\"attachment_296\" style=\"width: 427px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-296\" class=\"wp-image-296 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155205\/Fig5_1_16.png\" alt=\"Fig5_1_16\" width=\"417\" height=\"60\" \/><\/p>\n<p id=\"caption-attachment-296\" class=\"wp-caption-text\">Figure 11. Devangari, eighth century<\/p>\n<\/div>\n<div id=\"attachment_297\" style=\"width: 462px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-297\" class=\"wp-image-297 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155207\/Fig5_1_17.png\" alt=\"Fig5_1_17\" width=\"452\" height=\"55\" \/><\/p>\n<p id=\"caption-attachment-297\" class=\"wp-caption-text\">Figure 12. West Arab Gobar, tenth century<\/p>\n<\/div>\n<div id=\"attachment_298\" style=\"width: 443px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-298\" class=\"wp-image-298 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155208\/Fig5_1_18.png\" alt=\"Fig5_1_18\" width=\"433\" height=\"56\" \/><\/p>\n<p id=\"caption-attachment-298\" class=\"wp-caption-text\">Figure 13. Spain, 976 BCE<\/p>\n<\/div>\n<p>Finally, figure 14<a class=\"footnote\" title=\"Katz, page 231.\" id=\"return-footnote-193-21\" href=\"#footnote-193-21\" aria-label=\"Footnote 21\"><sup class=\"footnote\">[21]<\/sup><\/a>\u00a0shows various forms of these numerals as they developed and eventually converged to the fifteenth\u00a0century in Europe.<\/p>\n<div id=\"attachment_299\" style=\"width: 665px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-299\" class=\"wp-image-299 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155209\/Fig5_1_19.png\" alt=\"Fig5_1_19\" width=\"655\" height=\"556\" \/><\/p>\n<p id=\"caption-attachment-299\" class=\"wp-caption-text\">Figure 14.<\/p>\n<\/div>\n<h2>Roman Numerals<\/h2>\n<p>The numeric system represented by <b>Roman numerals<\/b> originated in ancient Rome (<b>753 BC\u2013476 AD)\u00a0<\/b>and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages (generally comprising the 14th and 15th centuries (c. 1301\u20131500)). Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols:<\/p>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<td><strong>Symbol<\/strong><\/td>\n<td>I<\/td>\n<td>V<\/td>\n<td>X<\/td>\n<td>L<\/td>\n<td>C<\/td>\n<td>D<\/td>\n<td>M<\/td>\n<\/tr>\n<tr>\n<td><strong>Value<\/strong><\/td>\n<td>1<\/td>\n<td>5<\/td>\n<td>10<\/td>\n<td>50<\/td>\n<td>100<\/td>\n<td>500<\/td>\n<td>1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Hindu-Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some minor applications to this day.<\/p>\n<p>The numbers 1 to 10 are usually expressed in Roman numerals as follows:<\/p>\n<dl>\n<dd><b><span class=\"times-serif\" title=\"Roman numeral\">I, II, III, IV, V, VI, VII, VIII, IX, X<\/span><\/b>.<\/dd>\n<\/dl>\n<p>Numbers are formed by combining symbols and adding the values, so <span class=\"times-serif\" title=\"Roman numeral\">II<\/span> is two (two ones) and <span class=\"times-serif\" title=\"Roman numeral\">XIII<\/span> is thirteen (a ten and three ones). Because each numeral has a fixed value rather than representing multiples of ten, one hundred and so on, according to <i>position<\/i>, there is no need for &#8220;place keeping&#8221; zeros, as in numbers like 207 or 1066; those numbers are written as <span class=\"times-serif\" title=\"Roman numeral\">CCVII<\/span> (two hundreds, a five and two ones) and <span class=\"times-serif\" title=\"Roman numeral\">MLXVI<\/span> (a thousand, a fifty, a ten, a five and a one).<\/p>\n<p>Symbols are placed from left to right in order of value, starting with the largest. However, in a few specific cases,\u00a0to avoid four characters being repeated in succession (such as <span class=\"times-serif\" title=\"Roman numeral\">IIII<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">XXXX<\/span>), subtractive notation is used: as in this table:<\/p>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<td><strong>Number<\/strong><\/td>\n<td>4<\/td>\n<td>9<\/td>\n<td>40<\/td>\n<td>90<\/td>\n<td>400<\/td>\n<td>900<\/td>\n<\/tr>\n<tr>\n<td><strong>Roman