What is the square root of a number?

How it works

The square root of a number x is the number r that is 0 or more and gives x when squared:

√x = r, where r ≥ 0 and r × r = x

Fill in either box and the calculator finds the other (leave the other box empty):

• Square root: r = √x. Only x of 0 or more has a real square root.
• Number: x = r².

It also shows:

• Negative square root: −√x. Both √x and −√x square to x, but √x means the one that is 0 or more.
• Simplified radical form (whole numbers only): √x written as k√s (k and s without thousands separators), where s is a whole number with no square factor other than 1. The calculator splits x into prime factors. Each pair of equal prime factors moves out of the root as one factor of k, and each unpaired prime factor stays inside, in s. When nothing is left inside (s = 1), x is a perfect square and the form is just k.
• Perfect square? Yes when the whole number x is the square of a whole number (s = 1).

Assumptions

• Real numbers only: x must be 0 or more.
• Whole numbers are exact. A number typed as digits only (with optional thousands commas in groups of three, such as 12,345,678; a comma elsewhere, as in 1,5, has no answer; a true minus sign − reads as a minus) is read as an exact whole number, up to 100 digits. Longer whole numbers have no answer ("Enter a whole number of at most 100 digits."). The square root's whole part ⌊√x⌋ comes from Newton's method on whole numbers, so the perfect-square check is exact at any such size: 12,345,678,987,654,321 = 111,111,111², so its root is 111,111,111 and it is a perfect square. The root of a perfect square is shown with every digit. The root of a whole number that is not a perfect square is irrational and is shown as a 64-bit float, about 16 significant digits.
• The simplified radical form is shown for every perfect square, and for other whole numbers from 0 to 10¹⁸. Above 10¹⁸ it is left out for numbers that are not perfect squares. √0 = 0, and 0 is a perfect square.
• The number is shown in the answer as typed, with thousands separators: "The square root of 12,345,678,987,654,321 is 111,111,111." The link keeps the typed digits.
• Decimals (a number with a decimal point or in e-notation, such as 0.25 or 1e20) are read as 64-bit floats, about 16 significant digits, and get no radical form or perfect-square check. A number in e-notation is written out in full in the answer (2.50e3 shows as 2,500). One limit applies both ways: a typed number, a decimal, or a number worked out from the root has at most 100 digits before the decimal point (below 10¹⁰⁰); a larger decimal has no answer ("Enter a number with at most 100 digits before the decimal point.").
• Text that is not a number has no answer ("Enter a number, for example 1,200 or 3.5."), and so does a negative number ("Enter 0 or more: a negative number has no real square root.").
• From the root: leave the number empty and type a square root r of 0 or more. r² is worked out exactly from the decimal digits of r as typed (1.5 is 15 ÷ 10, so 1.5² = 225 ÷ 100 = 2.25) and rounded once. A whole r gives the exact whole number r², with its radical form and the perfect-square check (Yes). Another r gives the 64-bit float nearest to the exact r², shown as the shortest decimal that reads back as the same number (at most 17 significant figures). A result with more than 100 digits before the decimal point (r² ≥ 10¹⁰⁰, for example r = 10⁵⁰) has no answer ("The number would have more than 100 digits before the decimal point."), and so does a non-zero result below about 2.2 × 10⁻³⁰⁸. When both boxes are filled (an old link), the page answers from the number if the square root fits it (within 1 part in 10¹² of the square root worked out from the number), and otherwise has no answer ("These do not fit: clear the number or its square root.").

Worked examples by hand

√72. 72 = 2 × 2 × 2 × 3 × 3. The pairs are (2, 2) and (3, 3), so k = 2 × 3 = 6, and one 2 is left inside: √72 = 6√2. As a decimal, 6 × 1.41421356 = 8.48528137. The negative root is −8.48528137. 72 is not a perfect square.

√144. 12 × 12 = 144, so √144 = 12, and 144 is a perfect square.

√2. 2 is prime, so nothing comes out: the simplified form is √2 = 1.41421356.

The number whose square root is 1.5. 1.5 × 1.5 = 2.25.

√0.25. 0.5 × 0.5 = 0.25, so √0.25 = 0.5. It is not a whole number, so there is no simplified radical form.

√12,345,678,987,654,321. Newton's method gives ⌊√x⌋ = 111,111,111, and 111,111,111² = 12,345,678,987,654,321 exactly, so the root is 111,111,111, the simplified form is 111111111, and it is a perfect square.

√1,800. 1,800 = 2³ × 3² × 5². The pairs give k = 2 × 3 × 5 = 30, and one 2 is left inside: √1,800 = 30√2 = 42.42640687.