Deep dive

The 16 sutras and 13 sub-sutras

Each sutra is a short word formula that packs a whole method into a phrase. Flip a card for the short version, then open Learn more for the full story: animated diagrams, why it works, worked examples and further reading. Each card shows how far you are with its tricks. You have collected 0 of 29.

The 16 sutras

Use the number that is one more than the digit before. It powers instant squares of numbers ending in 5 and long decimals like 1/19.65² → 6 × 7 = 42, then 25 → 4225Learn more

Subtract each digit from 9 and the last one from 10. This turns subtracting from 100, 1000 or 10000 into a no-borrow job, and it is the engine of near-base multiplication.1000 − 368 → 6, 3, 2 → 632Learn more

Multiply straight down, then crosswise, then straight down again, one column at a time. It works for any numbers and even for algebra.21 × 13 → 2|7|3 → 273Learn more

Flip the sign as you move a term across, then apply it. It solves equations in one line and divides by numbers just above a base like 12 or 112.3x + 4 = x + 10 → 2x = 6 → x = 3Learn more

Spot the same expression on both sides of an equation and set it to zero. Many tricky-looking equations collapse instantly.(x + 3)(x + 8) = (x + 4)(x + 6) → x = 0Learn more

In two equations, if the x numbers are in the same ratio as the answers, then y must be zero. Look before you calculate.6x + 7y = 8 and 19x + 14y = 16 → 7:14 = 8:16, so x = 0Learn more

When two equations have swapped numbers, add them and subtract them. Both answers fall out at once.5x + 3y = 13 and 3x + 5y = 11 → x + y = 3, x − y = 1Learn more

Fill in the missing piece to make a perfect square or cube, then work with the complete shape. This is completing the square.x² + 6x = 7 → (x + 3)² = 16Learn more

Look at how things change step by step. It links to rates of change, the heart of calculus.Change of x² as x grows is about 2xLearn more

See how far a number is from the base, go that far again, then add the square of the gap. Squares near 100 become easy.96² → 96 − 4 = 92, 4² = 16 → 9216Learn more

Use the average or the whole group to understand each part. Handy for factoring and for symmetric problems.Two numbers add to 20 and multiply to 96 → around 10: 10 ± 2Learn more

Write the remainders of long division, then use the last digit to turn them into decimal digits. Great for 1/7 and friends.1/7 remainders 3, 2, 6, 4, 5, 1 → 0.142857...Learn more

A pattern for special equations where the last term and twice the one before it add up to zero.Used for equations with four consecutive factorsLearn more

Multiplying by 9, 99 or 999? Take one less than the number, then the rest is all from 9.67 × 99 → 66 | 33 → 6633Learn more

Digit sums (or coefficient sums) multiply the same way the numbers do. A two-second way to check an answer.12 × 13 = 156 → 3 × 4 = 12 → 3, and 1 + 5 + 6 = 12 → 3Learn more

Check a factoring by adding the coefficients of each factor. If the factors are right, the totals match.x² + 5x + 6 = (x + 2)(x + 3) → 12 = 3 × 4Learn more

The 13 sub-sutras

Scale a hard base into an easy one. Working base 50 is 100 ÷ 2; multiplying by 25 is × 100 ÷ 4.Learn more

Some remainders repeat in a fixed pattern, which makes long divisions predictable.Learn more

When factoring, the first terms multiply to the first term and the last terms to the last term.Learn more

7 × 143 = 1001, which explains the repeating pattern of 1/7 = 0.142857...Learn more

A quick divisibility test: chop off the last digit, multiply it by a magic number and add it to the rest.Learn more

If a number is short of the base by some amount, take that same amount away again.Learn more

The full recipe for squares near a base: go short again, then attach the square of the gap.Learn more

Same first part and last digits adding to 10? Multiply the first part by one more, then multiply the last digits.Learn more

In some equations, comparing only the last terms is enough to find the answer.Learn more

Add coefficients to check products of expressions quickly.Learn more

Remove one letter at a time to factor big expressions piece by piece.Learn more

Look closely and the answer appears. Square roots and cube roots of perfect powers come from patterns in the digits.Learn more

The companion check for multiplying expressions: sums of coefficients multiply consistently.Learn more

A mystery from the old texts

The Lost Śulba ScrollA math mystery in chaptersKatapayadi ciphers, rope-and-peg locks and a scroll torn into pieces. New chapter every week.

Where Vedic math comes from

Vedic mathematics was popularized by Bharati Krishna Tirtha, a scholar and monk who wrote the book "Vedic Mathematics", published in 1965 after his death.

The book presents 16 short sutras (word formulas) and 13 sub-sutras that turn many calculations into one-line mental steps.

Historians and mathematicians have pointed out that these sutras do not appear in the Vedas themselves. The author said he reconstructed them from appendices, and the methods are best seen as a 20th century system inspired by Indian tradition.

Either way, the methods are real mathematics: each trick can be proved with ordinary algebra, which is a great way to see why it works.

Older roots: what you see in the background

The 1965 sutras stand on a much older tradition of Indian mathematics. The moving background draws real pieces of it.

Devanagari numerals. The faint constellations are the digits ० to ९ as they are written in Devanagari today. The Nikhilam Family in the logo is three of them: १, ० and ९.
Śulba Sūtras. Rope geometry for building Vedic fire altars, written down by Baudhāyana around 800 to 500 BCE. It states the diagonal rule now called Pythagoras and gives √2 as 1 + 1/3 + 1/12 − 1/408.
Falcon altar. The śyenaciti, a fire altar shaped like a falcon, built from about a thousand bricks laid by exact rules.
Powers of ten. The Taittirīya Saṃhitā names every power of ten from eka (1) to parārdha (one trillion).
Meru-prastāra. Piṅgala's triangle for counting poetic meters (about 200 BCE), the pattern later called Pascal's triangle.
Brahmi numerals. Early Indian digits, ancestors of the 0 to 9 used worldwide today. The shapes here are stylized.
Nakshatras. The 27 lunar mansions of the Vedāṅga Jyotiṣa, the Vedic star calendar.

Mathematics in Sanskrit verse

The 16 sutras above are short phrases, not verses, and no older text contains them: the 1965 book's own editor says they point to the Swami's own discovery. But India has a long tradition of real mathematics written as verse (śloka), so that rules could be learned by heart and chanted. Here are 18 Sanskrit passages, from the Vedic texts to the Kerala school: most are verses, a few are prose rules (sutras) and mantras. Each card lists the editions and studies it was checked against.

Listen uses your device's Hindi voice when it has one, otherwise an English voice reading the respelling, so treat it as a guide to the sounds, not a recitation.

From the Vedic texts

Verses and rules from the Vedic texts and their helper texts: number names, the calendar, and the rope geometry of fire altars.

