Square Root Calculator

Definition

The square root √x, also written x^{1/2}, means in the reals the non-negative value y such that y² = x. For x > 0 there is exactly one such y > 0; for x = 0, √0 = 0. For x < 0 there is no y ∈ ℝ. The calculator computes √(x) and for x > 0 also shows exp((1/2) ln x) from NIST DLMF §4.2.

MathWorld (Square Root) and algebra texts stress the non-negative branch: although (−y)² = x when y = √x, the notation √x means only +y. This calculator reports an error for negative x — matching the ℝ model.

Historical context: Euclid and irrationality

Squares and roots were known in Babylon (tables approximating √2). In Euclid’s Elements, geometric constructions link √x to the side of a square of area x. The irrationality of √2 — traditionally associated with the Pythagorean school — shows that a root need not be rational: no lowest-terms p/q satisfies (p/q)² = 2.

Conceptually √x is the inverse of y ↦ y² on [0, ∞). In the complex numbers every x ≠ 0 has two square roots; the principal Arg branch selects one. Here we stay with the non-negative real root.

Heron’s (Babylonian) method

Heron of Alexandria’s classical method (and earlier Babylonian recipes) approximates √x by iteration: start from y₀ > 0, then y_{k+1} = (y_k + x/y_k)/2. Geometrically that is the arithmetic mean of y_k and x/y_k — the sides of a rectangle of area x. The sequence converges quickly to √x for x > 0.

For x = 2 and y₀ = 1: y₁ = 1.5; y₂ ≈ 1.4167; y₃ ≈ 1.414215 — already close to √(2). The calculator does not list Heron steps in the panel; the theory explains why the same formula appears in Newton’s method.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Newton’s method for f(y) = y² − x

Newton–Raphson for f(y) = 0 with f(y) = y² − x and f′(y) = 2y yields y_{k+1} = y_k − (y_k² − x)/(2y_k) = (y_k + x/y_k)/2. The identity with Heron’s method is classical: both are the same quadratic iteration.

Modern √ in floating-point arithmetic uses hardware or library approximations (often with a Newton / Goldschmidt correction), not hand-stepped Heron loops. The result panel compares √ with exp((1/2) ln x) for x > 0 to show agreement with the power definition.

Formal properties

For x ≥ 0: (√x)² = x and √(x²) = |x| in the reals. For a, b ≥ 0: √(a·b) = √a · √b and √(a/b) = √a / √b when b > 0. √x = exp((1/2) ln x) for x > 0. The function √ is concave and increasing on [0, ∞).

The identity √(x²) = |x| explains why √ never returns a negative even when x is the square of a negative number “in the history” of a calculation. The panel’s y² check returns to the entered x (within floating-point error).

Special cases and examples

Examples on this page: √81 = 9; √2 ≈ 1.41421356; √0.25 = 0.5; √0 = 0; √144 = 12; √256 = 16; √100 = 10; √0.01 = 0.1. √(−9) → error. Perfect squares yield an exact result.

The Pythagorean theorem and vector length use the same root but with a sum of squares — separate pages. Here there is one field x.

The calculator computes √x and marks perfect squares when the result is an integer in the y² ≈ x check. The panel compares x with y² on a shared bar scale and shows exp((1/2) ln x) for x > 0. That is not a separate Heron iteration — only a consistency check between the power definition and the root.

The nth root, a power with an exponent, and the Pythagorean √(a²+b²) are siblings: each task has different inputs. Here there is one field x ≥ 0. Negative x ends in an error in the real model; zero yields zero.

Examples: √81 = 9; √2 ≈ 1.41421356; √0.25 = 0.5; √0 = 0; √144 = 12; √256 = 16; √100 = 10; √0.01 = 0.1. The y² check returns to x within floating-point arithmetic noise.

Numerical notes and sources

√(x) for x ≥ 0 is stable in the usual floating-point arithmetic range. Comma and period are equivalent. The calculator does not return −√x or complex roots.

Sources: NIST DLMF §4.2; MathWorld — Square Root; Wikipedia — Square root / Heron’s method; Euclid’s Elements (geometric context). Educational tool.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

The classroom floating-point arithmetic reminder stays in the Heron section and below. The calculator does not return −√x. Sources: NIST DLMF §4.2; MathWorld — Square Root; Wikipedia — Square root; Euclid’s Elements.