Audited 06 Jun 2026·Last updated 27 Jul 2026·5 citations·Tier 1·0 uses

Square Root Calculator

Calculate square roots and nth roots with high precision. Enter any non-negative real number and root degree for exact mathematical results.

Square Root Calculator

Non-negative real number to take the root of
Integer degree: 2 for square, 3 for cube, etc.
Result
1
Principal nth root of the input number

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, with evidence of square root computation appearing in Babylonian clay tablets from approximately 1800 BCE. The YBC 7289 tablet from the Yale Babylonian Collection gives an approximation of √2 as 1.41421296 in sexagesimal notation, accurate to six decimal places.

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. The need for a dedicated square root calculator persists despite the ubiquity of scientific calculators and spreadsheet software because precision requirements vary dramatically across domains.

In financial modeling, square roots appear in the volatility term of the Black-Scholes option pricing formula, where small rounding errors can shift derivative prices by meaningful amounts. 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. A calculator that accepts arbitrary non-negative inputs and allows custom root degrees serves all of these use cases in a single interface.

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 nth root is the inverse operation of raising a number to the nth power.

How to use this calculator.

  1. Enter the number you wish to take the root of in the Number field.
  2. Enter the root degree in the Root degree field (2 for square root, 3 for cube root, etc.).
  3. Ensure the number is non-negative if the root degree is even.
  4. Click Calculate to compute the principal nth root.
  5. Review the result, which is rounded to the precision of double-precision floating-point arithmetic.
  6. For verification, raise the result to the power of the root degree; it should equal the original number.

The formula.

x = ⁿ√a ⇔ xⁿ = a
x · principal nth roota · input numbern · root degree

The square root function is defined implicitly by the equation x² = a for x ≥ 0 and a ≥ 0. There is no finite algebraic expression for √a in terms of elementary arithmetic operations for general a; the function is transcendental in the sense that it cannot be expressed using only addition, subtraction, multiplication, division, and integer-root extraction a finite number of times. For rational inputs that are perfect squares, such as a = 9 or a = 144, the square root is rational and can be computed exactly by factoring or by lookup. For irrational inputs such as a = 2, the decimal expansion is infinite and non-repeating, so computation proceeds by successive 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. 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. For double-precision arithmetic, starting with a guess accurate to one byte, Newton's method reaches machine precision in fewer than six iterations for any positive input. For the general nth root, Newton's method applied to f(x) = x^n − a yields x_{k+1} = ((n−1)x_k + a/x_k^(n−1))/n. Alternatively, the identity ⁿ√a = exp((ln a)/n) reduces the problem to logarithm and exponentiation, operations that are implemented via polynomial approximations and bit-manipulation tricks in standard mathematical libraries.

A worked example.

Example

To compute the square root of 2, the calculator applies an iterative refinement algorithm starting from an initial approximation. Using Newton's method with x₀ = 1.5, the first iteration gives x₁ = (1.5 + 2/1.5)/2 = (1.5 + 1.333333)/2 = 1.416667. The second iteration gives x₂ = (1.416667 + 2/1.416667)/2 = (1.416667 + 1.411765)/2 = 1.414216. The third iteration gives x₃ = (1.414216 + 2/1.414216)/2 = (1.414216 + 1.414211)/2 = 1.414213. After three iterations, the result agrees with the exact value √2 = 1.4142135623730951 to six decimal places. The Babylonian tablet YBC 7289 recorded the equivalent of 1.41421296, demonstrating that ancient mathematicians achieved comparable accuracy using geometric methods. In modern floating-point hardware, the same result is obtained in a single instruction with rounding error below 10⁻¹⁶. The square root of 2 is irrational, so no finite decimal representation is exact; the calculator displays sufficient digits for the user's precision requirements and truncates or rounds according to standard conventions.

number2
root Degree2

Frequently asked questions.

