How this calculator works
A perfect square is a whole number that equals another whole number multiplied by itself. The calculator finds the integer square root by exact integer arithmetic rather than a floating-point guess, compares the square with the input, and reports the neighbouring squares. Negative squares do not exist among the real numbers, so the input starts at zero.
Formula
n is a perfect square when n = k^2 for an integer k ≥ 0. Every perfect square has the form 4m or 4m + 1, and its last digit is 0, 1, 4, 5, 6 or 9.
Worked example
144 = 12^2, so 144 is a perfect square and its roots are 12 and -12. The number 150 is not a perfect square: it lies between 144 (12^2) and 169 (13^2).
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Frequently asked questions
Is zero a perfect square?
Yes. 0 = 0^2, so zero is a perfect square. It is the only perfect square with a single square root in the non-negative integers.
Does the calculator accept decimals or negatives?
No. Perfect squares are defined here for whole numbers from zero upwards. A negative number has no real square root, and a decimal such as 2.25 is a perfect square only in a looser sense (1.5^2), which this page does not test.
Why does the status mention two roots?
Every positive perfect square has a positive and a negative root, because multiplying two negative numbers gives a positive product. The integer square root output is the non-negative one.