Verse (śloka) · anuṣṭubh

Like the crest of a peacock

यथा शिखा मयूराणां नागानां मणयो यथा ।
तद्वद्वेदाङ्गशास्त्राणां ज्योतिषं मूर्धनि स्थितम् ॥

yathā śikhā mayūrāṇāṃ nāgānāṃ maṇayo yathā |
tadvad vedāṅgaśāstrāṇāṃ jyotiṣaṃ mūrdhani sthitam ||

Say it: YA-thaa SHI-khaa ma-yoo-RAA-naam naa-GAA-naam MA-na-yo YA-thaa TA-dvad vay-daang-ga-shaas-TRAA-naam JYO-ti-sham MOOR-dha-ni STHI-tam

Meaning. Like the crests on the heads of peacocks and the jewels on the hoods of cobras, jyotiṣa (in a widely quoted variant: gaṇita, calculation) stands at the head of the Vedāṅga sciences.

The math. An old book about the calendar says that counting and calculating is the shiny crown on top of all the helper subjects that support the Vedas. Praise of calculation (calendar astronomy) as the foremost auxiliary science

Word by word, source and notes

yathā: just as
śikhā: the crests
mayūrāṇāṃ: of peacocks
nāgānāṃ: of cobras
maṇayaḥ: the jewels (on their hoods)
yathā: just as
tadvat: in the same way
vedāṅgaśāstrāṇāṃ: of the sciences that are limbs of the Veda
jyotiṣaṃ: jyotiṣa, the science of time-reckoning by the sky (variant: gaṇitaṃ, calculation)
mūrdhani: at the head
sthitam: stands

Source. Vedāṅga Jyotiṣa of Lagadha: Ārca (Ṛgveda) recension verse 35, Yājuṣa (Yajurveda) recension verse 4. Date disputed: the sky it describes fits roughly 1400 to 1200 BCE, but its language points to the last centuries BCE (Witzel, Pingree, Plofker), and some argue for later still.

Good to know. The printed texts checked (GRETIL Ārca 35, K. V. Sarma ed.; sanskritdocuments Yājuṣa 4) read jyotiṣaṃ. The popular reading gaṇitaṃ ("mathematics") is the one quoted by Datta and Singh (History of Hindu Mathematics, citing "Veda. Jyot. 4") and by later writers; treat it as a variant. Jyotiṣa here means calendar astronomy for timing rituals, not astrology in the modern sense and not mathematics in general.

Checked against:

GRETIL: Lagadha, Ṛgveda-Vedāṅgajyotiṣa (K. V. Sarma ed.)
sanskritdocuments.org: Yājuṣa Jyotiṣa
Dharmawiki: Ganita Shastra (cites Datta and Singh, History of Hindu Mathematics, p. 7)
A. K. Dutta, Mathematics in Ancient India (Resonance), quotes the gaṇitam reading
arXiv 2506.02216: Mathematics, a missing factor in assessing the antiquity of Vedāṅga Jyotiṣa (dating survey)
Wikipedia: Vedanga Jyotisha (dating views)

Vedic mantra

Names for powers of ten

इमा मे अग्न इष्टका धेनवः सन्त्वेका च दश च दश च शतं च शतं च सहस्रं च सहस्रं चायुतं चायुतं च नियुतं च नियुतं च प्रयुतं चार्बुदं च न्यर्बुदं समुद्रश्च मध्यं चान्तश्च परार्धश्चैता मे अग्न इष्टका धेनवः सन्त्वमुत्रामुष्मिंल्लोके ॥

imā me agna iṣṭakā dhenavaḥ santv ekā ca daśa ca daśa ca śataṃ ca śataṃ ca sahasraṃ ca sahasraṃ cāyutaṃ cāyutaṃ ca niyutaṃ ca niyutaṃ ca prayutaṃ cārbudaṃ ca nyarbudaṃ samudraś ca madhyaṃ cāntaś ca parārdhaś caitā me agna iṣṭakā dhenavaḥ santv amutrāmuṣmiṃl loke ||

Say it: I-maa MAY AG-na ISH-ta-kaa DHAY-na-vah SANT VAY-kaa cha DA-sha cha DA-sha cha SHA-tam cha SHA-tam cha sa-HA-sram cha sa-HA-sram CHAA-yu-tam CHAA-yu-tam cha NI-yu-tam cha NI-yu-tam cha PRA-yu-tam CHAAR-bu-dam cha NYAR-bu-dam sa-MU-drash cha MA-dhyam CHAAN-tash cha pa-RAAR-dhash CHAI-taa MAY AG-na ISH-ta-kaa DHAY-na-vah SANT va-mu-traa-MUSH-miml LO-kay

Meaning. O Agni, may these bricks be milk cows for me: one and ten, ten and a hundred, a hundred and a thousand, a thousand and ten thousand, ten thousand and a hundred thousand, a hundred thousand and a million, ten million, a hundred million, an ocean, a middle, an end, a parārdha. O Agni, may these bricks be milk cows for me, there in that world.

The math. Thousands of years ago, priests building a fire altar chanted a ladder of number names where each step is ten times bigger, all the way up to a million million. Names for 1, 10, 100, ... up to 10^12 (parārdha), each ten times the one before

Word by word, source and notes

imāḥ: these
me: for me
agne: O Agni
iṣṭakāḥ: bricks (of the fire altar)
dhenavaḥ: milk cows
santu: may they be
ekā: one
ca: and
daśa: ten
śataṃ: a hundred
sahasraṃ: a thousand
ayutaṃ: ten thousand
niyutaṃ: a hundred thousand
prayutaṃ: a million
arbudaṃ: ten million
nyarbudaṃ: a hundred million
samudraḥ: a billion (literally: ocean)
madhyaṃ: ten billion (literally: middle)
antaḥ: a hundred billion (literally: end)
parārdhaḥ: a trillion (literally: the far half)
etāḥ: these
amutra amuṣmin loke: there, in that (other) world

Source. Śukla Yajurveda, Vājasaneyi Saṃhitā (Mādhyandina) 17.2; a parallel list is in Taittirīya Saṃhitā 4.4.11. Vedic mantra collection, roughly early first millennium BCE (dates approximate).

Good to know. This is a ritual prayer for the fire altar, not a lesson in arithmetic. The text gives the names in order; the values as powers of ten (samudra = 10^9 and so on) follow the standard scholarly reading (Datta and Singh). Number names above a thousand differ between later texts (compare Āryabhaṭa and Bhāskara). Accents omitted here; TITUS has the accented text. Some editions read "nyarbudaṃ ca samudraś ca".

