Ten symbols, a place, and a mark for nothing. It is the most consequential piece of notation ever devised, and every arithmetic operation you have ever performed depends on it. This chapter follows it from the Indus brick to the Gwalior inscription to Fibonacci — and then turns to the units in which the subcontinent measured length, time and weight, and to a treatise on poetry that invented binary.
There is a temptation to treat a numeral system as a labelling convention — arbitrary, like choosing a font. It is not. What you can compute depends on how you write.
Pierre-Simon Laplace put the point as well as anyone has:
The ingenious method of expressing every possible number using a set of ten symbols — each symbol having a place value and an absolute value — emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated.
Two things are being credited there, and they are separable. Absolute value: each of ten symbols stands for a quantity. Place value: the same symbol stands for a different quantity depending on where it sits. The second is the hard idea, and it does not work without a way of marking an empty place — which is why zero and place value arrive together, and why neither is useful alone.
Al-Bīrūnī, writing in 1030 CE after a long stay in India, noticed the practical consequence at once: “Whilst we use letters for calculation according to their numerical value, the Indians do not use letters at all for arithmetic.” Separating numerals from letters is what makes columnar calculation possible.
Abstract claims about notation become obvious the moment you try to compute in the alternative.
The idea is explained by analogy in the Śārīraka-bhāṣya, in a passage about identity and relation. One and the same Devadatta, says Śaṅkara, is called by many names — man, brāhmaṇa, learned, generous, child, youth, elder, father, son, grandson, brother, son-in-law — according to his own nature and his relations. And then:
यथा च एकापि सती रेखा स्थानान्यत्वेन निविशमाना एक-दश-शत-सहस्रादि-शब्दप्रत्ययभेदम् अनुभवति ।
yathā ca ekāpi satī rekhā sthānānyatvena niviśamānā eka-daśa-śata-sahasrādi-śabda-pratyaya-bhedam anubhavati
“Just as one and the same stroke, being placed in different positions, comes to be called one, ten, a hundred, a thousand.” The place-value system is being used here as the obvious, everyday thing that explains the difficult metaphysical thing — which tells you how thoroughly it had been absorbed.
Zero has to be invented three times before it is finished: as a gap, as a symbol, and as a number you can compute with.
In the Brāhmasphuṭa-siddhānta (628 CE) Brahmagupta states, correctly: a quantity minus itself is zero; zero added to or subtracted from a quantity leaves it unchanged; zero times anything is zero. He also states that zero divided by zero is zero — which is wrong, and was corrected later. Bhāskara II, five centuries on, argues that a quantity divided by zero is an unchanging infinite magnitude, and offers a genuinely modern-sounding image: like the infinite and unchanging Absolute, it neither increases nor decreases when things are added to or taken from it. The point is not that they got it right. It is that the question was being asked at all, which requires zero to have already become a number rather than a hole.
Claims about early Indian numeracy do not rest on texts alone. Three lines of physical evidence are worth stating precisely.
Excavations at Harappa, Mohenjo-daro, Dholavira and Lothal yield fired bricks in a consistent length : width : depth ratio of 4 : 2 : 1. A ratio maintained across sites hundreds of kilometres apart implies a shared standard and the means of enforcing it.
Streets measured at 1.8, 3.6, 5.4 and 7.2 metres — exactly 1, 2, 3 and 4 units of a common module. The Arthaśāstra later names two such units for roads and distances: the dhanus of 96 aṅgulas and the gārhapatya-dhanus of 108.
A legal document from Bharukaccha (Broach) dated 594 CE carries a number in place-value form. The Gwalior inscription of 876 CE has the zero. Datta and Singh catalogue 33 inscriptions and grant plates in decimal place-value notation between 595 and 975 CE.
Standardised brick ratios and modular street widths demonstrate systematic measurement and a shared standard. They do not by themselves demonstrate a place-value numeral system — the Indus script is undeciphered, and nobody can currently read its numbers. Keeping these two claims apart is the difference between a solid argument and one a specialist can dismantle in a sentence.
Before a numeral is written it has to be said. Sanskrit's naming scheme is fully systematic, which is more than can be said for English.
