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Number Systems

  One of the most important prerequisites for easy manipulation of numbers is a system of notation which is easily understood and which is clear and unambiguous. This need is reflected in the whole of mathematics (see symbolism). The history of the various different ways which human beings have written numbers is complex. The earliest number systems, like the earliest writing systems, consisted of pictorial representations of the numbers—rather similar to the tally marks used until very recently even in official records. This system is extremely cumbersome, but has the advantage of being easy to understand.

It did not take very long before the notation was made more complicated, but easier to use, usually by adopting a new symbol for the number 10. The reason that 10 was picked is not clear; any number could just as easily be used. It is possible that the number is a reflection of the way that hands were used in counting, so that instead of making a mark for every number, you made a mark every time you ran out of fingers to count with. It is also quite easy to recognize how many marks there are in a tally up to some such figure, but differentiating at sight between, say, 15 and 16 marks is quite difficult.

This paved the way for various ancient number systems which had large numbers of different symbols, such as the Roman (with symbols for 1, 5, 10, 50, 100, 500 and 1000 and complicated rules for picking which symbols to use for a given number) and Greek (which used letters of the alphabet for numbers, so had 27 different symbols altogether). All these systems were cumbersome to use and difficult to read—from that point of view they were worse than the tally systems they replaced. The major defect was their lack of perception of the idea that zero should have a representation.

This defect was first solved by Indian mathematicians a few hundred years later. They evolved the number system used today (it is known as the Arabic system because the Arabs introduced it to the West). In this system, there are ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. For numbers larger than nine, the symbols are combined so that the number that is read is (the number on the right hand end) + 10 × (the number on its left) + 100 × (the number on its left) and so on, moving up to a higher power of 10 with each new column used. This system has many important advantages over the ones known previously. It does not run out of symbols as the numbers get too big, or force large numbers of repetitions of the numbers; this makes them easier to read and uses fewer symbols (it is far easier to understand the number 2378 than MMCCCLXXVIII, its equivalent in Roman notation). Arithmetical operations become far easier; once a few tables are learnt, along with the techniques of long multiplication and long division, it is easy to add, multiply, subtract or divide large numbers. (At least, it is easy compared to earlier systems, in which arithmetic was an occult art confined to a few cognoscenti.) These advantages meant that, after an initial period of conservative scepticism, the Arabic system revolutionized mathematics and science. The work of such men as Kepler, Newton and the other men of science in the 17th and 18th centuries would have been impossible without the invention of the Arabic number system, as would the work of mathematicians who worked on the beginnings of number theory.

As mentioned before, 10 is not the only number which can give rise to a number system like the Arabic one. In fact, any number will do for this; the particular number which is used is known as the base. By far the most used base apart from 10 is 2. This is because there are only two symbols in the base 2, or binary, number system, 0 and 1. The possibility of representing these symbols with electric currents (on for 1 and off for 0) is the only reason why computers are capable of performing arithmetical operations, and therefore one of the major reasons for the success of the computer revolution in many areas. SMcL

See also magic numbers.Further reading John McLeish, Number.



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