Checked against:

TITUS: Vājasaneyi Saṃhitā (Mādhyandina), book 17
Some Sanskrit Verses in Ancient Math (BPRIM, 2026)

Sutra (prose rule)

The diagonal rule (the "Pythagorean" theorem)

दीर्घचतुरस्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति ॥

dīrghacaturasrasyākṣṇayā rajjuḥ pārśvamānī tiryaṅmānī ca yat pṛthagbhūte kurutas tad ubhayaṃ karoti ||

Say it: deer-gha-cha-tu-ra-sras-YAAK-shna-yaa RAJ-juh paar-shva-MAA-nee tir-yang-MAA-nee cha YAT pri-thag-BHOO-tay KU-ru-tas TAD U-bha-yam ka-RO-ti

Meaning. The diagonal cord of a rectangle produces both (areas) which the cords along the long side and the short side produce separately.

The math. Make a square on each side of a rectangle and one on its diagonal: the two side squares together cover exactly as much ground as the diagonal square, so 3 and 4 give a diagonal of 5. a² + b² = c² for the sides a, b and diagonal c of a rectangle (right triangle)

Word by word, source and notes

dīrghacaturasrasya: of a rectangle (literally: a long four-cornered figure)
akṣṇayā rajjuḥ: the diagonal cord
pārśvamānī: the cord measuring the long side
tiryaṅmānī: the cord measuring the short side
ca: and
yat: what (area)
pṛthagbhūte: separately
kurutaḥ: the two produce
tat: that
ubhayaṃ: both together
karoti: it (the diagonal) produces

Source. Baudhāyana Śulbasūtra 1.12 (numbering of Sen and Bag 1983, used by Plofker) = 1.48 (Thibaut's numbering). Same wording in Āpastamba Śulbasūtra 1.4. Baudhāyana is dated roughly 800 to 500 BCE, with wide uncertainty.

Good to know. A rule for laying out brick fire altars with cords, stated without proof. The next sūtra lists triples such as 3, 4, 5 and 5, 12, 13. Babylonian tablets (for example Plimpton 322, around 1800 BCE) show such triples were known even earlier, and the name "Pythagorean" is a later Greek attribution; this is one of several independent early statements. Spelling varies between caturasra and caturaśra.

Checked against:

Wikipedia: Baudhayana sutras (text of the diagonal rule, Thibaut translation)
Wikipedia: Shulba Sutras (Sen and Bag / Plofker numbering 1.12, 2.12; dating)
GRETIL: Āpastamba Śulbasūtra with commentary (1.4 to 1.6)
arXiv 2406.10147: From Concrete to Abstract in Indian Mathematics (cites BŚ 1.48 in Thibaut numbering)

Sutra (prose rule)

How long is a square's diagonal? (√2)

समस्य द्विकरणी ।
प्रमाणं तृतीयेन वर्धयेत्तच्चतुर्थेनात्मचतुस्त्रिंशोनेन सविशेषः ॥

samasya dvikaraṇī |
pramāṇaṃ tṛtīyena vardhayet tac caturthenātmacatustriṃśonena saviśeṣaḥ ||

Say it: sa-MAS-ya dvi-KA-ra-nee pra-MAA-nam tri-tee-YAY-na VAR-dha-yayt TACH cha-tur-thay-naat-ma-cha-tus-trim-sho-NAY-na sa-vi-SHAY-shah

Meaning. (The diagonal) of a square is the doubler. Increase the measure by its third, and this (third) by its own fourth less its thirty-fourth part: this is the saviśeṣa (the value with a small excess).

The math. If a square has side 1, its diagonal is a little more than 1.414; the old rule says 1 + 1/3 + 1/12 - 1/408, which is 1.414216, off by only about 2 millionths. √2 ≈ 1 + 1/3 + 1/(3·4) - 1/(3·4·34) = 577/408 ≈ 1.414216 (true value 1.414214)

Word by word, source and notes

samasya: of a square
dvikaraṇī: "the doubler", the diagonal (it makes a square of double area)
pramāṇaṃ: the measure (the side)
tṛtīyena: by a third (of itself)
vardhayet: one should increase
tat: that (third)
caturthena: by its fourth
ātmacatustriṃśonena: less its own thirty-fourth part
saviśeṣaḥ: (this is) the saviśeṣa, "with a little extra", a slightly-too-long value

Source. Baudhāyana Śulbasūtra 1.9 and 2.12 (Sen and Bag / Plofker numbering) = 1.61 to 1.62 (Thibaut's numbering); identical wording in Āpastamba Śulbasūtra 1.5 to 1.6. Roughly 800 to 500 BCE, uncertain.

Good to know. The text gives the rule, not the method that produced it; how it was found is reconstructed by historians. The word saviśeṣa signals that the value is slightly too large, so the authors seem to have known it was approximate; it does not show they knew √2 is irrational. The quoted Sanskrit is the Āpastamba text from GRETIL; translations follow Thibaut and Datta.

Checked against:

Wikipedia: Baudhayana sutras (text of the diagonal rule, Thibaut translation)
Wikipedia: Shulba Sutras (Sen and Bag / Plofker numbering 1.12, 2.12; dating)
GRETIL: Āpastamba Śulbasūtra with commentary (1.4 to 1.6)

Vedic mantra · anuṣṭubh

Whole from whole (pūrṇam adaḥ)

पूर्णमदः पूर्णमिदं पूर्णात्पूर्णमुदच्यते ।
पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ॥

pūrṇam adaḥ pūrṇam idaṃ pūrṇāt pūrṇam udacyate |
pūrṇasya pūrṇam ādāya pūrṇam evāvaśiṣyate ||

Say it: POOR-nam A-dah POOR-nam I-dam POOR-naat POOR-nam u-DA-chya-tay poor-NAS-ya POOR-nam aa-DAA-ya POOR-nam ay-vaa-va-SHISH-ya-tay

Meaning. That is whole, this is whole; from the whole the whole comes forth. Taking the whole from the whole, the whole alone remains.

The math. A very old prayer about wholeness: you can take everything out of the whole and it is still whole, which reminds many people today of infinity, even though it was not written as math. Often compared today with infinity (∞ - ∞), though in modern arithmetic ∞ - ∞ is undefined

Word by word, source and notes

pūrṇam: full, whole, complete
adaḥ: that (the unseen)
idaṃ: this (the seen world)
pūrṇāt: from the whole
udacyate: rises, comes forth
pūrṇasya: of the whole
ādāya: taking away
eva: still, only
avaśiṣyate: remains

Source. Bṛhadāraṇyaka Upaniṣad 5.1.1 (Kāṇva); also recited as the opening invocation (śānti-pāṭha) of the Īśa Upaniṣad. Probably 7th to 6th century BCE (approximate).

