Aggregating Systems

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

Need for Large Numbers & Number Systems

After the establishment of river valley civilizations & the development of agriculture, 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 and taxing land & produce speeded up the need for representing large magnitudes.

Archaelogical excavations in Egypt have provided a wealth of documentation of every aspect of their society. They also show that scribes were held in high esteem.

Hence apart from numerals for small numbers, cultures had to invent many independent ideas to represent large numbers, of which two have survived till date. One of them can be called the "aggregating systems" and the other "place value systems."

Aggregating Systems were easier in concept and were the first to be invented.

In this chapter we will briefly see the Egyptian, Greek & the Roman number systems, which were aggregating systems. They never went beyond Natural Numbers & simple fractions.

Aggregating Number Systems

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 of these were also "decimal" or based on "collections of ten." 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 an "aggregating system" there is no need for a number Zero or a separate symbol for zero, as zero does not change the value of any aggregate, whatever be its size.

Egyptian Number System

It had symbols/ hieroglyphs for ten and multiples of ten up to a million.

Much of our understanding of Egyptian math comes from the discovery of the Rhind Papyrus.

Rhind Papyrus is also called the “Ahmes Papyrus” in honor of the scribe who compiled it. The papyrus is from the Egyptian Middle Kingdom and dates to around 1650 BCE. It was purchased by Henry Rhind in Egypt in 1858 and placed in the British Museum in 1864 by the estate of Henry Rhind, thus it bears his name.

It can be called the oldest surviving math textbook. Ahmes himself says it is a compilation of math knowledge from the previous 500 years.

It contains about 84 problems with solutions. It is not surprising that 6 of the problems are to do with calculating the slopes of pyramids. It contains problems on calculation of area & volumes and multiplication & division.

It also contains problems which indicate an understanding of the theorem was later known as Pythagoras Theorem.

Greek Number System

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

But Greeks discovered that ideas of arithmetic and geometry can be dealt with in the abstract. They gave spiritual meanings to numbers and the geometric configurations in which they could be arranged.

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.

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

The first nine numbers (what we write as one to nine) were represented by the first nine Greek alphabets. The next nine alphabets represented "tens" (what we call ten to ninety) and the next nine alphabets represented hundreds (from hundred to nine hundred).

They never accepted that the concept of "nothing" is a mathematical idea. Hence in spite of their advances in geometry, they did not consider zero as a number. This possibly hindered their invention of the idea of a place value system.

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.

After the work of Greeks, there was very little advancement in mathematics in Europe 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 was an improvement on the Greek system. Whereas Greeks used a large number of their alphabets to represent numerals, Roman system used only 7 alphabets/ numerals.

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. M, D, C etc are the numerals of the system. It is only a convention that the numerals are written in descending order of their magnitude.

It can very well be also written as VCCCCMMM. The value of "X" does not change wherever it occurs. 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.

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.

Limitations of Aggregating Systems

Limitations on the magnitudes

With an existing set of numerals, an aggregating system can, for practical purposes, represent only numbers up to a certain magnitude.

In theory we can use a large number of numerals to represent any magnitude. For example a quadrillion can be represented by a string of 'M's. But the number of 'M's used will make the number difficult to even remember!

The magnitude of numbers normally required by a society depends on its economic, social and scientific needs, and this may keep on increasing. New numerals can be added as and when the need arises.

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 & 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 & 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. This is impractical in practice.

Computations are Complicated

An Aggregating system like 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 became a tool for those in the positions of power.

Results Cannot be Easily Documented

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

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