Bhāskarācārya opens the Līlāvatī by naming the places, each ten times the last, up to 1017:
एक-दश-शत-सहस्र-अयुत-लक्ष-प्रयुत-कोटयः क्रमशः ।
अर्बुदम्-अब्जं-खर्व-निखर्व-महापद्म-शङ्कवस्तस्मात् ॥
जलधिश्च-अन्त्यं-मध्यं-परार्धमिति दशगुणोत्तरं संज्ञाः ।
संख्यायाः स्थानानां व्यवहारार्थं कृताः पूर्वैः ॥
Līlāvatī, opening verses — “these names of the places of number, each ten times the last, were made by the ancients for the conduct of affairs”
Mahāvīrācārya, in the Gaṇita-sāra-saṅgraha of about 850 CE, states the result of a certain squaring not as digits but as a description: एकादिषडन्तानि क्रमेण हीनानि — “beginning with one and going up to six, then decreasing by one”. The number is 12345654321, and it is the square of 111111111. You can only describe a number that way if you and your reader both hold the place-value picture firmly in mind.
The tradition has an unusual appetite for the very large. Numbers appear in Indian texts that had no counterpart elsewhere for a thousand years.
You need names for large numbers when you need to talk about them, and you need to talk about them when your cosmology is large. Indian time-reckoning runs to the kalpa of 4.32 billion years (section 14), and Jaina cosmology is larger still. Compare Greek and Roman practice, where the largest ordinarily named number is the myriad, 104 — Archimedes had to construct an extension specially, in the Sand Reckoner, to say how many grains of sand would fill the universe.
A number written in digits cannot be fitted to a metre. A number written as a chain of ordinary nouns can. That is the whole motivation.
Bhūta is an entity, saṃkhyā a number. The system replaces each digit with a word for something the world reliably contains in that quantity, and — because synonyms are allowed — the composer picks whichever word scans.
| Category | Examples | Value |
|---|---|---|
| The number word itself | śūnya, eka, dvi, tri … nava | 0–9 |
| The physical world | kha, ākāśa (sky) candra, indu, śaśi, vidhu, soma (moon) netra (eyes), pakṣa (lunar fortnights) agni (the ritual fires) diś (directions) | 0 1 2 3 10 |
| Animals | gaja, vāraṇa, nāga (elephant — the eight of the quarters) ahi (serpent — the eight nāgas) aśva (horse) | 8 8 7 |
| The body | bāhu (arms) indriya (senses) dhātu (bodily tissues) randhra (apertures) | 2 5 7 9 |
| Deities and sages | vibudha, deva (the thirty-three gods) muni, ṛṣi (the seven sages) rudra (the elevens) | 33 7 11 |
| Concepts and cycles | veda, yuga (world-ages) guṇa (the three strands) ṛtu (seasons), rasa (tastes) bha, nakṣatra (asterisms) | 4 3 6 27 |
अङ्कानां वामतो गतिः — aṅkānāṃ vāmato gatiḥ, “of digits, the movement is leftward”. The first word named is the units digit, the second is the tens, and so on. Read a bhūta-saṃkhyā number left to right and you get it backwards, every time.
The finest demonstration of the system: Mādhava of Saṅgamagrāma's value for π, given to eleven decimal places, encoded entirely in nouns.
विबुधनेत्रगजाहिहुताशनत्रिगुणवेदभवारणबाहवः ।
नवनिखर्वमिते वृत्तिविस्तरे परिधिमानमिदं जगदुर्बुधाः ॥
vibudha-netra-gaja-ahi-hutāśana-tri-guṇa-veda-bha-vāraṇa-bāhavaḥ |
nava-nikharva-mite vṛtti-vistare paridhi-mānam idaṃ jagadur budhāḥ ||
“The wise have declared this to be the circumference of a circle whose diameter is nine nikharva.”
