Béda révisi "Akar kuadrat"

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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 (&radic;) munggaran dipaké dina [[abad ka-16]]. Diduga asalna tina bentuk singget pikeun [[r]], tina [[Basa Latin]] ''radix'' (hartina "[[akar (matematik)|akar]]").
== Sipat ==
:<math>\sqrt{x} = x^{\frac{1}{2}}</math>
[[Fungsi (matematik)|Fungsi]] akar kuadrat umumna metakeun [[rational number|wilangan rasional]] ka [[algebraic number|wilangan aljabar]]; &radic;''x'' is rational if and only if ''x'' is a rational number which, after cancelling, is a [[fraction (mathematics)|fraction]] of two [[perfect square|perfect squares]]s. In particular, &radic;2√2 is [[irrational number|irrational]].
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'' = &radic;''a''. This is incorrect, because the square root of ''x''<sup>2</sup> is not ''x'', but the absolute value |''x''|, one of our above rules. Thus, all we can conclude is that |''x''| = &radic;''a'', or equivalently ''x'' = &plusmn;&radic;±√''a''.
In [[calculus]], for instance when proving that the square root function is [[continuous]] or [[derivative|differentiable]] or when computing certain [[limit (mathematics)|limitlimits]]s, the following identity often comes handy:
:<math>\sqrt{x} - \sqrt{y} = \frac{x-y}{\sqrt{x} + \sqrt{y}}</math>
It is valid for all non-negative numbers ''x'' and ''y'' which are not both zero.
The function ''f''(''x'') = &radic;''x'' has the following graph, made up of half a [[parabola]] lying on its side:
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|&infin;]]). Its derivative is given by
:<math>f'(x) = \frac{1}{2\sqrt x}</math>
Its [[Taylor series]] about ''x'' = 1 can be found using the [[binomial theorem]]:
=== Calculators ===
[[calculator|Pocket calculatorcalculators]]s typically implement good routines to compute the [[exponential function]] and the [[natural logarithm]], and then compute the square root of ''x'' using the identity
:<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 &radic;''x'' is known as the "Babylonian method" and is based on [[Newtons method|Newton's method]]. It proceeds as follows:
# 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.
[[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.
=== Continued fraction methods ===
Quadratic irrationals, that is numbers involving square roots in the form (''a''+&#8730;b√b)/''c'', have periodic [[continued fraction]]s. This makes them easy to calculate recursively given the period. For example, to calculate &#8730;2√2, we make use of the fact that &#8730;2√2-1 = [0;2,2,2,2,2,...], and use the recurrence relation
: ''a''<sub>''n+1''</sub>=1/(2+a<sub>''n''</sub>) with ''a''<sub>0</sub>=0
to obtain &#8730;2√2-1 to some specific precision specified through ''n'' levels of recurrence, and add 1 to the result to obtain &#8730;2√2.
== 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 &radic;''z'' is as follows: if ''z'' = ''r'' exp(''i''&phi;φ) is represented in polar coordinates with -&pi;π < &phi;φ &le; &pi;π, then we set &radic;''z'' = &radic;''r'' exp(''i''&phi;φ/2). Thus defined, the square root function is [[holomorphic function|holomorphic]] everywhere except on the non-positive real numbers (where it isn't even [[continuous]]). The above Taylor series for &radic;(1+''x'') remains valid for complex numbers ''x'' with |''x''| < 1.
When the number is in rectangular form the following formula can be used:
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 &radic;(''zw'') = &radic;(''z'')&radic;(''w'') is in general '''not true'''. Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that -1 = 1:
:<math>-1 = i \times i = \sqrt{-1} \times \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1} = 1</math>
The third equality cannot be justified. (See [[invalid proof]].)
However the law can only be wrong up to a factor -1, &radic;(''zw'') = &plusmn;&radic;±√(''z'')&radic;(''w''), is true for either &plusmn;± as + or as - (but not both at the same time). Note that &radic;(''c''<sup>2</sup>) = &plusmn;±''c'', therefore &radic;(''a''<sup>2</sup>''b''<sup>2</sup>) = &plusmn;±''ab'' and therefore &radic;(''zw'') = &plusmn;&radic;±√(''z'')&radic;(''w''), using ''a'' = &radic;(''z'') and ''b'' = &radic;(''w'').
== 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 &radic;''A'' = ''B''.
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.
== Square roots of the first 20 positive integers ==
&radic; 1 = 1<br />
&radic; 2 &asymp;1≈1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 78462<br />
&radic; 3 &asymp;1≈1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 16909<br />
&radic; 4 = 2<br />
&radic; 5 &asymp;2≈2.2360679774 9978969640 9173668731 2762354406 1835961152 5724270897 2454105209 25638<br />
&radic; 6 &asymp;2≈2.4494897427 8317809819 7284074705 8913919659 4748065667 0128432692 5672509603 77457<br />
&radic; 7 &asymp;2≈2.6457513110 6459059050 1615753639 2604257102 5918308245 0180368334 4592010688 23230<br />
&radic; 8 &asymp;2≈2.8284271247 4619009760 3377448419 3961571393 4375075389 6146353359 4759814649 56924<br />
&radic; 9 = 3<br />
&radic;10√10 &asymp;3≈3.1622776601 6837933199 8893544432 7185337195 5513932521 6826857504 8527925944 38639<br />
&radic;11√11 &asymp;3≈3.3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609<br />
&radic;12√12 &asymp;3≈3.4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818<br />
&radic;13√13 &asymp;3≈3.6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293<br />
&radic;14√14 &asymp;3≈3.7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307<br />
&radic;15√15 &asymp;3≈3.8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937<br />
&radic;16√16 = 4<br />
&radic;17√17 &asymp;4≈4.1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338<br />
&radic;18√18 &asymp;4≈4.2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386<br />
&radic;19√19 &asymp;4≈4.3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203<br />
&radic;20√20 &asymp;4≈4.4721359549 9957939281 8347337462 5524708812 3671922305 1448541794 4908210418 51276
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