Bhāskara II divides by zero — and names the result (1150 CE)

# Bhāskara II divides by zero — and names the result (1150 CE)

Type 3 ÷ 0 into a calculator and you get an error. Ask a modern mathematician and you'll be told the expression is undefined. Ask Bhāskara II, writing in Sanskrit in 1150 CE, and you get an answer — with a name, and rules for how it behaves.

In the *Bījagaṇita*, his treatise on algebra, Bhāskara works the problem directly. Dividend 3, divisor 0. The quotient, he writes, is the fraction 3/0:

> "This fraction, of which the denominator is cipher, is termed an > infinite quantity." > > — *Bījagaṇita*, rules for cipher (1150 CE), trans. Colebrooke (1817)

The technical term is *khahara* — from *kha* (a standard Sanskrit word-numeral for zero, literally "space") and *hara* ("divisor"). A quantity with zero as its divisor. Colebrooke's footnote records the commentators' gloss: *ananta-rāśi*, "unbounded quantity," which "cannot be determined how great it is."

Who was Bhāskara II?

Bhāskara (1114 – c. 1185 CE), often called Bhāskarāchārya ("Bhāskara the teacher"), headed the astronomical observatory at Ujjain, the premier mathematical institution of medieval India. His two best-known books — the *Līlāvatī* on arithmetic and the *Bījagaṇita* on algebra — became the standard mathematical textbooks of the subcontinent for the next seven centuries. He stands at the end of a five-hundred-year lineage that runs back through Mahāvīra to [Brahmagupta's arithmetic of zero](/c/d10352e6-dc8c-58ee-bbbf-9b1489efdc9e) (628 CE), and he inherited its unfinished business.

Division by zero was the loose end. Brahmagupta, who wrote the first explicit rules for adding, subtracting, and multiplying with zero, stalled here: he declared 0/0 = 0 (wrong, by modern lights) and left a/0 as an unresolved formal fraction. Mahāvīra (850 CE) did worse, asserting that a number divided by zero is unchanged. Division by zero was a known problem in the tradition Bhāskara trained in — his answer to it was not a stray remark but a considered position.

What Bhāskara actually says

Two things, and the pairing is the point.

First, the definition: 3/0 is *khahara*, an infinite quantity. Not "impossible," not "nothing" — a defined mathematical object.

Second, its behavior. The *khahara*, he writes, admits "no alteration, though many be inserted or extracted" — add any finite quantity to it or remove any finite quantity from it, and it stays what it is. Colebrooke's commentators illustrate with the shadow of a gnomon at sunrise: infinite, and equally infinite whatever the height of the gnomon. Bhāskara himself reaches for a more striking simile — no change takes place in the infinite and immutable deity when worlds are absorbed or put forth at the destruction or creation of the universe. Infinity, in the *Bījagaṇita*, is the quantity that swallows all finite arithmetic. (The theological simile is quoted here with repaired text; the 1817 printing's scan is OCR-damaged at this passage.)

That invariance rule — ∞ ± finite = ∞ — is exactly how infinity behaves in the projectively extended real line, and it is literally how every computer on Earth behaves today: IEEE-754 floating point, the arithmetic standard in every processor, defines 1.0/0.0 as `Infinity`, and `Infinity` plus any finite float is `Infinity`. Bhāskara's *khahara* rule, stated for the same reason — so that algebra can keep going instead of halting — is running in your phone.

The honest comparison

European mathematics had no working object for this until John Wallis introduced the ∞ symbol in 1655 and wrote 1/0 = ∞ — five centuries after the *Bījagaṇita*. A rigorous account of division by zero had to wait longer still, for the limit concept of Cauchy's 1821 *Cours d'analyse*, which resolved the question by dissolving it: modern real analysis leaves a/0 undefined and handles the underlying idea with limits.

And measured against that standard, Bhāskara's system has a genuine defect, which honesty requires stating plainly. In worked problems he allows a quantity that has been multiplied by zero and then divided by zero to recover its original finite value — treating 0/0 as if the zeros cancel. Modern mathematics calls 0 · ∞ indeterminate precisely because that cancellation is illegitimate; it is why calculus needed limits. Bhāskara defined infinity's additive behavior correctly and its multiplicative interaction with zero incorrectly. The definition was five centuries early; the full algebra of infinity needed another seven.

There are also non-European precedents to credit within his own tradition: the conceptual groundwork — zero as a number with rules, rather than a mere placeholder — is Brahmagupta's, and the willingness to declare an operation's result impossible rather than fudge it has a parallel in [Mahāvīra's flat statement that negatives have no square roots](/c/a1a67dc8-e4eb-5313-979e-fbaad1a7857f). Bhāskara's contribution is the *object*: infinity with a name and an algebra.

Legacy

The *Bījagaṇita* and *Līlāvatī* stayed in continuous use in Sanskrit mathematical education into the colonial period — Persian translations were made at Akbar's court in the 1580s, and Colebrooke's 1817 English translation (the source text for this claim) was produced precisely because British mathematicians wanted to know what the Indian algebra tradition contained. Through that translation, Bhāskara's khahara became one of the first pieces of Sanskrit mathematics European readers encountered in full.

The idea itself — treat the result of division by zero as a defined infinite element rather than an error — reappears wherever mathematics needs arithmetic to be total: the projective line, the Riemann sphere of complex analysis, floating-point hardware. None of these descend historically from the *Bījagaṇita*; all of them arrive at its move. A twelfth-century algebra textbook and a twenty-first century CPU agree on what 3/0 should be, and agree on why: the computation must be allowed to continue.

The man who wrote it down also solved [Pell-type equations Europe wouldn't crack until Fermat's era](/c/e9b99c06-981f-5f4d-8628-0357df5417f0) — khahara is one entry in a long ledger.

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Sources

- [Colebrooke, *Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara*, 1817](https://archive.org/details/1817-henry-colebrooke-algebra-with-arithmetic-and-mensuration-from-the-sanskrit-of-br) — Bījagaṇita, rules for cipher §14–16 cited; commentators' glosses from Colebrooke's footnotes. - Plofker, *Mathematics in India*, 2009 — secondary scholarly synthesis (Bhāskara II chapter; referenced for context only). - IEEE Std 754-2019 — modern floating-point behavior of division by zero (referenced for the modern comparison only).

Related claims

- [Brahmagupta writes the rules for zero (628 CE)](/c/d10352e6-dc8c-58ee-bbbf-9b1489efdc9e) - [Mahāvīra: negatives have no square roots (850 CE)](/c/a1a67dc8-e4eb-5313-979e-fbaad1a7857f) - [Bhāskara II's chakravala method (1150 CE)](/c/e9b99c06-981f-5f4d-8628-0357df5417f0)