Good to know. This is philosophy, not a mathematics text. Śaṅkara's commentary explains it as about brahman, the infinite reality, and the world. Reading it as a statement about the number infinity is a modern interpretation. Interestingly, the same passage continues "khaṃ brahma" (brahman is space), and kha, "empty space", later became a common word for zero.

Checked against:

GRETIL: Bṛhadāraṇyaka Upaniṣad (Kāṇva) with Śaṅkara's commentary, 5.1.1
GRETIL: Īśopaniṣad (Kāṇva) with commentary (invocation)

From India's great mathematicians

Rules written as short poems so they could be learned by heart, from Pingala to Bhaskara.

Sutra (prose rule)

Piṅgala's shortcut for powers of two

द्विरर्धे । रूपे शून्यम् । द्विः शून्ये । तावदर्धे तद्गुणितम् ॥

dvir ardhe | rūpe śūnyam | dviḥ śūnye | tāvad ardhe tadguṇitam ||

Say it: DVIR AR-dhay ROO-pay SHOON-yam DVIH SHOON-yay TAA-vad AR-dhay tad-GU-ni-tam

Meaning. (If the number can be halved, halve it and write) two. (If not,) take away one and write a zero. (Then, working back from the end, starting from one:) at a zero, double; at a halving mark, multiply the amount by itself.

The math. To count all the long and short beat patterns in a 6-syllable line, Piṅgala halves and subtracts down to nothing, then doubles and squares back up: 1, 2, 4, 8, 64, so there are 64 patterns. 2^n by repeated halving and squaring (the "square and multiply" method); e.g. 2^6 = 64

Word by word, source and notes

dviḥ: two (write the mark "2")
ardhe: when (the number) is halved
rūpe: when one is taken away
śūnyam: an empty mark, a zero
dviḥ: twice, double
śūnye: at the empty mark
tāvat: that much
ardhe: at the halving mark
tadguṇitam: multiplied by itself

Source. Piṅgala, Chandaḥśāstra (Chandaḥsūtra) 8.28 to 8.31 in the numbering used by A. Kulkarni. Commonly dated to the last centuries BCE, uncertain. Explained by the commentator Halāyudha (Mṛtasañjīvanī, usually dated to the 10th century CE).

Good to know. These are terse prosody rules for counting Sanskrit meters, and they only make sense with a commentary. Sūtra numbering and readings vary between editions. Whether śūnya here is already the numeral zero or just a written marker is debated. Popular claims that Piṅgala "invented binary numbers" overstate it: his listing of long and short syllable patterns is analogous to binary counting. The number triangle (meru-prastāra, now called Pascal's triangle) is described in Halāyudha's later commentary, not in these sūtras.

Checked against:

A. Kulkarni, Recursion and Combinatorial Mathematics in Chandaśāstra (arXiv math/0703658)
INDOLOGY list, July 2022: Sunya (Piṅgala 8.28 to 8.31 as transcribed by A. Kulkarni)

Verse (śloka) · āryā

Each place is ten times the last

एकं च दश च शतं च सहस्रं त्वयुतनियुते तथा प्रयुतम् ।
कोट्यर्बुदं च वृन्दं स्थानात्स्थानं दशगुणं स्यात् ॥

ekaṃ ca daśa ca śataṃ ca sahasraṃ tv ayutaniyute tathā prayutam |
koṭyarbudaṃ ca vṛndaṃ sthānāt sthānaṃ daśaguṇaṃ syāt ||

Say it: AY-kam cha DA-sha cha SHA-tam cha sa-HA-sram tva-yu-ta-NI-yu-tay TA-thaa PRA-yu-tam ko-TYAR-bu-dam cha VRIN-dam STHAA-naat STHAA-nam da-SHA-gu-nam SYAAT

Meaning. One, ten, a hundred, a thousand, ten thousand, a hundred thousand, a million, ten million, a hundred million and a billion: from place to place, each is ten times (the previous).

The math. Āryabhaṭa names the places in a number, ones, tens, hundreds and so on, and says each place is worth ten times the one to its right, which is exactly how we write numbers today. Decimal place value: each place is worth 10 times the place before (1 to 10^9)

Word by word, source and notes

ekaṃ: one
daśa: ten
śataṃ: a hundred
sahasraṃ: a thousand
ayutaniyute: ten thousand and a hundred thousand
tathā: likewise
prayutam: a million
koṭyarbudaṃ: ten million and a hundred million
vṛndaṃ: a billion (10^9)
sthānāt sthānaṃ: from place to place
daśaguṇaṃ: ten times
syāt: is

Source. Āryabhaṭa, Āryabhaṭīya, Gaṇitapāda 2; composed 499 CE (Kusumapura).

Good to know. The GRETIL e-text (from Shukla's edition, entered without sandhi) omits "ca" after ekaṃ and "tu" before ayuta; the fuller reading here is the one usually printed and is needed for the āryā meter. The verse names the places; it does not itself describe the written zero.

Checked against:

GRETIL: Āryabhaṭīya (K. S. Shukla ed., INSA 1976)
Some Sanskrit Verses in Ancient Math (BPRIM, 2026)

Verse (śloka) · āryā

Āryabhaṭa's value of π

चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम् ।
अयुतद्वयविष्कम्भस्यासन्नो वृत्तपरिणाहः ॥

caturadhikaṃ śatam aṣṭaguṇaṃ dvāṣaṣṭis tathā sahasrāṇām |
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ ||

Say it: cha-tu-RA-dhi-kam SHA-tam ash-TA-gu-nam dvaa-SHASH-tis TA-thaa sa-ha-SRAA-naam a-yu-ta-dva-ya-vish-kam-bhas-yaa-SAN-no vrit-ta-pa-ri-NAA-hah

Meaning. One hundred plus four, multiplied by eight, and sixty-two thousand added: this is the approximate circumference of a circle whose diameter is twenty thousand.

The math. (100 + 4) × 8 + 62,000 = 62,832, and a circle 20,000 units across goes about 62,832 units around, so π is about 3.1416. π ≈ 62832 / 20000 = 3.1416

Word by word, source and notes

caturadhikaṃ: increased by four
śatam: a hundred
aṣṭaguṇaṃ: multiplied by eight
dvāṣaṣṭiḥ: sixty-two
tathā: and also
sahasrāṇām: thousands
ayutadvaya: two ayutas, 20,000
viṣkambhasya: of a diameter
āsannaḥ: approximate, near
vṛttapariṇāhaḥ: circumference of the circle

Source. Āryabhaṭa, Āryabhaṭīya, Gaṇitapāda 10; 499 CE. Quoted in the Kriyākramakarī on Līlāvatī 199.

