Calculators
Integer Square Root Calculator
Find the exact integer square root, remainder, and perfect-square status of integers up to 1000 digits, privately in your browser.
Enter up to 1000 digits. The integer square root is the largest whole number whose square does not exceed n. All calculations stay in your browser.
How integer square roots work
The integer square root of a non-negative integer n is the largest integer r such that r² ≤ n. Binary search narrows an interval containing the root using exact BigInt comparisons. The remainder is n − r²; it is zero exactly when n is a perfect square. The next square is (r + 1)², which always exceeds n. For n = 12345, r = 111, r² = 12321, the remainder is 24, and the next square is 12544, a distance of 199 away. Zero has root zero and is a perfect square. This tool returns a floor root rather than a decimal approximation. Enter up to 1000 digits; leading zeros and surrounding whitespace are accepted. Negative numbers, plus signs, decimals, commas and scientific notation are rejected. All arithmetic is exact and runs locally without uploading or storing inputs.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is an integer square root?
It is the largest whole number r with r² ≤ n. For 20 the result is 4 with remainder 4, because 4² = 16 and 5² = 25.
How can I check for a perfect square?
A zero remainder means the input is a perfect square. Zero and one are both perfect squares.
Are large results rounded?
No. BigInt arithmetic and binary search preserve every digit for inputs of up to 1000 digits. This calculator returns a floor root, not a decimal approximation.
Which inputs are accepted?
Enter a non-negative integer with up to 1000 digits. Leading zeros and surrounding whitespace are accepted; signs, decimals, commas and scientific notation are rejected. Inputs stay in your browser.