Number Systems 2

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In contrast to the "aggregating system", Indians developed the "place value system".

Partial Place Value System of Sumerians & Egyptians

Sumerians & Egyptians and possibly Mayas had invented a place value system, which we can call as an "incomplete" one. It used a "number base" of sixty.

The central idea of a "place value system" was that the value of a symbol (numeral)would change depending on where it was placed with respect to the other symbols.

Sumerians used the concept of "zero or nothingness" only as a place holder and not as a number.

Hence in place of what we write as 0, a blank space was left. This caused confusions while representing numbers.

It also did not simplify number computations. We will look at these issues in detail in chapter 6.8 "Zero & the Place Value System".

There is no consensus on why the base sixty was chosen by Sumerians. But the system survives today in the way we measure daily time intervals in hours, minutes & seconds.

The Hindu Place Value System

In contrast the place value system invented by Hindus was "complete" in all respects.

It was a decimal or a Base Ten system. This made it unnecessary to invent separate symbols for numbers more than 9.

It also invented a symbol for zero and considered it as a number on par with numbers 1 to 9. Hence the number of numerals in the system became ten (0 to 9)

In a way the idea of place value was embedded in the Indian tradition & thinking for a long time before it was expressed in representing numbers.

The Hindu epics & puranas talk about very large numbers (equivalent to billions &amp; trillions) to describe time intervals associated with creation and Gods. Hindu philosophy had a very important role for the idea of "emptiness" or "Sunya".

How Hindu Numerals Became Hindu Arabic Numerals

This system with ten numerals was already in use in India by the 6thcentury AD. Brahmagupta had already given results of operation with 0, in his Brahmasputa Siddhantha.

In the 8thcentury, it reached Baghdad through Arab merchants & scholars. Baghdad was being developed by the then Calif as a centre of learning. Muhammad ibn Mūsā al Khwarizmi, a scholar from Persia, working here, wrote a book called al-Kitāb al-mukhtaṣar fī ḥisāb aljabr wal-muqābala (The Compendious Book on Calculation by Completion and Balancing) which introduced the Hindu number system and computations with it.

The name of the book also gave the name "algebra" to a method of solving equations.

Through Spain, where Islamic and European scholars were working together, Khwarizmi's book on the Hindu numerals reached medieval Europe. The oldest book on Arithmetic using these numerals & the place value systems, is supposed to be that of Avicenna around 1000 AD.

Leonardo of Pisa (of Fibonacci series fame) was introduced to algebra through al-Khwārizmī's book. He introduced these numerals along with the place value system, to Europe, through his book Liber Abaci in 1202.

Before this number system reached Europe, the roman number system was used. Because computations using this system were very difficult, the abacus was used for these purposes. There were abacists who had mastered the use of abacus in computations.

Initially the unfamiliar numeral shapes for numbers 1 to 9 and the strange 0 created a lot of resistance to the use of these numbers. Added to it was that these numbers (ultimately called Hindu-Arabic) were received through Arabs, who were considered as enemies by Christians!

Over a period of time, when their enormous advantage became obvious, the Hindu-Arabic numerals were accepted. What were these advantages?

'Algebraic Structure & Ease of Documentation of Computations

The algebraic structure of the system made it easy to develop simple algorithms for doing all the 4 operations. The algorithms were simple enough for a lay person to learn to perform computations. It is because of the simplicity that all the 4 operations are taught all over the world in schools at the primary school itself!

The biggest advantage for merchants was that the computations could be documented on paper for future reference. This is very much like a teacher correcting the math answer papers of her students several days after the examination!

Hence the Indian system was adopted rapidly in European commerce. It has been eventually adopted all over the world. Ultimately the algorists won the battle of the number systems over the abacists.

Printing Press & Computations With Hindu-Arabic Numerals

The elite in any society is interested in controlling the spread of knowledge, which they see as a threat to their entrenched positions. Mastery of numerical computations, using the abacus and other means, were seen as as secrets to be preserved by a few.

The invention of printing helped in spreading Hindu-Arabic numerals & computational algorithms with them, rapidly in the Middle Ages. Since computational procedures could be demonstrated with written examples, the same could be printed as textbooks also!

So plenty of arithmetic textbooks were written, printed and widely sold.

Before dealing with place value in the numbers, we will first see the idea of place value as is used in daily life contexts other than numbers. This will make the idea easier to understand when applied to numbers.

< 6.1 Number Systems 1 | Topic Index | 6.3 Place Value in Daily Life >