Numeral<\/strong><\/td>\n<td>IV<\/td>\n<td>IX<\/td>\n<td>XL<\/td>\n<td>XC<\/td>\n<td>CD<\/td>\n<td>CM<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In summary:<\/p>\n<ul>\n<li><span class=\"times-serif\" title=\"Roman numeral\">I<\/span> placed before <span class=\"times-serif\" title=\"Roman numeral\">V<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">X<\/span> indicates one less, so four is <span class=\"times-serif\" title=\"Roman numeral\">IV<\/span> (one less than five) and nine is <span class=\"times-serif\" title=\"Roman numeral\">IX<\/span> (one less than ten)<\/li>\n<li><span class=\"times-serif\" title=\"Roman numeral\">X<\/span> placed before <span class=\"times-serif\" title=\"Roman numeral\">L<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">C<\/span> indicates ten less, so forty is <span class=\"times-serif\" title=\"Roman numeral\">XL<\/span> (ten less than fifty) and ninety is <span class=\"times-serif\" title=\"Roman numeral\">XC<\/span> (ten less than a hundred)<\/li>\n<li><span class=\"times-serif\" title=\"Roman numeral\">C<\/span> placed before <span class=\"times-serif\" title=\"Roman numeral\">D<\/span> or <span class=\"times-serif\" title=\"Roman numeral\">M<\/span> indicates a hundred less, so four hundred is <span class=\"times-serif\" title=\"Roman numeral\">CD<\/span> (a hundred less than five hundred) and nine hundred is <span class=\"times-serif\" title=\"Roman numeral\">CM<\/span> (a hundred less than a thousand)<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the Hindu-Arabic numeral for\u00a0<span class=\"times-serif\" title=\"Roman numeral\">MCMIV.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q559978\">Show Solution<\/span><\/p>\n<div id=\"q559978\" class=\"hidden-answer\" style=\"display: none\">One thousand nine hundred four, 1904 (M is a thousand, CM is nine hundred and IV is four)<\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=86577&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3><span id=\"Modern_use\" class=\"mw-headline\">Modern use<\/span><\/h3>\n<p>By the 11th century, Hindu\u2013Arabic numerals had been introduced into Europe from al-Andalus, by way of Arab traders and arithmetic treatises. Roman numerals, however, proved very persistent, remaining in common use in the West well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Replacement by their more convenient &#8220;Arabic&#8221; equivalents was quite gradual, and Roman numerals are still used today in certain contexts. A few examples of their current use are:<\/p>\n<div class=\"thumb tright\">\n<div class=\"thumbinner\">\n<p><a class=\"image\" href=\"https:\/\/en.wikipedia.org\/wiki\/File:Carlos_IV_Coin.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"thumbimage\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8b\/Carlos_IV_Coin.jpg\/360px-Carlos_IV_Coin.jpg\" srcset=\"\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8b\/Carlos_IV_Coin.jpg\/540px-Carlos_IV_Coin.jpg 1.5x, \/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8b\/Carlos_IV_Coin.jpg\/720px-Carlos_IV_Coin.jpg 2x\" alt=\"\" width=\"360\" height=\"188\" data-file-width=\"1022\" data-file-height=\"535\" \/><\/a><\/p>\n<div class=\"thumbcaption\"><\/div>\n<\/div>\n<\/div>\n<div class=\"thumb tright\">\n<div class=\"thumbinner\">\n<div class=\"thumbcaption\">Spanish Real using &#8220;IIII&#8221; instead of IV<\/div>\n<\/div>\n<\/div>\n<ul>\n<li>Names of monarchs and popes, e.g. Elizabeth II of the United Kingdom, Pope Benedict XVI. These are referred to as regnal numbers; e.g. <span class=\"times-serif\" title=\"Roman numeral\">II<\/span> is pronounced &#8220;the second&#8221;. This tradition began in Europe sporadically in the Middle Ages, gaining widespread use in England only during the reign of Henry VIII. Previously, the monarch was not known by numeral but by an epithet such as Edward the Confessor. Some monarchs (e.g. Charles IV of Spain and Louis XIV of France) seem to have preferred the use of <span class=\"times-serif\" title=\"Roman numeral\">IIII<\/span> instead of <span class=\"times-serif\" title=\"Roman numeral\">IV<\/span> on their coinage (see illustration).<\/li>\n<li>Generational suffixes, particularly in the US, for people sharing the same name across generations, for example William Howard Taft IV.