Square root

In mathematics, a square root of a number x is a number y such that y² = x; in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is x.^[1] For example, 4 and −4 are square roots of 16, because 4² = (−4)² = 16.

Every non-negative real number x has a unique non-negative square root, called the principal square root or simply the square root, which is denoted by √x, where the symbol √  is called the radical sign or radix.^[2] For example, while the roots to the algebraic polynomial y² = 9 are ±3, the principal square root is strictly √9 = 3.

Square roots of negative numbers can be discussed within the framework of complex numbers, where the imaginary unit i is defined by the relation i² = −1.

Properties and identities [edit]

The principal square root function f(x) = √x maps the set of non-negative real numbers ℝ_≥0 onto itself. For all non-negative real numbers x and y, the radical operation satisfies fundamental algebraic identities:

Using the laws of exponents, radicals can be expressed as fractional powers: √x = x^1/2 and ⁿ√x = x^1/n.

Computation and algorithms [edit]

Because many square roots are irrational numbers (such as √2 ≈ 1.41421356...), numerical approximation methods have been developed throughout mathematical history.

Babylonian method (Heron's formula)

The Babylonian method (also known as Heron's method) is an iterative algorithm that converges quadratically to √S. Given an initial positive estimate x₀, subsequent approximations are calculated by:

At each step, if x_n is an overestimate of √S, then S / x_n is an underestimate, and their arithmetic mean yields an improved approximation.

Newton–Raphson approximation

In numerical calculus, finding √S is framed as finding the positive zero of the polynomial function f(x) = x² − S = 0. Applying the Newton–Raphson formula:

Interactive evaluation module [edit]

The following interactive module evaluates exact and approximate principal roots, simplifies radical expressions, and computes prime factorizations using JavaScript client-side execution.

History and ancient mathematics [edit]

Babylonian clay tablet YBC 7289

The Yale Babylonian Collection clay tablet YBC 7289 was created in southern Mesopotamia between 1800 BCE and 1600 BCE.^[3] The tablet shows a tilted square with two diagonals and three sexagesimal numbers. One number across the center is given as:

This estimate differs from the true value of √2 ≈ 1.41421356... by less than one part in a million (< 0.00005%), demonstrating extraordinary numerical proficiency in the Old Babylonian Empire.

Spiral of Theodorus

The Spiral of Theodorus (also called the square root spiral or Pythagorean spiral) is a geometric construction composed of contiguous right triangles, attributed to Theodorus of Cyrene in the 5th century BCE.^[4] Starting from an isosceles right triangle with unit legs 1 and 1 (hypotenuse √2), subsequent triangles are constructed by adding a unit leg perpendicular to the previous hypotenuse, yielding the square roots √3, √4, √5, &dots;, √17.

Pythagorean incommensurability

The discovery that the square root of 2 is an irrational number is traditionally credited to Hippasus of Metapontum, a Pythagorean philosopher. In ancient Greek geometry, numbers were understood as geometric lengths. Hippasus proved that the diagonal of a square cannot be measured as a rational fraction a/b of its side, contradicting the core Pythagorean philosophy that "all things are numbers".

References [edit]

1. ^ Gel'fand, I. M.; Shen, A. (2003). Algebra. Birkhäuser. p. 120. ISBN 978-0-8176-3677-7.
2. ^ Rudolff, Christoff (1525). Behend und hübsch Rechnung durch die kunstreichen regeln Algebre so gemeincklich die Coss genent werden.
3. ^ Fowler, David; Robson, Eleanor (1998). "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context". Historia Mathematica. 25 (4): 366–378.
4. ^ Plato, Theaetetus, 147d.