Akar kuadrat: Béda antarrépisi

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''.

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]].

The following important properties of the square root functions are valid for all positive realréal numbers ''x'' and ''y'':

[[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]].

Suppose that ''x'' and ''a'' are realsréals, and that ''x''²=''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''² 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''.

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.

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:

# 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.

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]].

Based on Pell's equation there is a methode to calculate the square root in the headhéad, by simply subtraction of odd numbers.

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

To every non-zero [[complex number]] ''z'' there exist precisely two numbers ''w'' such that ''w''² = ''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.

If ''A'' is a [[positive definite]] matrix or operator, then there exists precisely one positive definite matrix or operator ''B'' with ''B''² = ''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''² = ''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.