## The Indian Decimal Place Value System

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India’s Inspiring Scientific Heritage

A primer series into the development of Jyotiṣa & Gaṇitā in India

Through a series of short articles, we have set forth to cover a few highlights in the development of Jyotiṣa (Astronomy) and Gaṇitā (Mathematics). Having had delved into the Vedic period, we had moved ahead in the timeline and presented an introductory glimpse into the phenomenal breakthroughs in Chandas-śāstra composed by Piṅgala-nāga around 3rd century BCE.  Then we had an article that shall cut across the timelines of the three eras of development of mathematics in India in order to appreciate how India’s contribution to approximating π is immensely significant. Before delving into great Indian mathematicians and their marquee works in this series, we will cover the ingenious number systems of our Gaṇitā tradition.

Appreciating the decimal place value system

The contribution of “zero” by Indians to the world of mathematics is often said as a statement without much thought to it or even used as a sly remark sometimes. This stems from a certain lack of perspective. Zero plays a highly crucial role in the “base-10” numbers or the decimal place value system and it is this system that thoroughly enables easy computation and it is foundational to all of mathematics as we know it today.

Imagine doing arithmetics in Roman Numerals or other systems such as base-2, base-12 or even base-60! While all other systems have value, it is the decimal place value system that has unshackled humanity from inaccess to easy computation and zero place a vital role in the design of this base-10 system that is the bedrock of all divisions of math as we learn it today!

The Indian Mathematicians developed the decimal place value system along with the notion of the zero-number. The place value system is essentially an algebraic concept:

 5203 = 5.10 3 + 2.10 2 + 0.10 1 + 3 is analogous to 5x 3 + 2x 2 + 0x 1 + 3x 0

It is this algebraic technique of representing all numbers as polynomials of a base number, which makes all the calculations systematic and simple. The algorithms developed in India for multiplication, division and evaluation of square, square-root, cube and cube-root, etc., have become practically easy only because of the design of the number system. We must appreciate that they have contributed immensely to the simplification and popularisation of mathematics the world over.

Antiquity of the decimal place value system

Generally most of us do not get to know or have opportunities to get to know answers to questions like:

• When did we start counting?
• Were there other systems of counting?
• What are the different ways of representing numbers? etc.,

As we keep using the decimal system of numeration right from our childhood, we are so familiar with it that we tend to think that it has been there forever. It is indeed pretty old. But how old?

One of the most ancient literature Ṛg-veda presents the number 3,339 using word numeration:

 विबुधनेत्रगजाहिहुताशनत्रिगुणवेदभवारणबाहवः। Three thousand three hundred and thirty-and-nine deities worshipped Agni … [Ṛg-veda 3.9.9]

From such quotes it is evident that the decimal place value system has been in vogue amongst the Vedic seers.

Enumeration of powers of ten

In the Vedic era, there is abundant evidence of usage of the decimal number system along with enumeration of powers of tens that indicate an active representation of very large numbers.

In the Yajurveda-saṃhitā (Vājasaneyi, XVII.2) we have the following list of numeral denominations proceeding in the ratio of 10, that gives the name of the various place values in the decimal system:

 eka (1), daśa (10), śata (100), sahasra (1000), ayuta (10000), niyuta(10 5 ), prayuta (10 6 ), arbuda (10 7 ), nyarbuda (10 8 ), samudra (10 9 ), madhya (10 10 ), anta (10 11 ), and parārdha (10 12 ).

The same list occurs in the Taittirīya-saṃhitā (IV.40.11.4 and VII.2.20.1), and with some alterations in the Maitrāyaṇī (II.8.14) and Kāṭhaka (XVII.10) Saṃhitās and other places.

The numbers were classified into even ~ yugma, literally meaning ‘pair’ and odd ~ ayugma, literally meaning ‘not pair’. The zero has been called kṣudra (trifling). The negative number is indicated by the term anṛca, while the positive number by ṛca, in the Atharvaveda (XIX.22, 23).

It is indeed enthralling to know in depth about the rich scientific heritage of the Indian civilization. We shall continue to see other breakthroughs and developments in Indian mathematics in the following editions of Parnika.

Aum Tat Sat!