Brahmagupta in 628 CE: the world's first explicit arithmetic of zero (including a wrong-but-pioneering division-by-zero attempt)

# Brahmagupta in 628 CE: the world's first explicit arithmetic of zero (including a wrong-but-pioneering division-by-zero attempt)

In modern math class, you learn the rules for zero as if they're self-evident:

- $0 + x = x$ - $0 \cdot x = 0$ - $x - x = 0$ - Division by zero: **undefined**

Those rules feel obvious. They aren't. They're propositions someone had to write down for the first time. The first person to write them down — explicitly, as a coherent system of arithmetic rules — was Brahmagupta of Bhillamala (modern Bhinmal, Rajasthan), in 628 CE.

The verse:

> The sum of two affirmative quantities is affirmative; of two > negative is negative; of an affirmative and a negative is their > difference; or, if they be equal, nought. > > — *Brahmasphutasiddhanta* XVIII.19, trans. H. T. Colebrooke (1817)

"Affirmative" is Colebrooke's English for *dhana* (positive number). "Negative" is *rina* (negative number, etymologically related to "debt"). "Nought" is *śūnya* — what we now call **zero**.

The rule says: add two positives, get positive. Add two negatives, get negative. Mix a positive and a negative, you get their difference — and if they're equal in magnitude, the result is zero itself.

That last clause is the move. Brahmagupta treats *śūnya* as a quantity that arithmetic produces, manipulates, and yields. Not a placeholder for "nothing here," not an absence — a **number**.

The other rules in the same chapter

The XVIII.19 stanza is just one piece. Brahmagupta gives a full suite of rules for arithmetic involving cipher (zero) and signed quantities in the same chapter:

- $\text{cipher} + \text{affirmative} = \text{affirmative}$ - $\text{cipher} + \text{negative} = \text{negative}$ - $\text{cipher} + \text{cipher} = \text{cipher}$ - $\text{affirmative} - \text{cipher} = \text{affirmative}$ - $\text{cipher} - \text{negative} = \text{affirmative}$ - $\text{affirmative} \times \text{cipher} = \text{cipher}$ - $\text{cipher} \times \text{cipher} = \text{cipher}$ - $\sqrt{\text{cipher}} = \text{cipher}$ (square root of zero)

All of these match modern arithmetic exactly. Brahmagupta isn't deriving them or proving them; he's *codifying* them, in a way that suggests the rules were already in working use in Indian mathematical practice by the early 7th century. The contribution is making them explicit.

The Indian decimal place-value system — the foundation that makes zero behave as a number — was already in use by Brahmagupta's time. The **Bakhshali manuscript**, carbon-dated by Oxford in 2017 to between 224 and 383 CE, shows a dot symbol for zero used in calculation. By 628 Brahmagupta is working with a fully matured numerical notation; the rules for arithmetic just need to be written down.

The division-by-zero attempt

In Brahmasphutasiddhanta XVIII.35 (per Colebrooke's translation), Brahmagupta extends the arithmetic to division by cipher. The quantity $x \div 0$ he calls *khachhada* — "cipher-cut" — and treats as a finite quantity that obeys ordinary arithmetic rules.

**This is mathematically wrong.** Division by zero is undefined in the real numbers, and treating it as a finite quantity leads to contradictions (e.g., $1/0 = 2/0$ would imply $1 = 2$).

But the attempt itself is the achievement. Brahmagupta is the first mathematician anywhere to seriously **ask** what happens when you divide by zero. He recognizes that division by zero is a special case that needs its own rule. He gives the wrong rule, yes — but the conceptual move of treating "what is $x \div 0$?" as a well-formed question is itself pioneering.

European mathematics didn't seriously engage with division by zero until the development of calculus in the 17th century, and the modern formal answer ("undefined") wasn't established until 19th- century real analysis. From 628 to 1830 is twelve centuries. Brahmagupta tried, got it wrong, and moved on. Europe wasn't ready to even try until the 1600s.