<\/li>\n<li>In the French Republican Calendar, initiated during the French Revolution, years were numbered by Roman numerals &#8211; from the year I (1792) when this calendar was introduced to the year XIV (1805) when it was abandoned.<\/li>\n<li>The year of production of films, television shows and other works of art within the work itself. It has been suggested \u2013 by BBC News, perhaps facetiously \u2013 that this was originally done &#8220;in an attempt to disguise the age of films or television programmes.&#8221;<sup id=\"cite_ref-23\" class=\"reference\">[23]<\/sup> Outside reference to the work will use regular Hindu\u2013Arabic numerals.<\/li>\n<li>Hour marks on timepieces. In this context, 4 is usually written <span class=\"times-serif\" title=\"Roman numeral\">IIII<\/span>.<\/li>\n<li>The year of construction on building faces and cornerstones.<\/li>\n<li>Page numbering of prefaces and introductions of books, and sometimes of annexes, too.<\/li>\n<li>Book volume and chapter numbers, as well as the several acts within a play (e.g. Act iii, Scene 2).<\/li>\n<li>Sequels of some movies, video games, and other works (as in <i>Rocky II<\/i>).<\/li>\n<li>Outlines that use numbers to show hierarchical relationships.<\/li>\n<li>Occurrences of a recurring grand event, for instance:\n<ul>\n<li>The Summer and Winter Olympic Games (e.g. the <span class=\"times-serif\" title=\"Roman numeral\">XXI<\/span> Olympic Winter Games; the Games of the <span class=\"times-serif\" title=\"Roman numeral\">XXX<\/span> Olympiad)<\/li>\n<li>The Super Bowl, the annual championship game of the National Football League (e.g. Super Bowl <span class=\"times-serif\" title=\"Roman numeral\">XXXVII<\/span>; Super Bowl 50 is a one-time exception<sup id=\"cite_ref-24\" class=\"reference\">[24]<\/sup>)<\/li>\n<li>WrestleMania, the annual professional wrestling event for the WWE (e.g. WrestleMania <span class=\"times-serif\" title=\"Roman numeral\">XXX<\/span>). This usage has also been inconsistent.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\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-193\">\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>Mathematics for the Liberal Arts I. <strong>Provided by<\/strong>: Extended Learning Institute of Northern Virginia Community College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/eli.nvcc.edu\/\">http:\/\/eli.nvcc.edu\/<\/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 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-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Inca Counting Boards. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/fL1N_V89g78\">https:\/\/youtu.be\/fL1N_V89g78<\/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 86557. <strong>Authored by<\/strong>: Abert, Rex. <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>Roman Numerals. <strong>Authored by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/en.wikipedia.org\/wiki\/Roman_numerals\">https:\/\/en.wikipedia.org\/wiki\/Roman_numerals<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">Public domain content<\/div><ul class=\"citation-list\"><li>Image of Finger. <strong>Authored by<\/strong>: geralt. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/finger-touch-hand-structure-769300\/\">https:\/\/pixabay.com\/en\/finger-touch-hand-structure-769300\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/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-193-1\">Eves, Howard; An Introduction to the History of Mathematics, p. 9. <a href=\"#return-footnote-193-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-193-2\">Eves, p. 9. <a href=\"#return-footnote-193-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-193-3\">McLeish, John; The Story of Numbers\u2014How Mathematics Has Shaped Civilization, p. 7. <a href=\"#return-footnote-193-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-193-4\">Bunt, Lucas; Jones, Phillip; Bedient, Jack; The Historical Roots of Elementary Mathematics, p. 2. <a href=\"#return-footnote-193-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-193-5\"><a href=\"http:\/\/www.math.buffalo.edu\/mad\/Ancient-Africa\/mad_zaire-uganda.