Béda révisi "Akar kuadrat"

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Dina [[matematik]], '''akar kuadrat''' [[real number|wilangan riil]] [[non-negative|non-negatip]] ''x'' dilambangkeun ku <math>\sqrt x</math> sarta ngagambarkeun wilangan riil non-négatip nu ngarupakeunmangrupa ''kuadrat'' (hasil kali tina wilangan éta sorangan) nyaéta ''x''.
 
Contona, <math>\sqrt 9 = 3</math> saprak <math>3^2 = 3 \times 3 = 9</math>.
Ngalegaan tina konsep akar kuadrat keur wilangan riil négatip nyaéta dina [[wilangan imajinér]] jeung [[wilangan kompléks]].
 
Akar kuadrat mindeng mangrupa ''[[wilangan irasional]]'', requiring an infinite, non-repeatingrepéating series of digits in their [[decimal]] representation. For example, <math>\sqrt 2</math> cannot be written exactly in finite or repeatingrepéating 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 (√) munggaran dipaké dina [[abad ka-16]]. Diduga asalna tina bentuk singget pikeun [[r]], tina [[Basa Latin]] ''radix'' (hartina "[[akar (matematik)|akar]]").
== Sipat ==
 
The following important properties of the square root functions are valid for all positive realréal numbers ''x'' and ''y'':
 
:<math>\sqrt{xy} = \sqrt{x} \sqrt{y}</math>
:<math>\sqrt{x} = x^{\frac{1}{2}}</math>
 
[[Fungsi (matematik)|Fungsi]] akar kuadrat umumna metakeun [[rational number|wilangan rasional]] ka [[algebraic number|wilangan aljabar]]; √''x'' is rational if and only if ''x'' is a rational number which, after cancelling, is a [[fraction (mathematics)|fraction]] of two [[perfect square]]s. In particular, √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 realsréals, 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'' = √''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''| = √''a'', or equivalently ''x'' = ±√''a''.
 
In [[calculus]], for instance when proving that the square root function is [[continuous]] or [[derivative|differentiable]] or when computing certain [[limit (mathematics)|limits]], the following identity often comes handy:
# replace ''r'' by the average of ''r'' and ''x/r''
# go to 2
This is a quadratically convergent algorithm, which meansméans that the number of correct digits of ''r'' roughly doubles with eachéach step.
 
This algorithm works equally well in the [[p-adic numbers]], but cannot be used to identify realréal square roots with p-adic square roots; it is easyéasy, for example, to construct a sequence of rational numbers by this method which converges to +3 in the realsréals, but to -3 in the 2-adics.
 
=== An exact "long-division like" algorithm ===
 
Write the number in decimal and divide it into pairs of digits starting from the decimal
point. The numbers are laid out similar to the long division algorithm and the final square root will appearappéar above the original number.
 
For eachéach iteration:
# Bring down the most significant pair of digits not yet used and append them to any remainder. This is the ''current value'' referred to in steps 2 and 3.
# If <math>r</math> denotes the part of the result found so far, determine the greatestgréatest digit <math>x</math> that does not makemaké <math>y = x(20r + x)</math> exceed the current value. Place the new digit <math>x</math> on the quotient line.
# Subtract <math>y</math> from the current value to form a new remainder.
# If the remainder is zero and there are no more digits to bring down the algorithm has terminated. Otherwise continue with step 1.
 
 
Example: What is the square root of 152.2756?
 
Although demonstrated here for base 10 numbers, the procedure works for
any [[numeral system|base]], including [[Binary numeral system|base 2]]. In the description above, '''20''' meansméans double
the number base used, in the case of binary this would reallyréally be
'''100'''. The algorithm is in fact much easieréasier to perform in base 2, as in every step only the two digits 0 and 1 have to be tested. See [[Shifting nth-root algorithm]].
 
=== Pell's equation ===
 
=== Finding square roots in the head ===
Based on Pell's equation there is a methode to calculate the square root in the headhéad, by simply subtraction of odd numbers.
 
Ex: Square root of 27 is:
 
=== Continued fraction methods ===
Quadratic irrationals, that is numbers involving square roots in the form (''a''+√b)/''c'', have periodic [[continued fraction]]s. This makes them easyéasy to calculate recursively given the period. For example, to calculate √2, we makemaké use of the fact that √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 √2-1 to some specific precision specified through ''n'' levels of recurrence, and add 1 to the result to obtain √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 √''z'' is as follows: if ''z'' = ''r'' exp(''i''φ) is represented in polar coordinates with -π < φ ≤ π, then we set √''z'' = √''r'' exp(''i''φ/2). Thus defined, the square root function is [[holomorphic function|holomorphic]] everywhere except on the non-positive realréal numbers (where it isn't even [[continuous]]). The above Taylor series for √(1+''x'') remains valid for complex numbers ''x'' with |''x''| < 1.
 
When the number is in rectangular form the following formula can be used:
== 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 √''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 realréal numbers, and normal operators are akin to complex numbers.
 
== Square roots of the first 20 positive integers ==