Example research essay topic: Place Value Value System - 1,626 words

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The Development and Use of Zero as a Number The history of the number "zero" is not easy to track, since we have little written evidence. Books were not widely used, even for study, until after the invention of the printing press, which is why we have so little recorded history prior to the fifteenth century. However, it seems that there were several areas of the world involved with using a "zero" by the third century. The Babylonians are generally thought to be the earliest to use a zero, around 250 BC. However, the Chinese, the Hindus, and the Mayans were using the zero at different times not too distant. Western civilization did not incorporate the zero into their culture until at least the twelfth century.

The number zero has two uses: as a null character or place holder, and as a quantity. As a null character it makes counting easier and mathematical calculation by place possible. As a quantity, zero simplifies mathematical calculation and makes it possible to work with large numbers. Using zero simply as a place holder limits calculation by the width of the medium on which the numbers are written, because this is, essentially base one, using no character by putting in a place holder, or one character. Using zero as a quantity makes it possible to use other number bases than base one. Ths simplest would be base two, which uses zero and one.

Our system uses nine characters and the zero for base ten. The zero as a quantity makes this possible. Place value notation was the first use of the concept of zero, in that several civilizations developed a character as a place holder when no number was entered. "While four civilizations seem to have discovered place-value notation, three of them never quite reached the simplicity of our current Arabic numerals. For this notation only becomes highly efficient in conjunction with three other inventions: a symbol for "zero, " a unique base number, and the discarding of the addition principle for the digits I through 9. Consider, for instance, the oldest place-value system known, devised by Babylonian astronomers eighteen centuries before Christ. Their base number was 60.

Hence a number such as 43, 345, which is equal to 12? / 602 2? 60 25, was expressed by concatenating the symbols for 12, 2, and 25. (Dehaene 99) That is, the zero is used to occupy a space in a number system which uses place as a value. Ours is an example of such a system, where each place has the value of ten. The Babylonians used a "base sixty" system, (as opposed to our base ten), where each place was worth sixty. In principle, sixty distinct symbols would have been needed, one for each of the "digits" 0 to 59. Yet obviously it would have been impractical to learn sixty arbitrary symbols. Instead, the Babylonians wrote down these numbers using an additive base- 10 notation.

For instance, the "digit" 25 was expressed as 10 10 1 1 1 1 1. Eventually, the number 43, 345 was thus rendered by an obscure sequence of cuneiform characters that literally meant 10 1 1 [implication? 602 1, 1 1 [implication? 60 ], 10 10 1 1 1 1 1. Such a mixture of additive and place-value coding, with two bases 10 and 60, turned the Babylonian notation into an awkward system understandable only to a cultivated elite. " (Dehaene 99) Using this system a character was often used to denote a place where no number was used. This was a rather complicated method for coding, but it used only two symbols for a place holder.

Originally they simply left a space, but that was confusing, so later we can see the use of a tiny hollow dot as a place holder. "About 130 A. D. , Ptolemy in Alexandria used in his Almagest the Babylonian sexagesimal fractions, and also the omicron o to represent blanks in the sexagesimal numbers. This o was not used as a regular zero. It appears therefore that the Babylonians had the principle of local value, and also a symbol for zero, to indicate the absence of a figure, but did not use this zero in computation. Their sexagesimal fractions were introduced into India and with these fractions probably passed the principle of local value and the restricted use of the zero. " 7 (Cajori 1919, 5) One might wonder why base 60. Well, we can infer that it was due to the development of their calendar and the sundial.

When we talk about time we use base 60. There are 60 minutes to an hour and 24 hours in a day. These are both interesting multiples of five (the number of fingers on one hand). In western mathematics we developed a base ten system, since we stared by using our fingers. Who is to say that the Babylonians may not have simply used only one hand? There are some remote Slavic cultures (Lemon, for example) whose peasants still count on eight fingers, not using the thumbs, and they developed a base eight system.

However, due to the constant political change over the centuries, this system never got written down, and only a few still remember it. (This information was gathered from direct observation of a friend's grandfather who counted only on his fingers. When asked, he said he had always done that, because his father did, but it was never taught in the schools. ) It becomes clear when we look at the history of zero that using it as a quantity developed separately, and later. We have evidence of zero from tablets in the Selucid era (4 th to 1 st C BC) in Babylon, but there is no evidence that they ever used the zero as an actual number signifying no quantity. (Mc Quillin 1997) They used clay tablets to write on and used a wedge shaped character, a writing system called cuniform. On tablets recovered and dated between 1700 BC and 600 BC we see the wedge used sideways, alone or in pairs, and another symbol at time, which looks like a comma, to mark a place where no number is entered. However, because there is no case where these characters are found on the end of a line, we know that they were not using zero as a quantity, but only as a place holder. We can assume they used context to determine the actualy number, much like we do.

If you ask the price of a house and get a reply of four-fifty, you know it means $ 450, 000, and not $ 4. 50. The Mayan, famous for their calendar, also developed the use of a place holder zero. The Mayans, native inhabitants of Central America, were highly skilled mathematicians, astronomers, artists and architects. However, they failed to make other key discoveries and inventions that might have helped their culture survive. They never used the plow or metal tools and their civilization collapsed mysteriously around 900 CE. They had a very complex calendar system and needed a placeholder in their elaborate date system.

This lead to their invention of zero 600 years and 12, 000 miles removed from the Babylonians. The Mayans had several calendars. There was a 365 day civil year, a 260 day religious year and, key to their invention of zero, the complicated Long Count calendar which measured time from the start of the Mayan civilization (August 12, 3113 B. C. ) and completes a full cycle on December 21, 2012.

Mayan Long Count Units kin day until month 20 days tun year 360 days (18 months) kate 20 tuns 7200 days (20 years) baton 20 kahuna 144, 000 days (400 years) picture 20 batons 2, 880, 000 days (8, 000 years) calabtun 20 pictures 57, 600, 000 days (160, 000 years) kinchiltun 20 calabtuns 1, 152, 000, 000 days (3, 200, 000 years) al autun 20 kinchiltuns 23, 040, 000, 000 days (64 million years) It is the formal Long Count calendar that brought about the zero. The Mayan numerals were very complex in formal use painted or carved heads or even full figures were used to represent numbers. When using these ornate carvings on a style, or stone tablet, the Mayans had a rather rigid graphic layout; each period of time had a space and all the spaces needed to be filled in. So a date that was 8 batons, 14 kahuna, 3 tuns, 0 units and 12 kins had to have one figure for each place.

The zero was often represented by a shell shape. Despite the use of zero in the place value system, it was never used for calculations. Once again, this stems back to the calendar. You may have noticed in the chart above that a 360 day year is 18 months (20 days to a month).

This irregularity messed up an otherwise tidy vegisimal (base 20) system: Decimal 10 is (1 x 10) + (0 x 1) = 10 Vegisimal 10 is (1 x 20) + (0 x 1) = 20 Mayan 10 is (1 x 20) + (0 x 1) = 20 Decimal 100 is (1 x 10 exp 2) + (0 x 10) + (0 x 1) = 100 Vegisimal 100 is (1 x 20 exp 2) + (0 x 20) + (0 x 1) = 400 Mayan 100 is (1 x (18 x 20) ) + (0 x 20) + (0 x 1) = 360 Decimal 1000 is (1 x 10 exp 3) + (0 x 10 exp 2) + (0 x 10) + (0 x 1) = 1000 Vegisimal 1000 is (1 x...

Research essay sample on Place Value Value System