Good to know. The word āsanna ("near") shows Āryabhaṭa knew this was an approximation. Some modern writers suggest it hints that π is irrational; that is a modern interpretation, not something the text proves. The Kriyākramakarī commentary explains āsanna in a similar way.

Checked against:

GRETIL: Āryabhaṭīya (K. S. Shukla ed., INSA 1976)
Wikipedia: Aryabhata (Gaṇitapāda 10 text and translation)
Līlāvatī with Kriyākramakarī, ed. K. V. Sarma, VVRI 1975 (archive.org scan)

Verse (śloka) · āryā

Adding with fortunes, debts and zero

धनयोर्धनमृणमृणयोर्धनर्णयोरन्तरं समैक्यं खम् ।
ऋणमैक्यं च धनमृणधनशून्ययोः शून्ययोः शून्यम् ॥

dhanayor dhanam ṛṇam ṛṇayor dhanarṇayor antaraṃ samaikyaṃ kham |
ṛṇam aikyaṃ ca dhanam ṛṇadhanaśūnyayoḥ śūnyayoḥ śūnyam ||

Say it: DHA-na-yor DHA-nam RI-nam RI-na-yor dha-NAR-na-yor AN-ta-ram sa-MAI-kyam KHAM RI-nam AI-kyam cha DHA-nam ri-na-dha-na-SHOON-ya-yoh SHOON-ya-yoh SHOON-yam

Meaning. (The sum) of two positives is positive, of two negatives is negative; of a positive and a negative it is their difference, and if they are equal, zero. The sum of a negative and zero is negative, of a positive and zero positive, and of two zeros zero.

The math. Think of money: two gifts make a bigger gift, two debts make a bigger debt, a gift and a debt cancel down to the difference, and adding nothing changes nothing. Rules for adding positive and negative numbers and zero: (+)+(+)=+, (-)+(-)=-, a+(-a)=0, a+0=a, 0+0=0

Word by word, source and notes

dhanayoḥ: of two fortunes (positives)
dhanam: a fortune
ṛṇam: a debt (negative)
ṛṇayoḥ: of two debts
dhanarṇayoḥ: of a fortune and a debt
antaraṃ: the difference
samaikyaṃ: the sum of equal (fortune and debt)
kham: zero (literally: empty space)
ṛṇam aikyaṃ ca dhanam: the sum is a debt, and a fortune (respectively)
ṛṇadhanaśūnyayoḥ: of a debt and zero, of a fortune and zero
śūnyayoḥ: of two zeros
śūnyam: zero

Source. Brahmagupta, Brāhmasphuṭasiddhānta 18.30 (chapter on algebra, kuṭṭaka); 628 CE.

Good to know. One of the earliest known statements treating zero as a number with its own arithmetic rules. Brahmagupta's words for positive and negative are dhana (wealth) and ṛṇa (debt). In 18.34 to 18.35 he also says zero divided by zero is zero, which modern mathematics does not accept. Text checked in the Dvivedin (GRETIL) and R. S. Sharma editions.

Checked against:

GRETIL: Brāhmasphuṭasiddhānta (S. Dvivedin ed., 1902)
Brāhmasphuṭasiddhānta, ed. R. S. Sharma, 1966, vol. 4 (archive.org scan)
Wikipedia: Brāhmasphuṭasiddhānta (rules for zero and negatives)

Verse (śloka) · āryā

A debt times a debt is a fortune

ऋणमृणधनयोर्घातो धनमृणयोर्धनवधो धनं भवति ।
शून्यर्णयोः खधनयोः खशून्ययोर्वा वधः शून्यम् ॥

ṛṇam ṛṇadhanayor ghāto dhanam ṛṇayor dhanavadho dhanaṃ bhavati |
śūnyarṇayoḥ khadhanayoḥ khaśūnyayor vā vadhaḥ śūnyam ||

Say it: RI-nam ri-na-DHA-na-yor GHAA-to DHA-nam RI-na-yor dha-NA-va-dho DHA-nam BHA-va-ti shoon-YAR-na-yoh kha-DHA-na-yoh kha-SHOON-ya-yor VAA VA-dhah SHOON-yam

Meaning. The product of a negative and a positive is negative; of two negatives, positive; the product of positives is positive. The product of zero and a negative, of zero and a positive, or of two zeros, is zero.

The math. Multiplying a positive by a negative gives a negative, two negatives make a positive, and anything times zero is zero, written down about 1,400 years ago. (-)×(+) = -, (-)×(-) = +, (+)×(+) = +, a×0 = 0

Word by word, source and notes

ṛṇam: a debt (negative)
ṛṇadhanayoḥ: of a debt and a fortune
ghātaḥ: the product
dhanam: a fortune (positive)
ṛṇayoḥ: of two debts
dhanavadhaḥ: the product of fortunes
bhavati: is
śūnyarṇayoḥ: of zero and a debt
khadhanayoḥ: of zero and a fortune
khaśūnyayoḥ: of zero and zero
vā: or
vadhaḥ: the product
śūnyam: zero

Source. Brahmagupta, Brāhmasphuṭasiddhānta 18.33; 628 CE.

Good to know. Stated as rules without proof. Text checked in the Dvivedin (GRETIL) and R. S. Sharma editions, which agree.

Checked against:

GRETIL: Brāhmasphuṭasiddhānta (S. Dvivedin ed., 1902)
Brāhmasphuṭasiddhānta, ed. R. S. Sharma, 1966, vol. 4 (archive.org scan)
Wikipedia: Brāhmasphuṭasiddhānta (rules for zero and negatives)

Verse (śloka) · āryā

Numbers written as words: Śaka 427

सप्ताश्विवेदसंख्यं शककालमपास्य चैत्रशुक्लादौ ।

saptāśvivedasaṃkhyaṃ śakakālam apāsya caitraśuklādau |

Say it: sap-taash-vi-vay-da-SAM-khyam sha-ka-KAA-lam a-PAAS-ya chai-tra-shuk-LAA-dau

Meaning. Subtract (from the given year) the Śaka year numbered seven, twins (Aśvins), Vedas, that is 427, at the beginning of the bright fortnight of Caitra.

The math. Instead of digits, the poet used words: "seven", "the twins" (2) and "the Vedas" (4); reading them right to left gives 427, a year that is 505 CE. Bhūta-saṃkhyā (object-number words) read right to left: 7, 2, 4 → 427

Word by word, source and notes

sapta: seven: 7
aśvi: the Aśvin twins: 2
veda: the Vedas: 4
saṃkhyaṃ: numbering
śakakālam: the Śaka era (years)
apāsya: having subtracted
caitraśuklādau: at the beginning of the bright half of the month Caitra

Source. Varāhamihira, Pañcasiddhāntikā 1.8, first half (Romaka method); epoch Śaka 427 = 505 CE; work of the 6th century CE.

