Introduction We All Know the Ten Digits of the Hindu-Arabic Counting System: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
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History of Numbers 1a. I am familiar with the early evolution of counting. Introduction We all know the ten digits of the Hindu-Arabic counting system: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. And when we read a number like 352, because of place value, we know it stands for three groups of a hundred, Iive groups of ten, and two units. Our numerals are usually arranged according to a positional base 10 (or decimal) system. PRACTICE. Are you familiar with our base 10 or decimal system and place value? In Canvas, work through these 7 practice problems: Identify value of a digit Yet while there are still Hindu-Arabic numerals involved, telling time reQuires a different system. There are 60 seconds in every minute and 60 minutes in every hour. So if your watch displays 12:59:59 right now, then you expect it to read 1:00:00 a second later: We are so used to telling time in groups of 60 that it seems natural. But why there are not 100 seconds in each minute, or 100 minutes in an hour? In the late 1700s, a French attorney suggested a system of decimal time measurement in which each day has 10 hours, each hour 100 minutes, and each minute 1000 seconds. The actual duration of these decimal hours, minutes, and seconds would be different; however, time conversions would be simpler. For example, 6 decimal hours = 600 decimal minutes = 600,000 decimal seconds. So why does our system of telling time not conform to the usual base 10 counting system that governs most other aspects of our life? BCE means "Before Common Era". Also known as BC which means "Before Christ". So these are times before the year 0, or before the Christ supposedly died. CE means "Common Era". Also known as AD which means "After Death". So these are times after the year 0, or after the year Christ supposedly died. Babylonian Numbers The Babylonians were one of the =irst cultures to develop a positional numeral system. Babylonia was an ancient cultural region occupying southeastern Mesopotamia between the Tigris and Euphrates rivers (modern southern IraQ from around Baghdad to the Persian Gulf). Because the city of Babylon was the capital of this area for so many centuries, the term Babylonia has come to refer to the entire culture that developed in the area from the time it was Iirst settled, about 4000 BCE. 1 This positional system originated around 2000 BCE. Symbols for numerals were written in cuneiform, using a wedge-tipped tool to press marks into a soft clay tablet, which would be dried in the sun to create a permanent record. Instead of having only 10 digits and groups in powers of 10, their system was based on groups and powers of 60 (which is called a sexigesimal system). Below are the 59 digits used in this system (the 60th digit, a zero, was at Iirst only represented by an empty space): The Babylonian system spread throughout most of Mesopotamia, but it eventually faded into history, allowing other number systems such as Roman numerals and the Hindu-Arabic system to take its place. There are still remnants of this ancient sexigesimal counting system in the way we keep time, and also in how we measure angles in degrees. There are 360 degrees in a full circle, and 360 = 6 ⋅ 60. Furthermore, there are 60 arc minutes in one degree and 60 arc seconds in one arc minute. This system is used to locate any point on the surface of the Earth by its latitude and longitude. So even though our numerals are Hindu-Arabic, we still rely on the Babylonian base 60 system every second of the day and everywhere on the globe! PRACTICE. Are you familiar with the units of time, which are in base 60 or the sexigesimal system? Work through these 4 practice problems: Convert units of time. The Need for Simple Counting People at Iirst used objects used to track the numbers of separate things that must be counted. With this method, each stick (or pebble, or whatever counting object used) represents one item. One to One Correspondence. This method uses a "one to one correspondence", where each individual item that is being counted is linked with one object (such as a stick or pebble). In the picture below, each individual stick corresponds to one horse. The collection of sticks shows how many animals should be present: 2 Example of one-to-one correspondence: one stick represents one horse. The Tally Stick. Another possible way of using one-to-one correspondence is to make marks or cut notches into pieces of wood or bone. Instead of objects such as sticks or pebbles, each mark or notch corresponds to one of the things being counted. The advantage here is that people now had a more permanent record of the things they were counting. One of the earliest examples in the archeological record is a notched wolf bone, found in the Czech Republic, and thought to be thirty thousand years old: This paleolithic wolf bone tally stick had Iifty-ive notches carved into it. First came 25 in groups of Iive, then a notch of double length, followed by a similar double notch that began a series of 30. This would prove the mammoth hunters ability to count, and groups of ive suggest the ive ingers of a hand. Perhaps the most famous example of this is the Ishango Bone, discovered in 1960 in the Congo, in Central Africa. It is thought to be around twenty thousand years old, and seems to have been much more than a simple counting tool. In fact, it may be the world's oldest evidence of more advanced mathematics. NOTE: The markings on rows (a) and (b) each add up to 60. Row (a) seems consistent with a base 10 number system, because the notches are grouped as 20 + 1, 20 - 1, 10 + 1, and 10 - 1. Row (b) contains the prime numbers between 10 and 20. Row (c) seems to illustrate the method of doubling and multiplication used by the ancient Egyptians. It is believed that this may also represent a lunar phase counter/tracker: In our modern system, we have replaced sticks (or pebbles or whatever is being used), marks or notches with more abstract symbols. For example, one stick is replaced with our symbol “1,” two sticks with a “2”, three with the symbol “3”, and so on. But these modern symbols took many centuries to emerge. 3 Spoken Words As methods for counting developed, and as language progressed, it is perhaps natural that spoken words for numbers would appear. Unfortunately, the developments of these number names (in English) for our modern Hindu-Arabic counting system (especially those for the numbers one through ten), are not easy to trace. Past ten, however, we do see some patterns that reIlect the fact that the number system was base ten: • Eleven comes from “ein lifon,” meaning “one left over.” • Twelve comes from “twe lif,” meaning “two left over.” • Thirteen comes from “Three and ten” as do fourteen through nineteen. • Twenty appears to come from “twe-tig” which means “two tens.” • Hundred probably comes from a term meaning “ten times.” Of course, other languages have their own words for numbers! In Canvas, see: Counting to a thousand in 14 different languages Number names in Africa Even though there are over a thousand languages on the entire continent, the traditional words for 1, 2, 3, and 4 are similar for 50% of the continent, across multiple language groups. Many systems use Iive as their base. When Iive is the base, the number name for six translates to the "sum of Iive and one"; seven to the "sum of Iive and two", and so on. Some use twenty. When twenty is the base (as is the case with the Igbo people), the number word for thirty is ohu na iri which means "twenty and ten", and for ifty is ohu na iri na otu which means "twenty times two and ten". In many traditional African number systems, standardized hand gestures accompany or even replace number names. The Malinke or Mandinka people of Western Africa have some fascinating examples of how certain African people have conceptualized and developed number names. For example the word for nine is kononto, which literally means "to the one of the belly", a reference to the nine months of pregnancy. The number word for Iifteen means "three Iists", and the word for twenty refers to "a complete man", because twenty conceptualizes the ten Iingers AND ten toes. The number name for forty literally means "a mat", where a couple sleeps together, thus referring to the counting of the Iingers and toes of both the man and woman, which eQual forty. Written Numbers When we speak of “written” 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 “writing” in times past often took on forms that might look Quite unfamiliar to us today. Wooden sticks or bones with notches carved in them could be considered writing, as these are ways of recording information onto something that can be “read” by someone else. Other 4 mediums on which “writing” may have taken place include carvings in stone or clay tablets, markings on papyrus (invented by the ancient Egyptians), on actual paper made by hand*, or on parchments from animal skins.