Compute the principal square root of any non‑negative number. Get instant decimal approximation, simplest radical form (e.g., √50 = 5√2), and explore the sqrt function on an interactive Cartesian graph.
The principal square root of a non‑negative real number x is the unique non‑negative number y such that y² = x. It is denoted √x. Square roots are fundamental in algebra, geometry (Pythagorean theorem), statistics (standard deviation), physics (kinematics), and engineering (signal processing).
For any a ≥ 0, √a · √a = a and √(ab) = √a · √b (a,b ≥ 0).
Simplification rule: √(a²·b) = a√b, where a, b ≥ 0 and b is square‑free.
The concept of square roots traces back to Babylonian clay tablets (c. 1800 BCE), which contained algorithms for approximating √2. The symbol "√" was introduced by Christoph Rudolff in 1525. Modern real analysis defines √x as the inverse of the squaring function on [0, ∞). The irrationality of √2 was proved by the Pythagoreans, demonstrating that not all lengths are commensurable.
Simplest Radical Form: Our calculator decomposes the radicand into prime factors. For each pair of identical prime factors, one factor moves outside the radical. Example: √72 = √(2³·3²) = √(2²·2·3²) = 2·3√2 = 6√2.
Babylonian Method: Starting with an initial guess g₀ = max(1, x/2), iterate gₙ₊₁ = (gₙ + x/gₙ)/2. The algorithm converges rapidly.
| Radicand (x) | Simplest Radical Form | Decimal √x (approx) | Type |
|---|---|---|---|
| 4 | 2 | 2.0 | Perfect square / integer |
| 18 | 3√2 | 4.2426406871 | Simplifiable surd |
| 2 | √2 | 1.4142135624 | Irrational (prime radicand) |
| 0.25 | 0.5 | 0.5 | Rational square root |
| 75 | 5√3 | 8.6602540378 | Mixed radical |
A civil engineer needs to determine the diagonal length of a rectangular foundation with sides 12 m and 16 m. Diagonal d = √(12² + 16²) = √(144 + 256) = √400 = 20 m. Using our square root calculator, one can quickly verify perfect square radicand.