The monkey problem with two right answers: Bhāskara on quadratic roots (1150 CE)

# The monkey problem with two right answers: Bhāskara on quadratic roots (1150 CE)

Here is problem §139 of the *Bījagaṇita*, Bhāskara II's 1150 CE algebra, in Colebrooke's translation:

> "The eighth part of a troop of monkeys, squared, was skipping in a > grove and delighted with their sport. Twelve remaining were seen > on the hill, amused with chattering to each other. How many were > they in all?"

Set it up: (x/8)² + 12 = x, which rearranges to x² − 64x + 768 = 0. The quadratic formula gives x = (64 ± 32)/2 — that is, **48 or 16**.

Now check both. A troop of 48: one-eighth is 6, squared is 36 monkeys in the grove, plus 12 on the hill — 48. ✓ A troop of 16: one-eighth is 2, squared is 4 in the grove, plus 12 on the hill — 16. ✓ Two different troops, both perfectly consistent with the story. Bhāskara chose the numbers deliberately: the problem exists to teach that a quadratic has **two roots**, and that sometimes both of them are true.

And when they aren't

The companion lesson is the deeper one. In the neighboring problems, the second root is *not* valid — it comes out negative, or violates the story's terms — and the text teaches the student to throw it out. Colebrooke's volume preserves the reasoning in the commentary layer (Kṛṣṇa, glossing Bhāskara's rule): a value from such an equation "must be incongruous; because it is negative: **for people do not approve a negative absolute number**." A negative count of monkeys is not a solution; it is an artifact of the algebra, and the student is taught to recognize the difference.

Put the two lessons together and you have something genuinely modern: the equation is one thing, the problem is another, and solving means finding *all* the equation's roots and then *testing each against the problem's conditions*. Every algebra teacher today calls this "checking for extraneous solutions." It is standard curriculum in the *Bījagaṇita*, in 1150, taught through monkeys.

The honest comparison

Quadratic technique itself is ancient and multi-civilizational, and the claim says so: Babylonian scribes solved quadratic problems nearly three millennia before Bhāskara; Brahmagupta (628 CE) had already given the general Indian method; al-Khwārizmī's *Algebra* (9th c.) classified and solved the cases geometrically. Technique is not the claim. The claim is the *theory of the solutions* — the explicit doctrine that roots come in twos, exhibited with a both-roots-valid example, paired with a rejection criterion for roots that fail.

Europe's path to the same comfort was long. Al-Khwārizmī, working without negatives, treated the cases separately and took positive roots. Cardano (1545) handled negative and even complex roots but called them "fictitious." Descartes, in *La Géométrie* (1637) — half a millennium after the *Bījagaṇita* — was still calling negative roots *racines fausses*, "false roots." The mathematics was available to everyone; the *attitude* — treating the second root as information about the problem rather than an embarrassment — is where the traditions differed, and where the Indian textbook tradition (which had [embraced negative numbers as "debts" since Brahmagupta](/c/956cdea3-3b11-535c-844b-de8c73a129aa)) was unusually relaxed.

One more honesty note: the "people do not approve" phrasing sits in the 16th-century commentary expanding Bhāskara's terser rule; the article distinguishes text from gloss, as the claim notes do. The doctrine is Bhāskara's; the memorable sentence is his commentator's.

Legacy

The two-root doctrine flowed into the tradition's crown jewels: [the chakravala method's](/c/e9b99c06-981f-5f4d-8628-0357df5417f0) handling of solution families, and ultimately the general theory of equations. When European algebra finally normalized negative and multiple roots in the 17th–18th centuries, it converged on the position the Sanskrit textbooks had taught for half a millennium: count every root, then interrogate each one.

And the monkeys kept their job. Bhāskara's menagerie problems — monkeys, bees, [the whole poetic bestiary](/c/9078a2cf-c645-5145-a8b8-41ec1fd1c89a) — survived into modern Indian schoolbooks and puzzle collections, still teaching the same lesson: the algebra gives you candidates; the world picks the winners. A student who has met the two-troops-of-monkeys problem never again confuses solving the equation with answering the question — which may be the most transferable mathematical skill ever smuggled inside a joke about monkeys. He also, elsewhere in the same book, [divided by zero and named the result](/c/bc2f3ce7-9321-5405-8fae-9e56ff80f6b2); §139 shows the same mind insisting that even correct algebra must answer to reality.

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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 §139 cited; Kṛṣṇa's commentary on root rejection. - Plofker, *Mathematics in India*, 2009, ch. 6 — Bhāskara's algebra (secondary synthesis). - Descartes, *La Géométrie*, 1637 — "false roots" (referenced for the comparison only).

Related claims

- [Bhāskara II divides by zero — khahara (1150 CE)](/c/bc2f3ce7-9321-5405-8fae-9e56ff80f6b2) - [The Līlāvatī: an algebra textbook written as poetry (1150 CE)](/c/9078a2cf-c645-5145-a8b8-41ec1fd1c89a) - [Brahmagupta's sign rules for negatives (628 CE)](/c/956cdea3-3b11-535c-844b-de8c73a129aa)