Square Root Calculator

Background.

The square root of a non-negative real number is the unique non-negative real number that, when multiplied by itself, yields the original value. This operation is among the oldest in mathematics. The Old Babylonian tablet YBC 7289 in the Yale Babylonian Collection, usually dated to roughly 1800–1600 BCE, draws a square with its diagonals and writes √2 in base 60 as 1;24,51,10. In decimal that is 1.41421296, within about six ten-millionths of the true value.

Today, square root computation is embedded in every branch of quantitative science, from calculating standard deviations in statistics to determining signal magnitudes in electrical engineering and solving quadratic equations in physics. Many everyday questions are square roots in disguise: a square plot of 2,500 square metres has sides of 50 metres, and a variance of 0.04 means a standard deviation of 0.2.

In financial modeling, square roots appear in the volatility term of the Black-Scholes option pricing formula, where volatility is scaled by the square root of the time to expiry. In computer graphics, square root operations normalize vectors and compute Euclidean distances in three-dimensional space. In civil engineering, square roots determine the resultant of perpendicular force components and the geometric mean of material properties.

The Root degree field reaches past square roots. Enter 3 for a cube root, which turns the volume of a cube into its edge length, 4 for a fourth root, and any larger whole number for the corresponding nth root. Answers appear to four decimal places, and switch to scientific notation when they fall below 0.001 or reach 10¹⁵, so the square root of 0.0000001 reads 3.162 × 10⁻⁴ rather than a misleading 0.0003.

What is square root calculator?

The square root of a non-negative real number a is the non-negative real number x such that x² = a. It is denoted by √a or a^(1/2). Every positive real number has exactly two square roots, one positive and one negative, but the symbol √a refers exclusively to the principal (non-negative) root. The square root of zero is zero, and the square root of one is one. Negative real numbers do not have real square roots; their square roots are pure imaginary numbers in the complex plane. The square root generalizes to the nth root, denoted ⁿ√a or a^(1/n), where n is an integer greater than or equal to 2. For even n, the nth root is defined only for a ≥ 0 and yields a non-negative result. For odd n, the nth root is defined for all real a, and the sign of the result matches the sign of the input. The Number field accepts negative values too, because an odd root of a negative number is real: the cube root of −27 is −3. An even root of a negative number is refused. The nth root is the inverse operation of raising a number to the nth power.

How to use this calculator.

1. 01Enter the number you want the root of in the Number field. It may be negative only if the root degree is odd; decimals such as 0.75 or 12.5 are fine.
2. 02Enter the root degree as a whole number in the Root degree field: 2 for a square root, 3 for a cube root, 4 for a fourth root. A fractional degree such as 2.5 is refused; for that, raise the number to the power 1/2.5 = 0.4 with an exponent calculator instead.
3. 03Read the principal root in the Result box. It shows up to four decimal places, and very small or very large answers switch to scientific notation.
4. 04To check an answer, raise it to the root degree. Squaring the displayed 1.4142 gives 1.99996164 rather than exactly 2, and the gap comes from the four-decimal display, not from the root itself.

The formula.

The square root function is defined implicitly by the equation x² = a for x ≥ 0 and a ≥ 0. For perfect squares such as a = 9 or a = 144, the square root is a whole number and can be found exactly by factoring or by lookup. For a positive integer that is not a perfect square, such as a = 2, the root is irrational: its decimal expansion is infinite and non-repeating, so every computation produces an approximation. Newton's method provides a rapidly converging iterative scheme. Starting from an initial guess x₀ > 0, the update rule x_{k+1} = (x_k + a/x_k)/2 averages the current guess with its co-factor. If x_k is too large, a/x_k is too small by a similar amount, so their average lands much closer to the root. The error e_k = x_k − √a satisfies e_{k+1} ≈ e_k² / (2√a), demonstrating quadratic convergence: the number of correct digits roughly doubles with each iteration. Double precision carries 53 significant bits, so a starting guess good to about 8 bits reaches full precision in three steps (8, 16, 32, then 64 bits). One Newton step from the nearest perfect square also makes a good mental estimate: for √50, start at 7 and compute (7 + 50/7)/2 = 7.0714, against the true 7.0711. For the general nth root, Newton's method applied to f(x) = xⁿ − a yields x_{k+1} = ((n−1)x_k + a/x_k^(n−1))/n.

This calculator does not iterate. It evaluates a^(1/n) with the built-in double-precision power function, which is the identity ⁿ√a = exp((ln a)/n) written as a power. One side effect is worth knowing: the exponent 1/n can be stored exactly in binary only when n is a power of two, so for a cube root the stored exponent sits a hair below 1/3. The cube root of 64 is therefore computed as 3.9999999999999996 rather than exactly 4. The four-decimal display rounds this to 4, so it only surfaces if you copy a full-precision value into further arithmetic elsewhere.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Hardy, G.H. (1908). A Course of Pure Mathematics. Cambridge University Press.
2. [2]Knuth, D.E. (1997). The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Addison-Wesley.
3. [3]Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. NBS Applied Mathematics Series 55.
4. [4]NIST Digital Library of Mathematical Functions.
5. [5]Newton, I. (1671). Method of Fluxions. (Published 1736).