Aryabhata's Algorithm for the Square Root of an Integer

number of digits	8
calculate integer square root

random integer: 12 34 56 78

step 1

(20 × 0 + 3) × 3

12

-9

3

________

step 2

(20 × 3 + 5) × 5

3 34

-325

9

________

step 3

(20 × 35 + 1) × 1

9 56

-701

255

________

step 4

(20 × 351 + 3) × 3

255 78

-21069

4509

________

integer square root: 3513

remainder: 4509

This Demonstration shows the different steps in the "longhand" or manual calculation of integer square roots. The integer square root of a number

n

is the greatest integer less than or equal to

n

.

The algorithm used was discovered in 520 AD by the Indian mathematician Aryabhata. It is still used today for longhand calculations of integer square roots.

The main part of the algorithm is the function findLargest, which calculates the successive digits

b

i

of the integer square root. For the

th

i

group of two digits

n

i

, the number

b

i

is the largest integer for which

b

i

(20

a

i

+

b

i

)<

n

i

.

External Links

Square Root Algorithms (Wolfram MathWorld)

Permanent Citation

Erik Mahieu

"Aryabhata's Algorithm for the Square Root of an Integer"
http://demonstrations.wolfram.com/AryabhatasAlgorithmForTheSquareRootOfAnInteger/
Wolfram Demonstrations Project
Published: December 10, 2010