Integer square root calculator
The integer square root of 82 is 9, because 9 squared is 81 and 10 squared is already 100. Paste any non-negative whole number, even one with dozens of digits, and get that rounded-down root with every digit exact. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 123456789012345678901234567890
- Square root
- 351364182882014
The integer square root calculator returns the greatest whole number whose square does not exceed your input. Unlike an ordinary floating-point square root followed by rounding, it works with exact integer arithmetic, so very large decimal integers do not lose digits. The result is often called isqrt or floor sqrt. Enter any non-negative decimal integer, and the capability applies Newton iteration until the estimate stabilizes. It rejects negative values, fractions, exponents, and other ambiguous input instead of silently changing them.
What an integer square root means
For a non-negative integer n, the integer square root is the unique whole number r such that r squared is less than or equal to n, while the square of r plus one is greater than n. In everyday terms, it is the ordinary square root rounded down, never rounded to the nearest integer. If n is 81, the answer is 9 because the input is a perfect square.
If n is 82, the answer remains 9 because 9 squared is 81 and 10 squared is already 100. This definition also handles the boundary values cleanly: zero produces zero, and one produces one. The calculator returns both the normalized input and the result as decimal strings.
Supply digits only, with an optional leading plus sign. Negative integers are outside the stated domain and produce an invalid-input error rather than a complex result. Fractions and scientific notation are rejected because this operation is specifically defined here for exact integers.
How Newton iteration reaches the exact floor
The calculation uses an integer form of Newton's method. It begins with a power-of-two estimate chosen from the bit length of n, which guarantees an estimate at or above the true square root. Each step replaces the current estimate x with the floor of the average of x and n divided by x.
When x is too large, the quotient n divided by x is too small, so their average moves rapidly toward the root. The number of correct digits roughly doubles during the useful part of the process, making the method practical even for inputs with many decimal digits. All divisions are integer divisions and all intermediate values use arbitrary-precision integers.
No floating-point approximation participates in the answer. Iteration stops when the next estimate would no longer be smaller. At that point, the current estimate satisfies the floor-square-root definition.
This stopping condition also avoids oscillation around a non-perfect square. The algorithm is deterministic: the same accepted input always follows the same arithmetic rules and returns the same decimal result, without network calls, randomness, timestamps, or machine-dependent rounding.
Using the result safely in programs
Send an object with an n field containing a non-negative decimal integer string. A string avoids that damage and can be converted into a native big-integer type by clients that support one. The result is useful for algorithms that need exact bounds rather than approximate measurements.
You can determine the side length of the largest integer square fitting within an area, cap a divisor search at the exact square-root boundary, build integer geometry checks, or partition a count into the largest possible square grid. Do not send commas, spaces between digits, decimal points, or exponent notation.
What you can do with it
Bound a factor search
Find the exact final candidate needed when testing integer divisors up to the square root.
Size a square grid
Compute the largest whole-number side length whose square cells fit within a given item count.
FAQ
Does this return a rounded square root?
It always rounds down, returning the greatest integer whose square is no larger than n.
Can I submit a very large integer?
Yes. Submit it as a decimal string so every digit remains exact; the calculation uses arbitrary-precision integer arithmetic.
What is the integer square root of 82?
It is 9. Nine squared is 81, which does not exceed 82, while ten squared is 100, which does. The integer square root is always the ordinary square root rounded down.
Can I enter a negative number or a decimal?
No. Only whole numbers from zero upward are accepted, written as digits with an optional leading plus sign. Negative values, fractions and scientific notation are refused instead of being silently changed.
How can I check the result myself?
Square the result and confirm it is not larger than your number. Then add one to the result, square it again, and confirm that this time it is larger. Both checks together prove the answer.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/numth/integer-square-root \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":"123456789012345678901234567890"}'const res = await fetch("https://api.kit.forhosting.com/numth/integer-square-root", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": "123456789012345678901234567890"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/numth/integer-square-root",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": "123456789012345678901234567890"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/numth/integer-square-root", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":"123456789012345678901234567890"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":"123456789012345678901234567890"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/numth/integer-square-root", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": "123456789012345678901234567890"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "numth.integer_square_root",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
What an integer square root means
Strings preserve every digit across JSON clients, including values much larger than the safe integer range of JavaScript numbers.
Using the result safely in programs
The response includes n in normalized decimal form and square_root as a decimal string. Keeping both fields as strings is deliberate: many API environments parse JSON numbers through binary floating point, where integers above 9,007,199,254,740,991 can change before the calculation even begins. When verifying the result, compare square_root squared with n and also check that square_root plus one, squared, is greater than n. A negative input returns invalid_input, allowing callers to distinguish a domain mistake from an infrastructure failure. Each API request uses the published base price of $0.002.
Integration use cases
Process large identifiers exactly
Calculate floor square roots of decimal integers that cannot be represented safely as ordinary JSON numbers.
Technical questions
What happens for a negative number?
Negative values are outside the domain and return an invalid_input error.
Why are the output values strings?
Decimal strings preserve exact integers across JSON systems that may otherwise round large numeric values.
What does an API request cost?
The base price is $0.002 per request.