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Akar kuadrat mindeng mangrupa ''[[wilangan irasional]]'', requiring an infinite, non-repeating series of digits in their [[decimal]] representation. For example, <math>\sqrt 2</math> cannot be written exactly in finite or repeating decimal form. Equivalently, it cannot be represented by a [[fraction]] whose numerator and denominator are [[integer]]s. Nonetheless, it is exactly the length of the [[diagonal]] of a [[square]] with side length 1. The discovery that <math>\sqrt 2</math> is irrational is attributed to the [[Pythagoreans]].
[[Tabel lambang matematis|Lambang]] akar kuadrat (
== Sipat ==
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:<math>\sqrt{x} = x^{\frac{1}{2}}</math>
[[Fungsi (matematik)|Fungsi]] akar kuadrat umumna metakeun [[rational number|wilangan rasional]] ka [[algebraic number|wilangan aljabar]];
In [[geometry|geometrical]] terms, the square root function maps the [[area]] of a [[square]] to its side length.
Suppose that ''x'' and ''a'' are reals, and that ''x''<sup>2</sup>=''a'', and we want to find ''x''. A common mistake is to "take the square root" and deduce that ''x'' =
In [[calculus]], for instance when proving that the square root function is [[continuous]] or [[derivative|differentiable]] or when computing certain [[limit (mathematics)|
:<math>\sqrt{x} - \sqrt{y} = \frac{x-y}{\sqrt{x} + \sqrt{y}}</math>
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It is valid for all non-negative numbers ''x'' and ''y'' which are not both zero.
The function ''f''(''x'') =
[[
The function is continuous for all non-negative ''x'', and [[derivative|differentiable]] for all positive ''x'' (it is not differentiable for ''x''=0 since the [[slope]] of the [[tangent]] there is [[infinite|
:<math>f'(x) = \frac{1}{2\sqrt x}</math>
Its [[Taylor series]] about ''x'' = 1 can be found using the [[binomial theorem]]:
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=== Calculators ===
[[calculator|Pocket
:<math>\sqrt{x} = e^{\frac{1}{2}\ln x}</math>
The same identity is exploited when computing square roots with [[logarithm table]]s or [[slide rule]]s.
=== Babylonian method ===
A commonly used algorithm for approximating
# start with an arbitrary positive start value ''r'' (the closer to the root the better)
# replace ''r'' by the average of ''r'' and ''x/r''
# go to 2
This is a quadratically convergent algorithm, which means that the number of correct digits of ''r'' roughly doubles with each step.
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[[Pell's equation]] yields a method for finding rational approximations of square roots of integers.
=== Finding square roots in the head ===
Based on Pell's equation there is a methode to calculate the square root in the head, by simply subtraction of odd numbers.
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=== Continued fraction methods ===
Quadratic irrationals, that is numbers involving square roots in the form (''a''+
: ''a''<sub>''n+1''</sub>=1/(2+a<sub>''n''</sub>) with ''a''<sub>0</sub>=0
to obtain
== Square roots of complex numbers ==
To every non-zero [[complex number]] ''z'' there exist precisely two numbers ''w'' such that ''w''<sup>2</sup> = ''z''. The usual definition of
When the number is in rectangular form the following formula can be used:
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where the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number.
Note that because of the discontinuous nature of the square root function in the complex plane, the law
:<math>-1 = i \times i = \sqrt{-1} \times \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1} = 1</math>
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The third equality cannot be justified. (See [[invalid proof]].)
However the law can only be wrong up to a factor -1,
== Square roots of matrices and operators ==
If ''A'' is a [[positive definite]] matrix or operator, then there exists precisely one positive definite matrix or operator ''B'' with ''B''<sup>2</sup> = ''A''; we then define
More generally, to every [[normal operator|normal]] matrix or operator ''A'' there exist normal operators ''B'' such that ''B''<sup>2</sup> = ''A''. In general, there are several such operators ''B'' for every ''A'' and the square root function cannot be defined for normal operators in a satisfactory manner. Positive definite operators are akin to positive real numbers, and normal operators are akin to complex numbers.
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== Square roots of the first 20 positive integers ==
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