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NUMERATION SYSTEM
 

NUMERATION SYSTEM

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    NUMERATION SYSTEM NUMERATION SYSTEM Document Transcript

    • 2.0 THE FIRST TEN NUMBERS OF EACH NUMERATION SYSTEM 2.1 EGYPTIAN NUMERATION SYSTEM One of the earliest examples of a numeral system is the Egyptian numeral system, based on the following hieroglyphs: If we look at the diagram above, we will notice that the first nine numerals are pictographic in character, but the remaining ones are logographic in character. Notice also that this is a decimal system. However, it does not tell us how the Egyptians wrote compound numerals. As it turns out, the Egyptians used a simple additive system, as illustrated in the following diagrams. 2
    • Note carefully that although the Egyptian numeral system does not especially require a symbol for zero, the Egyptians nevertheless had a symbol for zero which they used for a variety of engineering and accounting purposes, including some rather astonishing projects, such as the Pyramids which were constructed during 2550 BC.HINDU-ARABIC EGYPTIAN NUMERATION SYSTEMNUMERATION SYSTEM 1 2 3 3
    • 45678910 4
    • 2.2 BABYLONIAN NUMERATION SYSTEM The Babylonians lived in Mesopotamia, which is between the Tigris and Euphrates rivers. They began a numbering system about 5,000 years ago. It is one of the oldest numbering systems. The first mathematics can be traced to the ancient country of Babylon, during the third millennium B.C. Tables were the Babylonians most outstanding accomplishment which helped them in calculating problems. The Babylonian numeration system was developed between 3000 and 2000 BCE. It uses only two numerals or symbols, a one and a ten to represent numbers and they looked these: To represent numbers from 2 to 59, the system was simply additives. The Babylonian number system began with tally marks just as most of the ancient math systems did. The Babylonians developed a form of writing based on cuneiform. Cuneiform means "wedge shape" in Latin. They wrote these symbols on wet clay tablets which were baked in the hot sun. Many thousands of these tablets are still around today. The Babylonians used a stylist to imprint the symbols on the clay since curved lines could not be drawn. The Babylonians had a very advanced number system even for todays standards. It was a base 60 system (sexagesimals) rather than a base ten (decimal). Base ten is what we use today. 5
    • The Babylonians divided the day into twenty-four hours, eachhour into sixty minutes, and each minute to sixty seconds. This form ofcounting has survived for four thousand years.Example#1:5 is written as shown:12 are written as shown:Notice how the ones, in this case two ones are shown on the right justlike the Hindu-Arabic numeration system45 is written as shown:For number bigger than 59, the Babylonian used a place value systemwith a base of 60 6
    • 62 is written as shown:Notice this time the use of a big space to separate the space valueWithout the big space, things look like this:However, what is that number without this big space? Could it be 2 ×60 + 1 or 1× 602 + 1 × 60 + 1 or???The Babylonians introduced the big space after they became aware ofthis ambiguity.The number 4871 could be represented as follow: 3600 + 1260 + 11 =4871 7
    • Even after the big space was introduced to separate place value, theBabylonians still faced a more serious problem?Since there was no zero to put in an empty position, the number 60would thus have the same representation as the number 1How did they make the difference? All we can say is that the contextmust have helped them to establish such difference yet the Babyloniannumeration system was without a doubt a very ambiguous numeralsystem.If this had become a major problem, no doubt the Babylonians weresmart enough to come up with a working system. 8
    • HINDU-ARABIC BABYLONIAN NUMERATION SYSTEMNUMERATION SYSTEM 1 2 3 4 5 6 9
    • 78910 10
    • 2.3 ROMAN NUMERATION SYSTEM Before Rome, the most developed civilization on the Italic Peninsula was the Etruscan civilization, who copied their numerals from the early Greek (Attic) system. These numerals were adopted and adapted by the Romans, who formulated the Roman numeral system, still in wide use today for a variety of purposes. There are other Roman numerals that most of us never learn, but can be found in Latin dictionaries – for example: 5000 I>> 10000 ==I>> 50000 I>>> 100000 ===I>>> 500000 I>>>> 1000000 ====I>>>> As every grade school child knows, the Roman numeral system is based on the following seven atomic numerals: IVXLCDM 1 5 10 50 100 500 1000 The Roman numeral system is not a simple additive system, but is rather an additive-subtractive system. In fact, the subtractive aspect is frequently a source of worry when reading large numerals – for example: MCMXCIX By saying that the Roman system is (partly) subtractive, we mean that some combinations of symbols require us to apply subtraction in order to interpret them. For example, IV stands for “one before five”, which is four [i.e., 5 minus 1]. Similarly, the numeral XC 11
