Square Root Calculator

Calculate the square root of any number instantly. Perfect squares return exact integers, irrational roots get 9-significant-digit precision, and negative inputs get a clear explanation.

Square Root

√16

4

Perfect square — exact integer

Verification4² = 16
Also equals16^0.5
ClassificationPerfect square

Number

Negative numbers are reported as undefined for real numbers (they need imaginary numbers).

How the Calculation Works

The engine computes IEEE 754 double-precision square roots, the same values Newton's method converges to. Newton's method refines a guess x by iterating x → (x + n/x) / 2: each iteration roughly doubles the number of correct digits, so a handful of steps reach full machine precision. Perfect squares are detected exactly (the root comes back as an integer) and displayed without decimals; irrational roots are rounded to 9 significant digits for display, e.g. √2 = 1.41421356.

Perfect Squares Reference

Quick Answer

The square root of n is the number x where x × x = n. Enter any non-negative number to get √n instantly: perfect squares return exact integers (√144 = 12), other values return the root to 9 significant digits (√2 = 1.41421356). Square roots of negative numbers are undefined for real numbers — they require imaginary numbers (i = √-1).

Key Facts

  • √n is the number x where x × x = n; √ is the inverse of squaring
  • Perfect squares (0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …) have exact integer roots
  • √2 = 1.41421356…, √3 = 1.73205081…, √5 = 2.23606798… — all irrational
  • √n is undefined for real numbers when n < 0 (√-1 = i, an imaginary unit)
  • √n equals n^0.5 — raising to the power ½ is the same operation
  • √(a·b) = √a · √b and √(a/b) = √a / √b for a, b ≥ 0
  • Every positive number has two square roots: ±√n; the √ symbol means the positive one
  • Calculators typically compute √ using Newton’s method or hardware sqrt instructions

Frequently Asked Questions

Yes. √n is exactly n^(1/2). For example, 25^0.5 = 5 = √25. This identity comes from the exponent rule (x^a)^b = x^(a·b): squaring n^0.5 gives n^(0.5·2) = n^1 = n. Fractional exponents generalize to all roots: cube root is n^(1/3), fourth root is n^(1/4).

Perfect squares are integers whose square roots are also integers: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, … They are the results 1², 2², 3², … For any other integer the square root is irrational — its decimal expansion never ends or repeats (√2 = 1.41421356…).

It is undefined for real numbers: no real number squared is negative, because a positive squared stays positive and a negative squared becomes positive. In higher math we extend to complex numbers using the imaginary unit i = √-1, so √-9 = 3i. This calculator shows an explanatory message instead of NaN for negative inputs.

Every positive number has two square roots: 2² = 4 and (-2)² = 4. By convention the radical symbol √ refers only to the principal (non-negative) root, so √4 = 2. If an equation asks "which numbers square to 4," the answers are ±2 — that is why quadratic equations produce two solutions.

It uses the hardware-accelerated square root available in JavaScript (Math.sqrt), which implements the IEEE 754 correctly-rounded square root — the same result Newton’s method converges to. Newton’s method iterates x → (x + n/x) / 2: starting from a guess, each step roughly doubles the number of correct digits, so 5–6 iterations give full double precision.

Perfect squares return exact integers (√144 = 12). Other inputs are computed at full double precision and displayed to 9 significant digits (√2 = 1.41421356). Note that irrational roots cannot be written exactly as decimals — 1.41421356 is a high-precision approximation, and the trailing digit can differ by 1 in other tools that round differently.