Good to know. Only the first half of the verse is given. The second half names the weekday of the epoch, and the editions disagree on that word (Fleet and Neugebauer and Pingree favour Tuesday; other manuscripts read Wednesday or Monday). The right-to-left order ("aṅkānāṃ vāmato gatiḥ") is the standard rule for word numerals.

Checked against:

Pañcasiddhāntikā, ed. T. S. Kuppanna Sastry and K. V. Sarma (archive.org scan)
Neugebauer and Pingree, The Pañcasiddhāntikā of Varāhamihira, vol. 1, 1970 (archive.org scan)
Indian Antiquary vol. 19 (1890), p. 54: Fleet on the epoch verse and its variant readings
Wikipedia: Pancha-siddhantika (Śaka 427 = 505 CE)

Verse (śloka) · anuṣṭubh

Nothing exists without mathematics

बहुभिर्विप्रलापैः किं त्रैलोक्ये सचराचरे ।
यत्किंचिद्वस्तु तत्सर्वं गणितेन विना न हि ॥

bahubhir vipralāpaiḥ kiṃ trailokye sacarācare |
yat kiṃcid vastu tat sarvaṃ gaṇitena vinā na hi ||

Say it: BA-hu-bhir vi-pra-LAA-paih KIM trai-LO-kyay sa-cha-RAA-cha-ray YAT KIM-chid VAS-tu TAT SAR-vam ga-ni-TAY-na VI-naa na hi

Meaning. What is the use of saying much? Whatever there is in the three worlds, with all their moving and unmoving beings, none of it exists apart from mathematics.

The math. A Jain teacher said: why argue? Everything in the whole world, moving or still, needs numbers and measuring to be understood. Praise of gaṇita (mathematics) as present in everything

Word by word, source and notes

bahubhiḥ: with many
vipralāpaiḥ: idle words
kiṃ: what is the use
trailokye: in the three worlds
sacarācare: with all moving and unmoving things
yat kiṃcit: whatever
vastu: thing
tat sarvaṃ: all of that
gaṇitena: mathematics, calculation
vinā: without
na hi: surely is not

Source. Mahāvīra (Mahāvīrācārya), Gaṇitasārasaṃgraha 1.16 (Saṃjñādhikāra); c. 850 CE, at the court of the Rāṣṭrakūṭa king Amoghavarṣa.

Good to know. A Jain mathematical textbook, not a Vedic one. Rangacharya's 1912 translation renders gaṇita here as "measurement". The online text is unproofread OCR (it prints "trailokaye"); the reading trailokye is standard.

Checked against:

Wisdom Library: Gaṇitasārasaṃgraha, ch. 1 Sanskrit (Rangacharya ed., 1912)
Wisdom Library: Gaṇitasārasaṃgraha, ch. 1 English (Rangacharya tr., 1912)

Verse (śloka) · upajāti (indravajrā family)

Bhāskara's squaring shortcuts

खण्डद्वयस्याभिहतिर्द्विनिघ्नी तत्खण्डवर्गैक्ययुता कृतिर्वा ।
इष्टोनयुग्राशिवधः कृतिः स्यादिष्टस्य वर्गेण समन्वितो वा ॥

khaṇḍadvayasyābhihatir dvinighnī tatkhaṇḍavargaikyayutā kṛtir vā |
iṣṭonayugrāśivadhaḥ kṛtiḥ syād iṣṭasya vargeṇa samanvito vā ||

Say it: khan-da-dva-yas-yaa-BHI-ha-tir dvi-NIGH-nee tat-khan-da-var-gai-KYA-yu-taa KRI-tir VAA ish-to-na-yu-graa-SHI-va-dhah KRI-tih SYAAD ish-TAS-ya var-GAY-na sa-MAN-vi-to VAA

Meaning. Or: the product of two parts (of a number), doubled and added to the sum of the squares of the parts, is the square. Or: the product of the number decreased and increased by any chosen amount, plus the square of that chosen amount, is the square.

The math. To square 297, Bhāskara picks 3: 294 × 300 = 88,200, then add 3 × 3 = 9 to get 88,209, which is much easier than multiplying 297 by 297. (a + b)² = 2ab + a² + b²; n² = (n - d)(n + d) + d², e.g. 297² = 294 × 300 + 9 = 88209

Word by word, source and notes

khaṇḍadvayasya: of two parts (of the number)
abhihatiḥ: the product
dvinighnī: multiplied by two
tatkhaṇḍavargaikyayutā: added to the sum of the squares of those parts
kṛtiḥ: the square
vā: or
iṣṭa: a chosen amount
ūnayugrāśivadhaḥ: the product of the number decreased and increased (by it)
syāt: is
iṣṭasya vargeṇa: by the square of the chosen amount
samanvitaḥ: increased

Source. Bhāskara II, Līlāvatī 20 (squaring, vargaparikarma); c. 1150 CE. The 297 example is in Bhāskara's own prose comment.

Good to know. The second rule is the same algebraic identity that makes Tirtha's Yāvadūnam squaring work (97² = 94 × 100 + 9 = 9409), so the connection is mathematical, not a source for the 1965 sūtra. K. V. Sarma's edition inserts an extra half-verse (khaṇḍadvayasyābhihatiś caturghnī ...) that other editions lack. The previous verse, Līlāvatī 19, defines the square: samadvighātaḥ kṛtiḥ, "the product of two equal numbers is the square".

Checked against:

GRETIL: Bhāskara, Līlāvatī (Āpaṭe 1937 and K. V. Sarma 1975 eds.)
Līlāvatī with Kriyākramakarī, ed. K. V. Sarma, VVRI 1975 (archive.org scan)
Līlāvatī with Tattvaprakāśikā, ed. L. Jha (Chowkhamba) (archive.org scan)

Verse (śloka) · śārdūlavikrīḍita

The bee puzzle

पञ्चांशोऽलिकुलात्कदम्बमगमत्त्र्यंशः शिलीन्ध्रं तयो-
र्विश्लेषस्त्रिगुणो मृगाक्षि कुटजं दोलायमानोऽपरः ।
कान्ते केतकमालतीपरिमलप्राप्तैककालप्रिया-
दूताहूत इतस्ततो भ्रमति खे भृङ्गोऽलिसंख्यां वद ॥

pañcāṃśo 'likulāt kadambam agamat tryaṃśaḥ śilīndhraṃ tayor
viśleṣas triguṇo mṛgākṣi kuṭajaṃ dolāyamāno 'paraḥ |
kānte ketakamālatīparimalaprāptaikakālapriyā-
dūtāhūta itas tato bhramati khe bhṛṅgo 'lisaṃkhyāṃ vada ||

Say it: pan-CHAAM-sho LI-ku-laat ka-DAM-bam A-ga-mat TRYAM-shah shi-LEEN-dhram TA-yor vish-LAY-shas TRI-gu-no mri-GAAK-shi KU-ta-jam do-laa-ya-MAA-no PA-rah KAAN-tay kay-ta-ka-maa-la-tee-pa-ri-ma-la-praap-tai-ka-kaa-LA-pri-yaa doo-taa-HOO-ta I-tas TA-to BHRA-ma-ti KHAY BHRING-go li-SAM-khyaam VA-da

Meaning. A fifth of a swarm of bees went to a kadamba flower, a third to a śilīndhra flower, and three times their difference, O fawn-eyed girl, to a kuṭaja flower. One more bee, called at the same moment by the fragrance of the ketaka and the mālatī, wanders here and there in the air. Tell me, lovely one, how many bees there are.