The alternative encoding, and the more flexible one: map consonants to digits, and a number becomes a pronounceable — sometimes meaningful — word.
| Word | Split | Digits | Number | Note |
|---|---|---|---|---|
| भवति | bha–va–ti | 4 – 4 – 6 | 644 | Vowels are ignored; each consonant contributes. |
| शक्त्यालोके | śa–ktyā–lo–ke | 5 – 1 – 3 – 1 | 1315 | In ktyā, k and t are dropped; only y, the last consonant before the vowel, counts. |
| सर्वार्थशीलस्थिरः | sa–rvā–rtha–śī–la–sthi–ra | 7 – 4 – 7 – 5 – 3 – 7 – 2 | 2,735,747 | Seven syllables carrying a seven-digit number, in a phrase that means something. |
| आयुरारोग्यसौख्यम् | ā–yu–rā–ro–gya–sau–khya–m | 0 – 1 – 2 – 2 – 1 – 7 – 1 | 1,712,210 | The initial ā stands alone, so it is 0; the final m has no vowel after it, so it is dropped. |
The Carnatic music tradition names its seventy-two parent scales by this code. The first two syllables of a melakartā name give its number: Dhīra-śaṅkarābharaṇam begins dhī–ra → 9, 2 → read right to left, 29. It is the twenty-ninth. A musician who knows the code can locate any of the seventy-two from its name alone, without a table.
The measurement systems share a striking design decision. Length, time and weight are each built up from a single indivisible unit of the same name — paramāṇu, “the utmost fine” — by repeated multiplication.
The primary sources for the systems that follow are the Līlāvatī, which opens by defining measures of length, volume and mass; chapters 19 and 20 of book two of the Arthaśāstra, which give units for space, time and weight and specify the inspection regime for enforcing them in trade; and the Āyurvedic pharmacopoeias, where weights matter because a preparation's proportions do.
Two ladders, meeting in the middle. The lower one climbs from the paramāṇu by repeated multiplication by seven; the upper one is the practical scale of carpenters, surveyors and road-builders, anchored on the aṅgula, a finger's breadth.
| Unit | × | Paramāṇus | Length (mm) | Sense of the name |
|---|---|---|---|---|
| Paramāṇu-raja | 1 | 1 | 2.878 × 10⁻⁷ | the finest mote |
| Reṇu | 7 | 7 | 2.015 × 10⁻⁶ | dust |
| Truṭi | 7 | 49 | 1.410 × 10⁻⁵ | a fragment |
| Vātāyana-raja | 7 | 343 | 9.871 × 10⁻⁵ | dust in a window's sunbeam |
| Śaśa-raja | 7 | 2,401 | 6.910 × 10⁻⁴ | dust on a hare |
| Eḍaka-raja | 7 | 16,807 | 4.837 × 10⁻³ | dust on a ram |
| Go-raja | 7 | 117,649 | 0.0339 | dust on a cow |
| Likṣā-raja | 7 | 823,543 | 0.237 | a nit |
| Sarṣapa | 7 | 5,764,801 | 1.659 | a mustard seed |
| Yava | 7 | 40,353,607 | 11.61 | a barley grain |
| Aṅguli-parva | 7 | 282,475,249 | 81.29 | a finger joint |
Notice the ladder's rhetoric: it climbs through things you can see, from motes of dust to seeds to a finger. Every step is checkable against something in the room.
| Unit | × previous | Aṅgulas | Metres | Use |
|---|---|---|---|---|
| Aṅgula | 1 | 1 | 0.0168 | a finger's breadth — the base module |
| Dhanurmuṣṭi | 8 | 8 | 0.134 | a bow's grip |
| Prājāpatya-hasta | 3 | 24 | 0.402 | a cubit — the standard building measure |
| Dhanus | 4 | 96 | 1.609 | a bow-length; roads and plots |
| Gārhapatya-dhanus | 1.125 | 108 | 1.811 | the alternative rod named in the Arthaśāstra |
| Goruta | 2,000 | 216,000 | 3,621 | “a cow's call” — the distance a lowing carries |
| Yojana | 4 | 864,000 | 14,484 | about 14.5 km — the long-distance unit |
The street widths measured at Kālibangan — 1.8, 3.6, 5.4 and 7.2 m — are 1, 2, 3 and 4 gārhapatya-dhanus of 1.81 m. A unit named in a text of the fourth century BCE appears to be the module of a city laid out two thousand years earlier. That is the kind of correspondence that makes the unit tables worth taking seriously.
