# Number Systems used in Ancient India

India’s contribution to the field of mathematics started before 1200 BC as recorded and it still prevails in the current century. As per the documented texts and archeological evidences available in hand, we have come up with a definite period to say that the contribution started around 1200 BC. But the fact is that there have been application of mathematical concepts and knowledge on mathematics seen even in Vedas. Rigveda being one of the oldest books in the world has various places or verses where application of concepts from mathematics can be seen. Though we will not be able to see detailed explanation about any of the concepts, we will be able to see verses describing about the calculations and concepts used. Some of the great scholars who came after 1200 BC have made enormous contributions to the field of mathematics. This article will describe about the number systems used in Ancient India.

We all know very well for the fact that modern discoveries and inventions have found out various methods to represent formulas and numbers. In ancient India, everything was represented only in the form of verses in Sanskrit and other regional languages. In order to represent numbers and formulas in the form of verses, a high level of perfection was required. There were three main number systems used in Ancient India to represent numbers in the verses. There are also references about other number systems but more details are not available about them. The meter in the verses were also followed strictly to maintain perfection in what they were trying to convey. Representing all the numbers in the form of verses was challenging. That is the reason why they had to strictly follow a number system to represent formulas and numbers. With the help of these number systems the ancient scholars have represented all the types of mathematics which include formulas, numbers, algebra etc., in the form of verses easily. There were three major types of number system with Aksharapalli contributing to total of 4 number systems used by Ancient Mathematicians namely:

- Aryabhateeya number system
- Bootha Sankya system
- Katapayadi number system
- Aksharapalli

In ancient India, the scientists, astronomers and mathematicians had to deal with huge numbers for their calculation. That is the reason why they had to invent these number systems making the calculation easier. Apart from the application of these number systems in representing formulas and calculations, these number systems were extensively used by the ancient astronomers and even poets as an encryption tool to silently represent a number or code. One best example is a verse that praises Lord Krishna but if Katapayadi numbers are substituted, the verse provides decimal value of pi. Scholars from ancient India have taken these number systems as a freedom to represent numbers in different ways along with verses that actually provides a different straight meaning.

**Aryabhateeya number system**

Aryabhata’s contribution to the field of mathematics is really commendable. He has shared his views with detailed explanation on some of the advanced concepts in mathematics and the field of astronomy. Aryabhata has used his own unique number system to represent his mathematical and astronomical calculations in the form of verses. It will take less than half hour for to learn Aryabhateeya number system for someone who know Sanskrit alphabets. Aryabhata has detailed about this number system clearly in the second stanza in his book Aryabhatiya.

**Bootha Sankya system**

Bootha sankya system was one of the interesting methods for representing numbers used by ancient mathematicians. The word Bootha Sankya can be split into Bootha + Sankya where the numbers were represented with the help of some elements. In other words it can be said that some elements of nature can were used to represent a set of numbers. Some of the examples for Bootha Sankya number system are

Eyes – 2

Veda – 4

Chandra – 1

Prithvi – 1

Bootha – 5

So in the above examples, if we take eyes, every human has two eyes. If we take the word veda, there are totally 4 vedas namely Rig, Yajur, Sama and Atharvana. If we take Chandra, there is only one moon for the world. If we take Prithvi, there is only one earth. If we take Bootha, there are 5 boothas namely air, water, land, space and fire. The list goes on. The representation doesn’t end with just one word. Any synonyms for the above words will also be used to represent the same number as Sanskrit is very rich in synonyms. In some of the verses the combination of such words may not make any sense as a sentence but when it is substituted with the actual numbers it will give wonderful values.

**Katapayadi Number system**

Katapayadi number system is another technique of using alphabets as numbers. From the history we can see that Katapayadi Sankhya was extensively used by the astronomers and mathematicians in ancient India. Though it is not still clear about the origin of Katapayadi number system, we can see that it is extensively employed in the schools of Kerala. Compared to Aryabhateeya number system, there are lots of advantages available in using Katapayadi number system wherein by combining vowels and consonants we will be able to create meaningful words and sentences with which a different meaning can also be used for representation. It is believed that Katapayadi number system was used extensively in ancient India for encryption purpose. For some of the normal verses if we substitute Katapayadi numbers, we will be able to get a number that might even provide more details about the period of the work.

**Aksharapalli**

This is another form of letter based number system used in ancient India. This system can be found in the manuscripts from 6th century AD. It is believed that this system was used extensively to save space when representing something in the form of manuscripts. Such manuscripts were mostly found in Nepal and Kerala region of India. It is also believed that the system might have evolved from the period of usage of Brahmi numeral system.