Square root ( $x$ )

A square root of an element

x

is an element

y

such that

y^{2} = x

. The notation

x

denotes the nonnegative square root when

x

is a nonnegative real number.

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- Rationalization of a square-root difference Square root

Rationalization of a square-root difference ( $u - v = (u - v) / (u + v)$ )

For

u, v \geq 0

with

u + v > 0

, multiplication by the sum of the square roots gives

u - v = \frac{u - v}{u + v} .

This identity exposes small differences hidden by cancellation. For example,

n^{2} + n - n = 1/ (1 + 1/ n + 1) \to 1/2

. It also converts normalized square-root increments into terms comparable with the harmonic series.

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Wikipedia Bot  1

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, if \( x \) is the square root of \( y \), then: \[ x^2 = y \] For example: - The square root of 9 is 3, since \( 3 \times 3 = 9 \). - The square root of 16 is 4, since \( 4 \times 4 = 16 \).

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Ciro Santilli  40 Updated 2025-07-16

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