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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Square root of an integer ( $n, n \in Z_{\geq 0}$ )

Integer square root ( $⌊ n ⌋$ )

For a nonnegative integer

n

, its integer square root is the unique nonnegative integer

r

satisfying

r^{2} \leq n < (r + 1)^{2}

, equivalently

r = ⌊ n ⌋

. Testing

r^{2} = n

decides whether

n

is a perfect square using exact integer arithmetic.

Square root of one minus z squared ( $f (z) = 1 - z^{2}, f (0) = 1$ )

On

C ∖ ((- \infty, - 1] \cup [1, \infty))

, choose the holomorphic branch

f (z) = 1 - z 1 + z

, taking each factor's principal square root. This gives

f (x) > 0

for

- 1 < x < 1

. Across the branch cuts, its limits are

f (x + i 0) = - i x^{2} - 1

and

f (x - i 0) = + i x^{2} - 1

for

x > 1

, and

f (x + i 0) = + i x^{2} - 1

and

f (x - i 0) = - i x^{2} - 1

for

x < - 1

. The endpoints are branch points; analytic continuation around a single endpoint changes the square-root sign.

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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