Square Root Calculator

What the square root of a number means

Squaring a number multiplies it by itself. Nine squared is 9 x 9 = 81. The square root runs that in reverse: it asks which number, multiplied by itself, produces the value you started with. The square root of 81 is 9, the square root of 100 is 10, and the square root of 0.09 is 0.3, since 0.3 x 0.3 = 0.09.

A helpful picture is area. If a square garden covers 144 square meters, each side is the square root of 144, or 12 meters. Area lives in two dimensions, so recovering one side from an area always calls for a square root. Cube roots do the same job one dimension up: a box holding 216 cubic inches has edges of 6 inches, because 6 x 6 x 6 = 216.

Every positive number has two numbers that square to it: 3 x 3 = 9 and (-3) x (-3) = 9. The square root symbol names the nonnegative one, called the principal root. So the square root of 9 is 3, never -3, even though both square to 9. Most roots are irrational numbers whose decimals never end or repeat; the square root of 2 starts 1.41421356 and has been computed to trillions of digits. The NIST Digital Library of Mathematical Functions treats powers and roots as the elementary functions they are, and MathWorld's square root entry collects the history and identities in one place.

Perfect squares and the simplified radical form

A perfect square is a whole number times itself. The first fifteen are:

A perfect square has a whole number root, so the widget answers 12 for 144 with no decimals at all. Numbers in between still simplify in a useful way. Since 72 = 36 x 2 and 36 is the largest perfect square that divides 72, the square root of 72 equals 6 x square root of 2. That exact form beats a decimal whenever you want to carry the value through later algebra, because no rounding ever enters the picture.

The widget finds the largest square factor by trial: it walks down from the floor of the root and tests whether the square divides the input. For 72 it lands on 36, leaves the 2 under the radical, and reports 6 x square root of 2. A number like 61, which is prime, keeps its root whole and gets reported as square root of 61.

How Newton's method homes in on the root

Calculators rarely look the root up. They iterate. To solve x squared equals n, take any guess x, divide n by it, and average the two. A guess that is too low makes n/x too high, and a guess that is too high makes n/x too low, so the average lands closer to the true root every time. Written out, the update rule is x next = (x + n/x) / 2, a scheme known to Babylonian mathematicians as a method of repeated averaging and analyzed today as Newton's method applied to f(x) = x^2 - n. Khan Academy's algebra 2 course covers radicals and their approximations in the same spirit.

Here is the widget's own loop for 72, starting from half the input:

The guesses settle on 8.485281 within a handful of rounds, and the widget stops after 7 steps in total, once two consecutive guesses agree to 12 decimal places. The convergence is quadratic, which means the count of correct digits roughly doubles on every pass, so even a poor first guess needs only a few rounds.

Square root rules that always hold

A short list of identities covers most algebra that involves roots. Each one is exact, not an approximation:

The last row is the classic trap. Because the product rule multiplies under the radical, people expect an addition version, and it does not exist. The check line in the widget exists for the same reason: every claimed root should square back to the input, and the widget shows that multiplication on screen so you never have to take a decimal on faith.

Estimating a root between two perfect squares

Before any iteration, a bracket gives you a fast mental estimate. For 72, the nearest perfect squares below and above are 64 and 81, so the root sits between 8 and 9. Since 72 is more than halfway from 64 to 81, the root sits closer to 9 than to 8, and 8.5 is a reasonable first guess. The widget prints the two nearby perfect squares, 64 and 81, for every non perfect input, which gives you that bracket without any work.

The same trick works downward for small numbers. The root of 0.5 lies between the roots of 0.49 and 0.64, that is between 0.7 and 0.8. For large numbers, note that the gap between consecutive perfect squares grows: 10000 and 10201 are adjacent squares, so a value near 10000 is pinned down to within about one percent by bracketing alone. Newton's method then refines the bracket to as many digits as you want.

Fractions, decimals and negative numbers

Fractions obey the quotient rule. The square root of 16/25 is the square root of 16 over the square root of 25, which is 4/5, a clean 0.8. Decimals are handled the same way as whole numbers: the widget returns 0.5 for 0.25, after 6 Newton steps starting from 1, and the check line confirms that 0.5 x 0.5 restores 0.25.

Negative inputs are a different case, because no real number squares to a negative value. The convention is to move to complex numbers: the square root of -9 is written i x 3, where i is the imaginary unit with i x i = -1. The widget reports not a real number for the decimal column, shows i x 3 as the principal root, and displays the message that 3 x 3 = 9 explains the form. Engineers meet these roots constantly in alternating current analysis, where impedances naturally take complex values.

Two inputs deserve their own mention. The square root of 0 is 0, and the square root of 1 is 1. They are the only numbers that are their own nonnegative square roots, and the Newton loop is skipped for 0 because the division n/x would be undefined.

Worked examples with the calculator

Every number below was produced by the widget on this page and confirmed in an automated test, so the outputs you see here are the outputs you will get:

The 144 row shows why perfect squares are easy to spot in the wild: the widget flags it, returns a whole number, and lists 144 and 169 as the neighboring squares. The 72 row is the typical case, where the decimal 8.485281 and the exact form 6 x square root of 2 describe the same number. For the negative input, the check and nearby square lines show a dash, since the real number line no longer contains the root.

What the calculator returns

Enter any number and press the button. The results table shows six lines: the decimal root rounded to six places, a yes or no flag on whether the input is a perfect square, the simplified radical form, the number of Newton steps the loop used together with its starting guess, a check line that squares the root back to the input, and the two nearest perfect squares for bracketing. A message line appears for negative inputs to explain the i notation.

Everything runs in your browser with plain JavaScript. No value is uploaded, no request leaves the page, and the same arithmetic works offline once the page has loaded. The rounding to six decimal places is a display choice; the exact radical form is available whenever you need a value with no rounding at all.