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.math.buffalo.edu\/mad\/Ancient-Africa\/mad_zaire-uganda.html<\/a> <a href=\"#return-footnote-193-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><li id=\"footnote-193-6\">Diana, Lind Mae; The Peruvian Quipu in <em>Mathematics Teacher, <\/em>Issue 60 (Oct., 1967), p. 623\u201328. <a href=\"#return-footnote-193-6\" class=\"return-footnote\" aria-label=\"Return to footnote 6\">&crarr;<\/a><\/li><li id=\"footnote-193-7\">Diana, Lind Mae; The Peruvian Quipu in <em>Mathematics Teacher, <\/em>Issue 60 (Oct., 1967), p. 623\u201328. <a href=\"#return-footnote-193-7\" class=\"return-footnote\" aria-label=\"Return to footnote 7\">&crarr;<\/a><\/li><li id=\"footnote-193-8\"><a href=\"http:\/\/wiscinfo.doit.wisc.edu\/chaysimire\/titulo2\/khipus\/what.htm\" target=\"_blank\" rel=\"noopener\">http:\/\/wiscinfo.doit.wisc.edu\/chaysimire\/titulo2\/khipus\/what.htm<\/a> <a href=\"#return-footnote-193-8\" class=\"return-footnote\" aria-label=\"Return to footnote 8\">&crarr;<\/a><\/li><li id=\"footnote-193-9\">Diana, Lind Mae; The Peruvian Quipu in <em>Mathematics Teacher, <\/em>Issue 60 (Oct., 1967), p. 623\u201328. <a href=\"#return-footnote-193-9\" class=\"return-footnote\" aria-label=\"Return to footnote 9\">&crarr;<\/a><\/li><li id=\"footnote-193-10\"><a href=\"http:\/\/www.cs.uidaho.edu\/~casey931\/seminar\/quipu.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.cs.uidaho.edu\/~casey931\/seminar\/quipu.html<\/a> <a href=\"#return-footnote-193-10\" class=\"return-footnote\" aria-label=\"Return to footnote 10\">&crarr;<\/a><\/li><li id=\"footnote-193-11\"><a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html<\/a> <a href=\"#return-footnote-193-11\" class=\"return-footnote\" aria-label=\"Return to footnote 11\">&crarr;<\/a><\/li><li id=\"footnote-193-12\"><a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/Mathematicians\/Al-Biruni.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/Mathematicians\/Al-Biruni.html<\/a> <a href=\"#return-footnote-193-12\" class=\"return-footnote\" aria-label=\"Return to footnote 12\">&crarr;<\/a><\/li><li id=\"footnote-193-13\"><a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html<\/a> <a href=\"#return-footnote-193-13\" class=\"return-footnote\" aria-label=\"Return to footnote 13\">&crarr;<\/a><\/li><li id=\"footnote-193-14\"><a href=\"http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www-groups.dcs.st-and.ac.uk\/~history\/HistTopics\/Indian_numerals.html<\/a> <a href=\"#return-footnote-193-14\" class=\"return-footnote\" aria-label=\"Return to footnote 14\">&crarr;<\/a><\/li><li id=\"footnote-193-15\">Ibid. <a href=\"#return-footnote-193-15\" class=\"return-footnote\" aria-label=\"Return to footnote 15\">&crarr;<\/a><\/li><li id=\"footnote-193-16\">Ibid. <a href=\"#return-footnote-193-16\" class=\"return-footnote\" aria-label=\"Return to footnote 16\">&crarr;<\/a><\/li><li id=\"footnote-193-17\">Ibid. <a href=\"#return-footnote-193-17\" class=\"return-footnote\" aria-label=\"Return to footnote 17\">&crarr;<\/a><\/li><li id=\"footnote-193-18\">Ibid. <a href=\"#return-footnote-193-18\" class=\"return-footnote\" aria-label=\"Return to footnote 18\">&crarr;<\/a><\/li><li id=\"footnote-193-19\">Katz, page 230 <a href=\"#return-footnote-193-19\" class=\"return-footnote\" aria-label=\"Return to footnote 19\">&crarr;<\/a><\/li><li id=\"footnote-193-20\">Burton, David M., <em>History of Mathematics, An Introduction<\/em>, p. 254\u2013255 <a href=\"#return-footnote-193-20\" class=\"return-footnote\" aria-label=\"Return to footnote 20\">&crarr;<\/a><\/li><li id=\"footnote-193-21\">Katz, page 231. <a href=\"#return-footnote-193-21\" class=\"return-footnote\" aria-label=\"Return to footnote 21\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Mathematics for the Liberal Arts I\",\"author\":\"\",\"organization\":\"Extended Learning Institute of Northern Virginia Community College\",\"url\":\"http:\/\/eli.nvcc.edu\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"pd\",\"description\":\"Image of 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