The math. 1/5 + 1/3 + 3 × (1/3 - 1/5) of the bees fly off and 1 bee is left over: that makes 3 + 5 + 6 + 1 = 15 bees. x/5 + x/3 + 3(x/3 - x/5) + 1 = x, so x = 15

Word by word, source and notes

pañcāṃśaḥ: a fifth
alikulāt: of a swarm of bees
kadambam agamat: went to a kadamba flower
tryaṃśaḥ: a third
śilīndhraṃ: (went) to a śilīndhra flower
tayoḥ viśleṣaḥ: the difference of these two
triguṇaḥ: three times (that)
mṛgākṣi: O fawn-eyed girl
kuṭajaṃ: (went) to a kuṭaja flower
dolāyamānaḥ: swinging to and fro
aparaḥ: one other (bee)
kānte: O lovely one
ketakamālatīparimala...dūtāhūtaḥ: called at once by the scents of the ketaka and mālatī flowers, like messengers from two sweethearts
itas tataḥ: here and there
bhramati khe: wanders in the air
bhṛṅgaḥ: a bee
alisaṃkhyāṃ: the number of bees
vada: tell

Source. Bhāskara II, Līlāvatī 55 (example for iṣṭakarma, the rule of false position) in the Āpaṭe numbering used by GRETIL; c. 1150 CE.

Good to know. Verse numbering differs between editions. Bhāskara's method is to assume any number, see what it gives, and scale (false position), not to solve an equation the modern way. The other famous bee problem (Līlāvatī 71, answer 72) involves a square root.

Checked against:

GRETIL: Bhāskara, Līlāvatī (Āpaṭe 1937 and K. V. Sarma 1975 eds.)
Līlāvatī, tr. Patwardhan, Naimpally and Singh (MLBD) (archive.org scan)

Numbers hidden in words

Kerala mathematicians wrote numbers as words and syllables. The last verse here is the one verse in the 1965 book.

Verse (śloka) · āryā

Mādhava's endless series for π

व्यासे वारिधिनिहते रूपहृते व्याससागराभिहते ।
त्रिशरादिविषमसंख्याभक्तमृणं स्वं पृथक्क्रमात्कुर्यात् ॥

vyāse vāridhinihate rūpahṛte vyāsasāgarābhihate |
triśarādiviṣamasaṃkhyābhaktam ṛṇaṃ svaṃ pṛthak kramāt kuryāt ||

Say it: VYAA-say vaa-ri-dhi-NI-ha-tay roo-PA-hri-tay vyaa-sa-saa-ga-raa-BHI-ha-tay tri-sha-raa-di-vi-sha-ma-sam-khyaa-BHAK-tam RI-nam SVAM PRI-thak KRA-maat KUR-yaat

Meaning. Multiply the diameter by four and divide by one. Then divide four times the diameter by the odd numbers three, five and so on, and subtract and add these in turn, one after another.

The math. Circumference = 4 × diameter × (1 - 1/3 + 1/5 - 1/7 + ...), a never-ending pattern that creeps closer and closer to π. C = 4D(1 - 1/3 + 1/5 - 1/7 + ...), i.e. π/4 = 1 - 1/3 + 1/5 - 1/7 + ...

Word by word, source and notes

vyāse: the diameter
vāridhinihate: multiplied by the oceans (4)
rūpahṛte: and divided by one
vyāsasāgarābhihate: the diameter multiplied by the seas (4)
triśarādiviṣamasaṃkhyābhaktam: divided by the odd numbers three, five (śara, arrows) and so on
ṛṇaṃ svaṃ: as minus and plus
pṛthak: separately
kramāt: in order
kuryāt: one should apply

Source. Attributed to Mādhava of Saṅgamagrāma (c. 1340 to 1425, Kerala); quoted in the Kriyākramakarī commentary on Līlāvatī 199 (Śaṅkara Vāriyar and Nārāyaṇa, c. 1534 to 1560) and, per K. V. Sarma, also in Jyeṣṭhadeva's Yuktibhāṣā; the series also appears in Śaṅkara Vāriyar's Yuktidīpikā on the Tantrasaṅgraha.

Good to know. No work by Mādhava containing this verse survives; it is known through later Kerala commentators who credit him. The same series was published in Europe by Gregory and Leibniz in the 1670s, independently as far as is known. The verse is followed by rules for correction terms that make the slow series much more accurate. The commentary notes "rūpahṛte" (divide by one) is there to show the odd numbers start from 1.

Checked against:

Līlāvatī with Kriyākramakarī, ed. K. V. Sarma, VVRI 1975 (archive.org scan)
Wikipedia: Kriyakramakari (Mādhava series, citing Plofker)
Wikipedia: Madhava of Sangamagrama (dates)

Verse (śloka) · drutavilambita

π to eleven decimal places, in words

विबुधनेत्रगजाहिहुताशनत्रिगुणवेदभवारणबाहवः ।
नवनिखर्वमिते वृतिविस्तरे परिधिमानमिदं जगदुर्बुधाः ॥

vibudhanetragajāhihutāśanatriguṇavedabhavāraṇabāhavaḥ |
navanikharvamite vṛtivistare paridhimānam idaṃ jagadur budhāḥ ||

Say it: vi-bu-dha-nay-tra-ga-jaa-hi-hu-taa-sha-na-tri-gu-na-vay-da-bha-vaa-ra-na-BAA-ha-vah na-va-ni-khar-VA-mi-tay vri-ti-VIS-ta-ray pa-ri-dhi-MAA-nam I-dam JA-ga-dur BU-dhaah

Meaning. When the diameter of a circle is nine nikharvas, the wise have stated the circumference to be: gods (33), eyes (2), elephants (8), serpents (8), fires (3), three (3), qualities (3), Vedas (4), lunar mansions (27), elephants (8), arms (2).

