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.

Need for Large Numbers & Number Systems

But human civilizations had to deal with large numbers to keep track of population, crop yields, land taxes, animal population etc. We can even say that the bureaucratic need for measuring & taxing speeded up the need for representing large magnitudes.

Hence apart from numerals for small numbers, they had to invent many independent ideas to represent large numbers, of which two have survived till date. Two of the systems which stood the test of time can be called the "aggregating systems" and the "place value systems"

In this chapter we will see the Greek number system, which never went beyond Natural Numbers & simple fractions and Roman Number System which was an aggregating system.

Greek Number System

Greeks seem to have focused 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 developed a system based on powers of ten to represent large numbers.

They also never accepted that the concept of "nothing" is a mathematical idea. So their number system did not have zero.

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.

In a previous chapter we have dealt in detail about the idea of magnitudes used by Greeks.

Greeks & Mathematics

At this point we will look very briefly at the contribution of Greeks to the discipline of math.

Egyptians, Sumerians & Babylonians used math mostly for "practical" purposes. The Hindus also continued this tradition.

Greeks discovered that ideas of arithmetic and geometry can be dealt with in the abstract.

But most of their efforts, as we will see in detail in later chapters, was in the development of geometry culminating in the compilation of "The Elements" by Euclid.

After the work of Greeks, there was very little advancement in mathematics until the Renaissance in the 15th century. In fact most of the Greek work was "lost" and regained through the work of Muslim scholars in Arab-occupied Spain.

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.

Roman Number System Does not need a Zero

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).

Limitations of Aggregating Systems

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.

Aggregating Systems Make Computations Complicated

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.

Let students actually experience this by trying to add two 3-digit numbers using the Roman Number System.

Using the abacus was a complicated process. It was also not easy for a common man to be well versed in the ancient number systems. Hence expert "abacists" were required. This "expertise in abacus" gave certain groups "social power" much like that of astrologers who could study horoscopes. It also meant that mathematics was a tool for those in the positions of power.

Computational Results Cannot be Easily Documented

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

Rudimentary Place-Value idea in the Roman Number System

Even the Roman system employs a rudimentary "place value system". For example, XI meant eleven whereas IX meant nine. When "I" is placed to the right of a number it 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 detail in the next chapter.

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