The time ladder is the most impressive of the three, because it can be checked — and it checks out.
| Unit | × previous | Paramāṇus | Seconds |
|---|---|---|---|
| Paramāṇu | 1 | 1 | 1.3133 × 10⁻⁵ |
| Aṇu | 2 | 2 | 2.6266 × 10⁻⁵ |
| Trasareṇu | 3 | 6 | 7.880 × 10⁻⁵ |
| Truṭi | 3 | 18 | 2.364 × 10⁻⁴ |
| Vedha | 100 | 1,800 | 0.0236 |
| Lava | 3 | 5,400 | 0.0709 |
| Nimeṣa | 3 | 16,200 | 0.213 — a blink |
| Kṣaṇa | 3 | 48,600 | 0.638 |
| Kāṣṭhā | 5 | 243,000 | 3.19 |
| Laghu | 15 | 3,645,000 | 47.87 |
| Nāḍikā | 15 | 54,675,000 | 718.0 ≈ 12 min |
| Muhūrta | 2 | 109,350,000 | 1,436.1 ≈ 24 min |
| Prahara | 7.5 | 820,125,000 | 10,770.5 ≈ 3 hours |
Eight praharas make a full day and night. Multiply out: 8 × 10,770.51 s = 86,164.1 seconds. The sidereal day — one rotation of the Earth relative to the fixed stars — is 86,164.09 seconds. The chain of thirteen multipliers, running from a unit of about thirteen microseconds all the way up, terminates on an astronomical constant to five significant figures. That is not a coincidence; it is what the ladder was built to do, and it means the small units are defined by division from an accurately observed astronomical period rather than guessed at from below.
In this scheme there are 60 muhūrtas in a day and a muhūrta is about 24 minutes. In the more familiar civil reckoning — the one used for ritual timing and in most Purāṇic passages — there are 30 muhūrtas in a day and each is 48 minutes. Both conventions are in the literature. When a text gives a duration in muhūrtas, check which scale it is using before converting.
The weight ladder is the one that had to be enforced, since it governed trade and taxation. The Arthaśāstra devotes attention not only to the units but to the office of the superintendent who verifies weights and the penalties for falsifying them.
| Unit | × previous | Paramāṇus | Grams | Anchor |
|---|---|---|---|---|
| Paramāṇu | 1 | 1 | 5.787 × 10⁻⁵ | |
| Vaṃśī | 30 | 30 | 1.736 × 10⁻³ | |
| Sarṣapa | 9 | 270 | 0.0156 | a mustard seed |
| Yava | 8 | 2,160 | 0.125 | a barley grain |
| Guñjā | 4 | 8,640 | 0.5 | the rati seed — still a jeweller's unit |
| Māṣaka | 6 | 51,840 | 3 | a bean |
| Karṣa | 4 | 207,360 | 12 | the standard dose unit in Āyurveda |
| Pala | 4 | 829,440 | 48 | |
| Tulā | 100 | 82,944,000 | 4,800 | a balance-load |
| Bhāra | 20 | 1,658,880,000 | 96,000 | a porter's load — 96 kg |
The guñjā deserves a note. It is the seed of Abrus precatorius, whose seeds are famously uniform in mass — which is exactly why a pre-industrial society would choose it as a standard. Goldsmiths in South Asia used the rati derived from it into the twentieth century.
A unit is only as good as the procedure for realising it. Here is one, given in a Purāṇa as a complete construction specification — with every dimension stated in the units of the preceding sections.
द्वादशार्धपलोन्मानं चतुर्भिश्चतुरङ्गुलैः ।
स्वर्णमाषैः कृतच्छिद्रं यावत् प्रस्थजलप्लुतम् ॥
dvādaśārdha-palonmānaṃ caturbhiś-caturaṅgulaiḥ |
svarṇa-māṣaiḥ kṛta-cchidraṃ yāvat prastha-jala-plutam ||
Around the third or second century BCE, a treatise on Sanskrit metre set up a two-valued classification of syllables and then enumerated all their combinations. That is binary notation, arrived at by way of poetry.
