Number Systems 1

< 5.8 Cardinal &amp; Ordinal Numbers | Topic Index | 6.2 Number Systems 2 >

We have seen the development of Number Sense in children and the representation of numbers up to 9 in various ways, starting from the concrete and proceeding towards abstract.

But human civilisations had to deal with large numbers to keep track of population, crop yields, land taxes, animal population etc. Hence apart from numerals for small numbers, they also invented many independent ideas to represent large numbers, of which two have survived till date. They can be called the &ldquo;aggregating system&rdquo; and the &ldquo;place value system&rdquo;.

We will deal in this chapter with the Greek number system & Roman Number Systems.

Greek Number System

Greeks seem to have focussed most of their math thinking on Geometry, where they made significant contributions.

Their Arithmetic was fairly elementary. For them, only the Natural numbers 1, 2, 3..... etc were "numbers". They did not develop a separate set of numerals. They used letters from their alphabet.

They did not consider fractions to be numbers, though they were aware of the use of fractions in other cultures. For them fractions were like what would call as ratios today. They were relations between magnitudes.

Roman Number System

The Roman Number System is an aggregating system.

In this system many symbols were invented to represent larger collections. Using these symbols, numbers for very large numbers could be written as a sum or aggregate of a few of these symbols. This idea could have arisen from the fact that intuitively we count large collections is by breaking them into smaller collections. We count the smaller collections first and later aggregate them.

Most civilisations seem to have thought of Ten or multiples of Ten or powers of Ten for these large magnitudes and invented symbols for these. The obvious reason for this decision seems to be that all humans generally are born with had ten fingers, which are used for any counting or tracking.

In the Roman Number System Thousand is written as M, Five Hundred as D, Hundred as C, Fifty as L, Ten as X, Five as V and One as I. Three Thousand Four Hundred Five is represented as MMMCCCCV. It can very well be written as VCCCCMMM. The order in which these symbols are written does not matter as each symbol represents a definite quantity, the number being the total of the individual values.

In an &ldquo;aggregating system&rdquo; there is no need for a separate symbol for 0, since 0 is not required in representing any of the smaller aggregates, the smallest being I (One).

An aggregating system can practically represent only numbers up to a certain magnitude with a combination of the difference standard values. But the magnitude of numbers required by a society depends on its economic, social and scientific needs, and this may keep on increasing.

The highest known number in the Roman System with a unique name or symbol was Myriad, which was actually equal only to Ten Thousand. But Myriad was also used in daily language to denote a very large uncountable magnitude. Hence given the state of their economic progress, Greeks &amp; Romans do not seem to have needed numbers which were more than Ten Thousand for their daily needs. Archimedes, while trying to estimate the number of sand grains in the Universe, had to invent a unit called Octad which was equal to a Myriad Myriad which in our system would be equal to a hundred million.

Hence as the need to handle larger and larger numbers increases, the only solution would be to invent more symbols &amp; names with values larger than M. Hence, we can say that in such a system, an endless number of symbols may be required to represent any magnitude. But this is impractical in practice.

The Roman number system is also not suited for paper-pencil computations. But for many centuries this was not a very significant handicap. Numerals were used mostly to document & communicate quantities. Computations were done mostly using manipulatives or physical devices like an abacus.

Using the abacus was a complicated process and hence expert &ldquo;abacists&rdquo; were required.

But with an abacus, the calculations could not be documented on a paper for future reference. For merchants this was a major drawback.

Even the Roman system employs a rudimentary &ldquo;place value system&rdquo;. For example, XI meant eleven whereas IX meant nine.&rdquo; I&rdquo; when placed to the right of a number acted as +1 and when placed to the left of a number, acted as -1. This idea of numbers reducing to the left and increasing to the right seems to a feature of the human brain. We can see its effect on the number line concept where numbers increase to the right. We also see it in the fact that most languages are written in a left to right manner.

We will see the Place value system in the next chapter.

< 5.8 Cardinal &amp; Ordinal Numbers | Topic Index | 6.2 Number Systems 2 >