    • stands for “ten before one-hundred”, which is ninety [i.e., 100 minus 10]. On the other hand, the string IC is officially ill-formed, although it could be understood to mean “one before one-hundred”, which would then be ninety-nine. So how do we interpret a Roman numeral such as „MCMXCIX‟? M is not before a larger numeral, so it reads: + 1000 1000 C is before a larger numeral, so it reads: - 100 M is after a negative prefix, so it reads: + 1000 900 X is before a larger numeral, so it reads: - 10 C is after a negative prefix, so it reads: + 100 90 I is before a larger numeral, so it reads: - 1 X is after a negative prefix, so it reads: + 10 9 Thus, „MCMXCIX‟ represents the number 1999.HINDU-ARABIC ROMAN NUMERATION SYSTEMNUMERATION SYSTEM 1 I 2 II 3 III 4 IV 5 V 6 VI 7 VII 8 VIII 9 IX 12
    • 10 X2.4 MAYAN NUMERATION SYSTEMThe Mayan number system dates back to the fourth century and wasapproximately 1,000 years more advanced than the Europeans of that time.This system is unique to our current decimal system, which has a base 10, inthat the Mayans used a base 20. This system is believed to have been used because, since the Mayanslived in such a warm climate and there was rarely a need to wear shoes, 20was the total number of fingers and toes, thus making the system workable.Therefore two important markers in this system are 20, which relates to thefingers and toes, and five, which relates to the number of digits on one handor foot. The Mayan numeration system evolved around A.D. 300. It uses 3basic numerals to represent any possible number: a dot for one, a horizontalbar for 5, and a conch shell for zero. The Mayans were also the first to symbolize the concept of nothing (orzero). The most common symbol was that of a shell ( ) but there were severalother symbols (e.g. a head). It is interesting to learn that with all of the greatmathematicians and scientists that were around in ancient Greece and Rome,it was the Mayan Indians who independently came up with this symbol whichusually meant completion as opposed to zero or nothing. They used the 3 symbols above to represent the numbers from 0through 19 as shown below: 13
    • For number bigger than 19, a number is written in a vertical position sothat it becomes a vertical place value system. Initially, the base used in theMayan numeration system was base 20 and their place values were 1, 20,202, 203,Then, they changed their place values to 1, 20, 20 × 18, 20 2 × 18, 203× 18, ...Using the base 20, 1, 20, 202,203, ..., we can write 20 as follow: In the ones place we have 0 and in the twenties place we have 1, sothe number is 0 × 1 + 1 × 20 = 0 + 20 = 20 Still using a base of 20, we can write 100 as follow: 0 × 1 + 5 × 20 = 0 + 20 = 100 Below is how to represent 2007 14
    • 5 × 202 + 0 × 20 + 7 = 5 × 400 + 0 + 7 = 2000 + 7 = 2007It is started from the bottom, a place value must have a number from the listabove. (1-19)Look carefully and see how it was separated into the place values.Again, it was separated according to numbers that are the list above from 1through 19The number is: 14 + 7 × 20 + 1 × 202 + 3 × 20 3 + 0 × 20 4 + 15 × 20 5 + 5 ×20 6The number is 14 + 140 + 1 × 400 + 3 × 8,000 + 0 + 15 × 3,200,000 + 5×64,000,000The number is = 14 + 140 + 400 + 24,000 + 0 + 48,000,000 + 320,000,000 = 15
    • 368024554 With the base 1, 20, 20 × 18, 202 × 18, 203× 18, ... computation is done the exact same way! Group as shown below: The number is 11 × 1 + 1 × 20 + 10 × 20 × 18 = 11 + 20 + 3600 = 3631 No doubt; the Mayan numeration system was sophisticated.HINDU-ARABIC MAYAN NUMERATION SYSTEMNUMERATION SYSTEM 1 2 3 4 16
    • 5678910 17
    • 2.5 WORD PROBLEM AND ITS SOLUTION Question: One of the natives on the island named Karu. One day, he went out to find food. He collected 41 carrots, 26 clams, 13 fishes and a dozen bananas in a big rattan basket. He felt hungry so he decided to eat 3 carrots and 5 fishes. As he was about to go back home, he fell onto the ground and lost 20 clams and 24 carrots. On seeing Karu was injured, a monkey quickly stole 3 bananas. How many carrots, clams, fishes and bananas left in the rattan basket at last? Answer: Carrots: 41- 3 – 24 = 14 Clams: 26 – 20 = 6 Fishes: 13 – 5 = 8 Bananas: 12 – 3 = 9 18
    • TRANSLATION INTO BABYLONIAN WRITING : . . 19
    • , ,. 20
    • . , . , 21
    • . , , ? 22
    • : : - - = : - = :- = :- = 23