Square root 1

Square root

Measured fall time of a small steel sphere falling from various heights. The data is ingood agreement with the predicted fall time of , where h is the height and g is

the acceleration of gravity.

In mathematics, a square root of anumber a is a number y such that y2 =a, or, in other words, a number ywhose square (the result of multiplyingthe number by itself, or yy) is a.[1]

For example, 4 is a square root of 16because 42 = 16.

Every non-negative real number a hasa unique non-negative square root,called the principal square root, whichis denoted by , where is calledthe radical sign or radix. For example,the principal square root of 9 is 3,denoted , because 32 = 3 3 =9 and 3 is non-negative. The termwhose root is being considered isknown as the radicand. The radicand isthe number or expression underneaththe radical sign, in this example 9.

Every positive number a has two square roots: , which is positive, and , which is negative. Together, thesetwo roots are denoted (see shorthand). Although the principal square root of a positive number is only one ofits two square roots, the designation "the square root" is often used to refer to the principal square root. For positivea, the principal square root can also be written in exponent notation, as a1/2.[2]

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally,square roots can be considered in any context in which a notion of "squaring" of some mathematical objects isdefined (including algebras of matrices, endomorphism rings, etc.)Square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers notexpressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where m and n areintegers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC.[3] Theparticular case is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Itis exactly the length of the diagonal of a square with side length 1.

Square root 2

Properties

The graph of the function , made up of half a parabola with a verticaldirectrix.

The principal square root function(usually just referred to as

the "square root function") is afunction that maps the set ofnon-negative real numbers onto itself.In geometrical terms, the square rootfunction maps the area of a square toits side length.

The square root of x is rational if andonly if x is a rational number that canbe represented as a ratio of two perfectsquares. (See square root of 2 forproofs that this is an irrational number,and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rationalnumbers into algebraic numbers (a superset of the rational numbers).

For all real numbers x

(see absolute value)

For all non-negative real numbers x and y,

and

The square root function is continuous for all non-negative x and differentiable for all positive x. If f denotes thesquare-root function, its derivative is given by:

The Taylor series of 1 + x about x = 0 converges for |x| 1 and is given by

which is a special case of a binomial series.

ComputationMost pocket calculators have a square root key. Computer spreadsheets and other software are also frequently usedto calculate square roots. Pocket calculators typically implement efficient routines to compute the exponentialfunction and the natural logarithm or common logarithm, and use them to compute the square root of a positive realnumber a using the identity

or The same identity is exploited when computing square roots with logarithm tables or slide rules.The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopher Heron of Alexandria, who first described it.[4] The method uses the same iterative scheme as the NewtonRaphson method yields when applied to the function

Square root 3

, using the fact that its slope at any point is , but predates it by many centuries.[5] It involves a simple algorithm,which results in a number closer to the actual square root each time it is repeated. The basic idea is that if x is anoverestimate to the square root of a non-negative real number a then will be an underestimate and so the average ofthese two numbers may reasonably be expected to provide a better approximation (though the formal proof of thatassertion depends on the inequality of arithmetic and geometric means that shows this average is always anoverestimate of the square root, as noted below, thus assuring convergence). To find x :1. Start with an arbitrary positive start value x (the closer to the square root of a, the fewer iterations will be needed

to achieve the desired precision).2. Replace x by the average between x and a/x, that is: , representing the NewtonRaphson method

resulting in ,(It is sufficient to take an approximate value of the average to ensure convergence)1. Repeat step 2 until x and a/x are as close as desired.If a is positive, the convergence is "quadratic," which means that in approaching the limit, the number of correctdigits roughly doubles in each next iteration. If a = 0, the convergence is only linear.Using the identity

the computation of the square root of a positive number can be reduced to that of a number in the range [1, 4). Thissimplifies finding a start value for the iterative method that is close to the square root, for which a polynomial orpiecewise-linear approximation can be used.The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying twon-digit numbers.Another useful method for calculating the square root is the Shifting nth root algorithm, applied for .

Square roots of negative and complex numbers

First leaf of the complex square root

Second leaf of the complex square root

Square root 4

Using the Riemann surface of the square root, one can see how the two leaves fit together

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative numbercan have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complexnumbers, that does contain solutions to the square root of a negative number. This is done by introducing a newnumber, denoted by i (sometimes j, especially in the context of electricity where "i" traditionally represents electriccurrent) and called the imaginary unit, which is defined such that i2=1. Using this notation, we can think of i as thesquare root of 1, but notice that we also have (i)2=i2=1 and so i is also a square root of 1. By convention,the principal square root of 1 is i, or more generally, if x is any positive number, then the principal square root of xis

The right side (as well as its negative) is indeed a square root of x, since

For every non-zero complex number z there exist precisely two numbers w such that w2=z: the principal square rootof z (defined below), and its negative.

Square root of an imaginary number

The square roots of i in the complex plane

The square root of i is given by

This result can be obtained algebraically by finding a and b such that

or equivalently

This gives the two simultaneous equations

with solutions

The choice of the principal root then gives

The result can also be obtained by using de Moivre's formula and setting

Square root 5

which produces

Principal square root of a complex numberTo find a definition for the square root that allows us to consistently choose a single value, called the principal value,we start by observing that any complex number x + iy can be viewed as a point in the plane, (x, y), expressed usingCartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair (r, ), where r 0 isthe distance of the point from the origin, and is the angle that the line from the origin to the point makes with thepositive real (x) axis. In complex analysis, this value is conventionally written r ei. If

then we define the principal square root of z as follows:

The principal square root function is thus defined using the nonpositive real axis as a branch cut. The principalsquare root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negativereals it isn't even continuous). The above Taylor series for 1 + x remains valid for complex numbers x with |x|

Square root 6

NotesBecause of the discontinuous nature of the square root function in the complex plane, the law zw = zw is ingeneral not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosenbranch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the valueof zw at all, leading to the equality's failure. A similar problem appears with the complex logarithm and therelation log z + log w = log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance thefollowing one showing that 1=1:

The third equality cannot be justified (see invalid proof). It can be made to hold by changing the meaning of so thatthis no longer represents the principal square root (see above) but selects a branch for the square root that contains(1)(1). The left-hand side becomes either

if the branch includes +i or

if the branch includes i, while the right-hand side becomes

where the last equality, 1 = 1, is a consequence of the choice of branch in the redefinition of .

