The Indian Decimal Place Value System

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:

1. When did we start count­ing?

2. Were there oth­er sys­tems of count­ing?

3. 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!