Why can I not compute the square root of a negative number?
The square root of a negative number is not a real number because the square of any real number, whether positive or negative, is always non-negative. In the real number system, there is no solution to the equation x² = −1. Complex analysis extends the number system to include imaginary numbers, where √−1 = i, and every negative real number has two complex square roots. For example, √−4 = ±2i. This calculator is designed for real-valued arithmetic as used in engineering, statistics, and everyday computation. If you need complex square roots, you must use a calculator or software that supports complex arithmetic, such as Python with the cmath module, MATLAB, or Wolfram Mathematica.
What is the difference between the square root and the principal square root?
Every positive real number has two square roots: one positive and one negative. For example, both 3 and −3 square to 9. The principal square root is the non-negative one, denoted by the radical sign √. Thus √9 = 3, not ±3. The equation x² = 9 has solutions x = ±3, but the expression √9 evaluates to 3 exclusively. This convention is essential for √ to be a well-defined function, which by definition assigns exactly one output to each valid input. In contexts where both roots are relevant, such as solving quadratic equations, mathematicians explicitly write ±√a.
How does the calculator compute nth roots?
For the square root, the calculator uses hardware floating-point instructions or a convergent iterative algorithm such as Newton's method, which doubles the number of correct digits with each step. For higher-order roots, the calculator may use the exponential-logarithmic identity ⁿ√a = exp((ln a)/n) or a generalized Newton iteration. The choice depends on the root degree and the magnitude of the input. The exponential form is numerically stable for a wide range of inputs and leverages highly optimized library implementations of ln and exp. For integer root degrees, direct Newton iteration can be faster and avoids the intermediate logarithmic step.
What is Newton's method for square roots?
Newton's method, also known as the Babylonian method or Heron's method, is an iterative algorithm for approximating roots. To find √a, start with any positive guess x₀. Each subsequent guess is the average of the previous guess and the quotient of the input divided by the previous guess: x_{k+1} = (x_k + a/x_k)/2. This formula arises from finding the x-intercept of the tangent line to the curve y = x² − a at the point (x_k, x_k² − a). The method converges quadratically, meaning that if the current guess has d correct digits, the next guess has approximately 2d correct digits.
Why does the calculator give 1 for the 1000th root of any number?
For any positive number a, the limit of ⁿ√a as n approaches infinity is 1. This follows from the identity ⁿ√a = exp((ln a)/n) and the fact that (ln a)/n approaches 0 as n grows, while exp(0) = 1. For very large n, the calculator may return exactly 1 because the difference from 1 is smaller than the machine epsilon of double-precision floating-point arithmetic, which is approximately 2.22 × 10⁻¹⁶. This is correct behavior: as the root degree increases, the result approaches 1, and for sufficiently large n, floating-point arithmetic rounds to 1.
Can I compute the square root of a fraction?
Yes. The square root of a fraction equals the square root of the numerator divided by the square root of the denominator, provided both are non-negative. For example, √(9/16) = √9 / √16 = 3/4 = 0.75. The calculator accepts decimal inputs, so you can enter the fraction as its decimal equivalent or compute the numerator and denominator separately. For exact rational results, the calculator returns a floating-point approximation. If both the numerator and denominator are perfect squares, the result is rational and terminates in decimal form.
What is the square root of zero?
The square root of zero is zero. This is the only case where the input and output are equal, and it is the fixed point of the square root function. Mathematically, 0² = 0, and by the definition of the principal square root, √0 = 0. The square root function is continuous at zero, with lim_{x→0⁺} √x = 0. However, the derivative of √x is 1/(2√x), which diverges to infinity as x approaches zero, meaning that the function becomes extremely sensitive to small perturbations near the origin.
How precise are the results from this calculator?
The calculator uses double-precision IEEE 754 floating-point arithmetic, which provides approximately 15 to 17 significant decimal digits. For square roots, hardware implementations typically return the correctly rounded result, meaning the output is the nearest representable floating-point number to the exact mathematical square root. For nth roots computed via software algorithms, the result is accurate to within one unit in the last place of the floating-point representation. This precision is sufficient for all applications in science, engineering, and finance.
What are some common applications of square roots?
Square roots appear in virtually every quantitative discipline. In statistics, the standard deviation is the square root of the variance, measuring dispersion in the original units of the data. In physics, the root-mean-square speed of gas molecules is the square root of the mean squared velocity. In electrical engineering, the magnitude of a complex impedance is the square root of the sum of the squares of resistance and reactance. In geometry, the diagonal of a square with side length s is s√2, and the distance between two points in a plane is the square root of the sum of squared coordinate differences. In finance, volatility is the square root of variance, and the Black-Scholes formula contains a √t term.
Is there a difference between √x and x^0.5?
For non-negative real x, there is no difference: √x and x^(1/2) represent exactly the same mathematical operation and yield the same numerical result. Both notations denote the principal non-negative square root. The radical notation √x is traditional in elementary algebra and geometry, while the exponent notation x^(1/2) is more convenient in calculus, analysis, and programming because it generalizes naturally to arbitrary rational exponents. In some programming languages and calculators, x^0.5 may be implemented using the general power function, which computes exp(0.5 × ln x), whereas √x may use a dedicated square root algorithm. For well-behaved inputs, both paths converge to the same correctly rounded result, but the dedicated square root instruction is typically faster and more accurate for values very close to 1.

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).
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