Piṅgala's Chandaḥ-śāstra classifies every syllable as one of two kinds:
A syllable with a short vowel and nothing heavy following it. Written | in the tradition's own notation.
A syllable with a long vowel; or a short vowel followed by a consonant cluster; or a short vowel with anusvāra ṃ or visarga ḥ; or optionally the last syllable of a line. Written S.
Take the most quoted verse in Sanskrit and run the rules over it.
Piṅgala groups syllables in threes. Three binary positions give eight patterns, and each gets a name:
1 · Which of the three “inventions” of zero is the distinctively Indian contribution?
Placeholder marks appear in several traditions. Brahmagupta's rules for computing with zero and with negative quantities — and Bhāskara's later question about division by it — require zero to have become a number.
2 · In bhūta-saṃkhyā, the sequence netra–agni denotes —
aṅkānāṃ vāmato gatiḥ. The first-named word is the units digit. This rule is what makes Mādhava's verse decode to 2,827,433,388,233 rather than its reverse.
3 · In kaṭapayādi, what happens to the k and t in ktyā?
One syllable yields at most one digit. Without this rule the encoding would not be reversible.
4 · What makes the Bhāgavata time ladder checkable?
The chain closes on an astronomical constant, which means the small units are defined by division from an accurately observed period rather than guessed from below.
5 · What is a De Bruijn sequence, and why does yamātārājabhānasalagam qualify?
And each window's first syllable names the pattern it begins — the mnemonic is simultaneously the enumeration and the naming scheme.
Almost nothing, and the honest position says so plainly. The kalpa is derived from a scheme of planetary cycles and a 4:3:2:1 division of the yugas; nothing in that derivation makes contact with geology. Presenting it as ancient knowledge of the Earth's age is exactly the “inflation” failure mode of chapter 1 — and it distracts from the sidereal-day result in section 14, which is a real, checkable correspondence.
Because the transmission medium was verse, not paper. A digit string cannot be set to a metre, cannot be checked by scansion, and is easy to corrupt in copying — a single stroke changes the value silently. A number encoded as meaningful words is protected by metre, by grammar and by sense all at once. It is error correction, purchased with cleverness.
He does not do arithmetic in base two, and claiming he anticipated computing overstates it. But he has a two-valued alphabet, positional strings over it, an ordering, an index-to-string procedure and a string-to-index procedure. That is a binary numeration system in everything but the name, and it is roughly eighteen centuries before Leibniz.
| IAST | Devanāgarī | Sense |
|---|---|---|
| aṅgula | अङ्गुल | A finger's breadth, about 16.8 mm; the base module of the practical length system. |
| bhūta-saṃkhyā | भूतसंख्या | Word-numerals: a digit named by a thing occurring in that quantity. |
| gaṇa | गण | A group of three syllables classified by weight; there are eight. |
| guru / laghu | गुरु / लघु | Heavy and light syllables — the two values of Piṅgala's binary classification. |
| kalpa | कल्प | 4.32 billion years; a day of Brahmā. |
| karṣa | कर्ष | A weight of about 12 g; the standard dose unit in Āyurveda. |
| kaṭapayādi | कटपयादि | The consonant-to-digit code that turns numbers into words. |
| koṭi | कोटि | 10⁷; the crore. |
| lakṣa | लक्ष | 10⁵; the lakh. |
| mahā-yuga | महायुग | 432 million years; four yugas in the ratio 4:3:2:1. |
| muhūrta | मुहूर्त | A time unit — 48 minutes in the common reckoning, about 24 in the Bhāgavata scheme. |
| nāḍikā | नाडिका | A time unit realised by a sinking-bowl water clock. |
| pala | पल | A weight of about 48 g. |
| paramāṇu | परमाणु | The indivisible; the base unit of length, weight and time alike. |
| parārdha | परार्ध | 10¹⁷; the last named place in the Līlāvatī's ladder. |
| prastāra | प्रस्तार | The systematic enumeration of all metrical patterns of a given length. |
| śūnya | शून्य | Empty; zero. |
| yojana | योजन | About 14.5 km; the long-distance unit. |