Bhaskara II's correction (~500 years later)

In 1150 CE, Bhaskara II — same Bhaskara who solved Pell's equation — revisits division by zero in his *Bijaganita* (algebra). He introduces the term *khahara* ("having cipher as denominator"). Where Brahmagupta treated $x/0$ as a finite number, Bhaskara argues that *khahara* is *unchanged* by addition or subtraction with finite numbers:

$$\text{khahara} + n = \text{khahara}$$

In modern terms, that's not exactly the limit-theory definition of infinity, but it's strikingly close. If you let $1/x$ as $x \to 0$, you get infinity (in the extended reals), and infinity plus any finite number is still infinity. Bhaskara is reaching for that intuition without the formal machinery.

Whether Bhaskara was "really" doing limit theory is debated by historians of mathematics. The conservative reading (Plofker 2009) is that he had an intuition about infinity-as-the-result-of-dividing- by-zero, expressed it in arithmetic terms, and stopped there. The generous reading (Gupta 1969, Jha & Jha 2005) treats his framing as a genuine infinity-as-limit conception. Either way, Bhaskara is closer to the modern position than Brahmagupta was.

The names "cipher" and "zero"

The Sanskrit *śūnya* means "empty" or "void." It was the placeholder symbol — the dot or small circle — used in Indian arithmetic to indicate an empty position in a decimal number. As Indian mathematics traveled west, the word came with it.

- Sanskrit: *śūnya* - Arabic translation (9th c., al-Khwarizmi and others): *ṣifr* - Medieval Latin (12th c., Fibonacci): *cifra*, later *zephirum* - Old Italian: *zefiro*, contracted to **zero** - English: "cipher" (older), "zero" (newer)

Colebrooke's 1817 translation uses "cipher" throughout. Modern English uses "zero." Both words descend from the same Sanskrit source. When you say "I have zero dollars," you're using a word that came down a 1,400-year linguistic chain from Brahmagupta's *śūnya*.

What Brahmagupta wasn't doing

Three honest caveats to keep this article in proportion:

**1. Brahmagupta didn't invent zero.** The place-value zero — a symbol for "nothing here, but the position matters" — predates him by centuries. The Bakhshali manuscript (3rd-4th c. CE) already shows zero in use. Brahmagupta's contribution is making zero a *number* with arithmetic rules, not inventing the placeholder.

**2. He didn't solve division by zero.** He attempted it and got the answer wrong. The right answer (it's undefined) wasn't reached in Indian mathematics either; it's a 19th-century real-analysis result.

**3. He didn't have negative numbers in the modern sense.** His "affirmative" and "negative" are debt-and-credit in commercial arithmetic context, not the negative integers of modern set theory. The arithmetic he writes down is consistent with both readings, but the conceptual reach is the practical-arithmetic one. The set- theoretic foundations come 1,200 years later.

What Brahmagupta *did* do, indisputably: he wrote down the world's first systematic arithmetic of zero as a number, including an attempt at division by zero. He treated negative numbers as arithmetically equivalent to positives (modulo sign rules). He gave coherent rules for sign, sum, product, and square root with zero. And he did it in 628 CE.

The next civilization to attempt anything as complete was the Arab mathematical tradition of the 9th-10th centuries (building directly on Brahmagupta via al-Khwarizmi). Europe didn't get there until the 17th century. From 628 to ~1600 is **almost a millennium** of intellectual head start.

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Sources

- [Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, H. T. Colebrooke trans., 1817](https://archive.org/details/1817-henry-colebrooke-algebra-with-arithmetic-and-mensuration-from-the-sanskrit-of-br) — Brahmasphutasiddhanta XVIII.19 cited above. - Plofker, K. (2009). *Mathematics in India*. Princeton University Press, ch. 4 — modern critical assessment of Brahmagupta's arithmetic of zero and Bhaskara II's later refinements. - Joseph, G. G. (2011). *The Crest of the Peacock: Non-European Roots of Mathematics* (3rd ed.). Princeton University Press, ch. 8. - Plofker, K. & Sarma, S. R. (2014). "Bhaskara II and his works." In *Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures*. — on the khahara concept.

Related claims

- [Aryabhata's pi approximation](/c/0b862684-d325-5002-b054-169bd2253ef9) — same Indian mathematical tradition, ~130 years before Brahmagupta. - [Bhaskara II's chakravala for Pell's equation](/c/e9b99c06-981f-5f4d-8628-0357df5417f0) — Bhaskara II is the same mathematician who refined Brahmagupta's division-by-zero attempt 500 years later.