Square Root

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For example, $\backslash sqrt\; 9\; =\; 3$ since $3^2\; =\; 3\; imes3\; =\; 9.$ This example suggests how square roots can arise when solving Quadratic Equation s such as $x^2=9$ or, more generally :$ax^2+bx+c=0.\; \backslash ,$ There are two solutions to the square root of a non-zero number. For a positive real number, the two square roots are the principal square root and the negative square root. For negative real numbers, the concept of Imaginary and Complex Number s has been developed to provide a mathematical framework to deal with the results. Square roots of positive Integer s are often '' Irrational Number s'', i.e., numbers not expressible as a Ratio of two integers. For example, $\backslash sqrt\; 2$ cannot be written exactly as ''m''/''n'', where ''n'' and ''m'' are integers. Nonetheless, it is exactly the length of the Diagonal of a Square with side length 1. The discovery that $\backslash sqrt\; 2$ is irrational is attributed to Hippasus , a disciple of Pythagoras . The square root Symbol ($\backslash sqrt\{\backslash \; \}$) was first used during the 16th Century . It has been suggested that it originated as an altered form of lowercase R , representing the Latin ''radix'' (meaning " Root "). PROPERTIES The principal square root function $f(x)\; =\; \backslash sqrt\{x\}$ is a Function which maps the Set of non-negative real numbers $\backslash mathbb\{R\}^+\; \backslash cup\; \backslash \{0\backslash \}$ onto itself. The principal square root function $f(x)\; =\; \backslash sqrt\{x\}$ always returns a unique value. To obtain both roots of a positive number, take the value given by the principal square root function as the first root (root1) and obtain the second root (root2) by subtracting the first root from zero (ie root2 = 0 − root1). The following important properties of the square root functions are valid for all positive real numbers $x$ and $y$: :$\backslash sqrt\{xy\}\; =\; \backslash sqrt\; x\; \backslash sqrt\; y\; \backslash qquad\; \backslash Rightarrow\; \backslash qquad\; \backslash sqrt\{\backslash left(\; 100y\; ight)\; \}\; \backslash ,\; =\; \backslash ,\; 10\; \backslash cdot\; \backslash sqrt\; y$ : :$\backslash sqrt\; x\; =\; x^\{1/2\}$ The square root Function maps Rational Number s to Algebraic Number s; also, $\backslash sqrt\; x$ is rational if and only if $x$ is a rational number which, after cancelling, is a ratio of two Perfect Square s. In particular, $\backslash sqrt\; 2$ is Irrational . In Geometrical terms, the square root function maps the Area of a Square to its side length.
	Suppose That <math>x</math> And <math>a</math> Are Real Numbers, And That <math>x^2	a</math>, and we want to find <math>x</math> A common mistake is to "take the square root" and deduce that <math>x = \sqrt a</math> This is incorrect, because the '''principal square root''' of <math>x^2</math> is not <math>x</math>, but the absolute value <math>\left x ight</math>, one of our above rules Thus, all we can conclude is that <math>\left x ight = \sqrt a</math>, or equivalently <math>x = \pm\sqrt a</math>
	\leq \pi</math>, Then We Set <math>\sqrt{z}	r^{i\phi \over 2}</math> Thus defined, the square root function is Holomorphic everywhere except on the non-positive real numbers (where it isn't even Continuous ) The above Taylor series for <math>\sqrt{1+x}</math> remains valid for complex numbers ''x'' with ''x'' < 1

represent rational numbers. This rational number can be found by realizing that ''x'' also appears under the radical sign, which gives the equation

:$x\; =\; \backslash sqrt\{2+x\}.$

If we solve this equation, we find that ''x'' = 2. More generally, we find that

:

Beware, however, of the discontinuity for n=0. The infinitely nested square root for n=0 does not equal one, as the "general" solution would indicate. Rather, it is (obviously) zero.

The same procedure also works to get

:

This method will give a rational $x$ value for all values of $n$ such that

:$\{n\}\; =\; \{x^2\}\; +\; \{x\}.$


SQUARE ROOTS OF THE FIRST 20 POSITIVE INTEGERS


:$\backslash sqrt\; \{1\}\; =$ 1
:$\backslash sqrt\; \{2\}\; \backslash approx$ 1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 78462
:$\backslash sqrt\; \{3\}\; \backslash approx$ 1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 16909
:$\backslash sqrt\; \{4\}\; =$ 2
:$\backslash sqrt\; \{5\}\; \backslash approx$ 2.2360679774 9978969640 9173668731 2762354406 1835961152 5724270897 2454105209 25638
:$\backslash sqrt\; \{6\}\; \backslash approx$ 2.4494897427 8317809819 7284074705 8913919659 4748065667 0128432692 5672509603 77457
:$\backslash sqrt\; \{7\}\; \backslash approx$ 2.6457513110 6459059050 1615753639 2604257102 5918308245 0180368334 4592010688 23230
:$\backslash sqrt\; \{8\}\; \backslash approx$ 2.8284271247 4619009760 3377448419 3961571393 4375075389 6146353359 4759814649 56924
:$\backslash sqrt\; \{9\}\; =$ 3
:$\backslash sqrt\; \{10\}\; \backslash approx$ 3.1622776601 6837933199 8893544432 7185337195 5513932521 6826857504 8527925944 38639
:$\backslash sqrt\; \{11\}\; \backslash approx$ 3.3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609
:$\backslash sqrt\; \{12\}\; \backslash approx$ 3.4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818
:$\backslash sqrt\; \{13\}\; \backslash approx$ 3.6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293
:$\backslash sqrt\; \{14\}\; \backslash approx$ 3.7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307
:$\backslash sqrt\; \{15\}\; \backslash approx$ 3.8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937
:$\backslash sqrt\; \{16\}\; =$ 4
:$\backslash sqrt\; \{17\}\; \backslash approx$ 4.1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338
:$\backslash sqrt\; \{18\}\; \backslash approx$ 4.2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386
:$\backslash sqrt\; \{19\}\; \backslash approx$ 4.3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203
:$\backslash sqrt\; \{20\}\; \backslash approx$ 4.4721359549 9957939281 8347337462 5524708812 3671922305 1448541794 4908210418 51276


GEOMETRIC CONSTRUCTION OF THE SQUARE ROOT


You can construct a square root with a compass and straightedge. This has been known at least since the time of the Pythagoreans. In his Elements , Euclid (fl. 300 BC) gave the construction of the Geometric Mean of two quantities in two different places: Proposition II.14 and Proposition VI.13 . Since the geometric mean of $a$ and $b$ is $\backslash sqrt\{ab\}$, you can construct $\backslash sqrt\{a\}$ simply by taking $b=1$.

The construction is also given by Descartes in his La Géométrie , see figure 2 on page 2 . However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.


SEE ALSO

• Quadratic Residue


• Radical (mathematics)


• Quadratic Irrational


• Cube Root


• Integer Square Root


• Root Of Unity


• Methods Of Computing Square Roots


• Square Root Of A Matrix

EXTERNAL LINKS

• Japanese soroban techniques - Professor Fukutaro Kato's method


• Japanese soroban techniques - Takashi Kojima's method


• Algorithms, implementations, and more - Paul Hsieh's square roots webpage


• Square root of positive real numbers with implementation in Rexx.