The math. Read the word-numbers backwards to get 2,827,433,388,233; divide by 900,000,000,000 and you get 3.14159265359, π correct to 11 decimal places. π ≈ 2827433388233 / (9 × 10^11) ≈ 3.14159265359

Word by word, source and notes

vibudha: the gods: 33
netra: eyes: 2
gaja: elephants (of the directions): 8
ahi: serpents: 8
hutāśana: sacred fires: 3
triguṇa: three qualities: 3, 3 (tri and guṇa)
veda: Vedas: 4
bha: lunar mansions: 27
vāraṇa: elephants: 8
bāhavaḥ: arms: 2
navanikharvamite: measuring nine nikharvas (9 × 10^11)
vṛtivistare: when the diameter of a circle
paridhimānam: the measure of the circumference
idaṃ: this
jagaduḥ: have declared
budhāḥ: the wise

Source. Attributed to Mādhavācārya (Mādhava of Saṅgamagrāma, c. 1340 to 1425) in the Kriyākramakarī commentary on Līlāvatī 199 (c. 1534 to 1560), ed. K. V. Sarma 1975.

Good to know. Uses bhūta-saṃkhyā: words for things with a known count, read right to left, so 33, 2, 8, 8, 3, 3, 3, 4, 27, 8, 2 gives 2,827,433,388,233. The value of nikharva (10^11 here) differs between texts. The commentary presents it as Mādhava's value; no independent work of Mādhava with this verse is known. Text read from the archive.org scan of Sarma's edition (OCR checked by hand).

Checked against:

Līlāvatī with Kriyākramakarī, ed. K. V. Sarma, VVRI 1975 (archive.org scan)
Wikipedia: Madhava of Sangamagrama (dates)

Verse (śloka) · sragdharā

A date hidden in a blessing (Kaṭapayādi)

स्फीतं लीलावतारैरिदमिह कुरुतामायुरारोग्यसौख्यम् ॥

sphītaṃ līlāvatārair idam iha kurutām āyurārogyasaukhyam ||

Say it: SPHEE-tam lee-laa-va-TAA-rair I-dam I-ha KU-ru-taam aa-yu-raa-ro-gya-SAU-khyam

Meaning. May this (hymn), rich with (the stories of) your playful incarnations, bring long life, health and happiness here.

The math. The last word, āyurārogyasaukhyam, also hides a number: in the Kaṭapayādi letter code it gives 1712210, the day count since the start of the Kali age on which the poem was finished, in late 1586 CE. Kaṭapayādi code read right to left: ā=0 yu=1 rā=2 ro=2 gya=1 sau=7 khyam=1 → 1712210

Word by word, source and notes

sphītaṃ: rich, full
līlāvatāraiḥ: with (stories of) the playful incarnations
idam: this (hymn)
iha: here
kurutām: may it bring about
āyuḥ: long life
ārogya: health
saukhyam: and happiness

Source. Nārāyaṇa Bhaṭṭa (Melpathur Nārāyaṇa Bhaṭṭatiri), Nārāyaṇīyam 100.11, last quarter-verse; Kerala, completed 1586 CE.

Good to know. In Kaṭapayādi, consonants stand for digits (ka, ṭa, pa, ya = 1 and so on), a bare vowel is 0, in a conjunct only the last consonant counts, and digits are read right to left. Converting Kali day 1712210 to a modern date gives late November or early December 1586 depending on the calendar conversion (Julian versus Gregorian). This is a devotional poem; the number is a chronogram, not mathematics in the text itself.

Checked against:

GRETIL: Nārāyaṇa Bhaṭṭa, Nārāyaṇīyam
Namboothiri Websites: Katapayaadi (decoding of āyurārogyasaukhyam)
INDOLOGY list, March 2014: Converting Kali chronograms (Nārāyaṇīyam date)
Wikipedia: Katapayadi system

Verse (śloka) · anuṣṭubh

The "gopī bhāgya" π verse (from Tirtha's book)

गोपीभाग्यमधुव्रात शृङ्गिशोदधिसन्धिग ।
खलजीवितखाताव गलहालारसन्धर ॥

gopībhāgyamadhuvrāta śṛṅgiśodadhisandhiga |
khalajīvitakhātāva galahālārasandhara ||

Say it: go-pee-bhaa-gya-ma-dhu-VRAA-ta shring-gi-sho-da-dhi-SAN-dhi-ga kha-la-jee-vi-ta-khaa-TAA-va ga-la-haa-laa-ra-SAN-dha-ra

Meaning. Tirtha says the verse can be read three ways: as a hymn to Kṛṣṇa, as a hymn to Śiva, and, through the letter-number code, as π/10 to 32 decimal places. He prints only the number, 0.31415926535897932384626433832792. (The word glosses above give the syllable-to-digit decoding rather than a translation, because the devotional readings are not spelled out in the book.)

The math. Each consonant stands for a digit, so the 32 syllables spell 3 1 4 1 5 9 2 6 ...: a clever memory rhyme for π, made in the 20th century. π/10 ≈ 0.31415926535897932384626433832792 (31 decimals correct; the true 32nd decimal is 5)

Word by word, source and notes

go pī bhā gya: 3 1 4 1
ma dhu vrā ta: 5 9 2 6
śṛ ṅgi śo da: 5 3 5 8
dhi sa ndhi ga: 9 7 9 3
kha la jī vi: 2 3 8 4
ta khā tā va: 6 2 6 4
ga la hā lā: 3 3 8 3
ra sa ndha ra: 2 7 9 2

Source. Bharati Krishna Tirtha, Vedic Mathematics (1965), final chapter "Miscellaneous Matters" (about pp. 360 to 361 in common editions); the code is explained in the chapter on the Vedic numerical code. 20th century.

Good to know. This is the only actual verse in Tirtha's book, and it is not ancient: no source before the book is known, and T. Hayashi called it "a quite new production" by Tirtha. Tirtha calls it well known and attaches no source. Standard Kaṭapayādi reads digits right to left ("aṅkānāṃ vāmato gatiḥ"), but this verse must be read left to right, and Tirtha's own key (with kṣa for zero) differs from the standard one. Our check: decoding gives 31 correct decimals, but the 32nd syllable (ra = 2) gives 2 where π/10 has 5 (0.31415926535897932384626433832795...). It is not the source of the 16 sūtras, which the book's editor V. S. Agrawala says were most likely the Swami's own.

Checked against:

Tirtha, Vedic Mathematics (archive.org OCR text: code chapter and Miscellaneous Matters)
Tirtha, Vedic Mathematics (second archive.org copy with the decoded value)
INDOLOGY list, Oct 1995: T. Hayashi on the verse as "a quite new production" by Tirtha
Wikipedia: Katapayadi system