Āryabhaṭa’s square-root rule extracts a root one decimal digit at a time: choose a digit, subtract the part of the square it contributes, then use the remaining value to find the next digit. For 54756, the process yields 234 with no remainder. Its steps follow the expansion of a square, while the familiar contracted long-division layout combines two subtractions that the historical procedure performs separately.
How the digit-by-digit method works
A square grows in predictable parts. If a root has three digits, write it as 100x + 10y + z. Then:
(100x + 10y + z)² = (10x + y)² × 100 + 2(10x + y)z × 10 + z²
This expansion explains the sequence: the first root digit contributes a leading square; the next digit adds a cross-term and its own square; each later digit adds a cross-term with the root already found and a square of its own. After subtracting those contributions, the remainder contains what is needed to determine the following digit. Bhāvanā’s explanation presents Āryabhaṭa’s rule as a decimal place-value algorithm for successive square-root digits.
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Worked example: finding √54756
Read the number in two-digit place-value groups from the right: 5 | 47 | 56. The first group sets the leading root digit; each subtraction then exposes the value needed for the next one. In the arithmetic below, the root is written as 100x + 10y + z, following Bhāvanā’s worked example.
- Find the hundreds digit, x. The leading block is 5, and the greatest square not exceeding it is 4 = 2². Thus
x = 2. Subtract4 × 10⁴ = 40000from 54756; the remainder is 14756. - Find the tens digit, y. Use the leading part of the remainder: 14 divided by
2x = 4gives 3, soy = 3. Subtract the cross-term contribution2xy × 10³ = 12000, leaving 2756. Then subtract the digit’s square at its place,y² × 10² = 900, leaving 1856. - Find the units digit, z. Divide the leading part 185 by
2(10x + y) = 46; the quotient is 4, soz = 4. Subtract the cross term46 × 4 × 10 = 1840, leaving 16. Subtract4² = 16, leaving zero.
The digits are 2, 3, and 4, and the remainder is zero, so √54756 = 234 exactly.
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Why each subtraction reveals another digit
Once a partial root is known, expanding its square shows the incremental cost of appending a digit. For example, if the partial root is 10x + y and the next digit is z, then (10x + y + z)² = (10x + y)² + 2(10x + y)z + z². The next digit must therefore fit the remainder after accounting for both the cross term and its own square. The divisor based on twice the partial root estimates that digit; subtracting the contributions tests and completes the choice. The zero remainder for 54756 confirms that the chosen digits account for the entire number.
For a nonsquare, integer digit extraction can leave a nonzero remainder; it does not make an irrational square root a finite exact integer. Ramasubramanian and Srinivas discuss separate approximation methods, including Śrīdhara’s method of scaling by a large square.
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Āryabhaṭa’s rule and the later contracted layout
The Āryabhaṭīya is dated to approximately 499 CE by Ramasubramanian and Srinivas, whose scholarly account describes the rule as a general method for finding successive square-root digits. They translate its instruction: “Always divide the non-square (even) place by twice the square-root [already found]. Having subtracted the square [of the quotient] from the square (odd) place, the quotient gives the [digit in the] next place in the square-root.” This is the authors’ translation of the verse, not English wording written by Āryabhaṭa.
The procedural distinction from the familiar contracted method is the handling of each new digit’s contribution:
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| Feature | Āryabhaṭa’s described procedure | Contracted layout |
|---|---|---|
| Place-value groups | Works through successive decimal places, selecting one root digit at a time. | Uses the same digit-by-digit place-value logic. |
| Next-digit divisor | Uses twice the root already found, adjusted to the relevant place. | Uses a trial divisor formed from the partial root, adjusted to the relevant place. |
| Subtractions | Subtracts the cross-term contribution and the new digit’s square separately. | Combines those contributions in a single trial-product subtraction. |
| Remainder | Feeds the next digit choice; zero at the end shows an exact square. | Has the same role in the digit extraction. |
Bhāvanā identifies the Yuktibhāṣā of Jyeṣṭhadeva, circa 1530, as an instance of the contracted form. It also cautions that the form’s precise earlier antiquity in India is unknown. The contracted layout is therefore a later attested presentation, not proof that Āryabhaṭa used exactly the school-style arrangement familiar today.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Historical context—and a different square-root result
Calculations in ancient India were often made on a sand-covered board called a pāṭī. Bhāvanā notes that figures could be erased easily and connects this practical setting with procedures designed around decimal place value and zero. The method is not established as dependent exclusively on that surface.
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The Baudhāyana Śulvasūtra belongs to an earlier geometric context and gives a √2-related approximation. That geometric result is distinct from Āryabhaṭa’s general digit-by-digit procedure for extracting square-root digits; the two should not be treated as the same algorithm.
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