Square roots of matrices and operatorsIf A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator Bwith B2 = A; we then define A1/2 = A = B. In general matrices may have multiple square roots or even an infinitudeof them. For example the 22 identity matrix has an infinity of square roots.[8]

Uniqueness of square roots in general ringsIn a ring we call an element b a square root of a iff b2 = a.In an integral domain, suppose the element a has some square root b, so b2 = a. Then this square root is notnecessarily unique, but it is "almost unique" in the following sense: If x too is a square root of a, then x2 = a = b2. Sox2 b2 = 0, or, by commutativity, (x + b)(x b) = 0. Because there are no zero divisors in the integral domain, weconclude that one factor is zero, and x = b. The square root of a, if it exists, is therefore unique up to a sign, inintegral domains.

To see that the square root need not be unique up to sign in a general ring, consider the ring from modulararithmetic. Here, the element 1 has four distinct square roots, namely 1 and 3. On the other hand, the element 2has no square root. See also the article quadratic residue for details.Another example is provided by the quaternions in which the element 1 has an infinitude of square rootsincluding i, j, and k.In fact, the set of square roots of 1 is exactly

Hence this set is exactly the same size and shape as the (surface of the) unit sphere in 3-space.

Square root 7

Principal square roots of the positive integers

As decimal expansionsThe square roots of the perfect squares (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots are irrationalnumbers, and therefore their decimal representations are non-repeating decimals.

0

1

1.414213562373095048801688724209698078569671875376948073176679737990732478462 (article) 1milliondigits [9], 2million[10], 5million [11], 10million [12]

1.732050807568877293527446341505872366942805253810380628055806979451933016909 (article) 1milliondigits [13]

2

2.236067977499789696409173668731276235440618359611525724270897245410520925638 (article) 1milliondigits [14]

2.449489742783178098197284074705891391965947480656670128432692567250960377457 1milliondigits [15]

2.645751311064590590501615753639260425710259183082450180368334459201068823230 1milliondigits [16]

2.828427124746190097603377448419396157139343750753896146353359475981464956924 1milliondigits [17]

3

3.162277660168379331998893544432718533719555139325216826857504852792594438639 1milliondigits [18]

3.316624790355399849114932736670686683927088545589353597058682146116484642609

3.464101615137754587054892683011744733885610507620761256111613958903866033818

3.605551275463989293119221267470495946251296573845246212710453056227166948293

3.741657386773941385583748732316549301756019807778726946303745467320035156307

3.872983346207416885179265399782399610832921705291590826587573766113483091937

4

4.123105625617660549821409855974077025147199225373620434398633573094954346338

4.242640687119285146405066172629094235709015626130844219530039213972197435386

4.358898943540673552236981983859615659137003925232444936890344138159557328203

4.472135954999579392818347337462552470881236719223051448541794490821041851276

4.582575694955840006588047193728008488984456576767971902607242123906868425547

Note that if the radicand is not square-free one can simplify, for example ; ; and .

Square root 8

As expansions in other numeral systemsThe square roots of the perfect squares (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots are irrationalnumbers, and therefore their representations in any standard positional notation system are non-repeating.The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothingup my sleeve numbers.

As periodic continued fractionsOne of the most intriguing results from the study of irrational numbers as continued fractions was obtained byJoseph Louis Lagrange circa 1780. Lagrange found that the representation of the square root of any non-squarepositive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeatsindefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, becausethey can be represented with a simple repeating pattern of integers.

[1; 2, 2, ...]

[1; 1, 2, 1, 2, ...]

[2]

[2; 4, 4, ...]

[2; 2, 4, 2, 4, ...]

[2; 1, 1, 1, 4, 1, 1, 1, 4, ...]

[2; 1, 4, 1, 4, ...]

[3]

[3; 6, 6, ...]

[3; 3, 6, 3, 6, ...]

[3; 2, 6, 2, 6, ...]

[3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]

[3; 1, 2, 1, 6, 1, 2, 1, 6, ...]

[3; 1, 6, 1, 6, ...]

[4]

[4; 8, 8, ...]

[4; 4, 8, 4, 8, ...]

[4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]

[4; 2, 8, 2, 8, ...]

The square bracket notation used above is a sort of mathematical shorthand to conserve space. Written in moretraditional notation the simple continued fraction for the square root of 11 [3; 3, 6, 3, 6, ...] looks like this:

Square root 9

where the two-digit pattern {3, 6} repeats over and over and over again in the partial denominators. Since 11 = 32+2,the above is also identical to the following generalized continued fractions:

Geometric construction of the square rootA square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave theconstruction of the geometric mean of two quantities in two different places: Proposition II.14 [19] and PropositionVI.13 [20]. Since the geometric mean of a and b is , one can construct simply by taking b = 1.The construction is also given by Descartes in his La Gomtrie, see figure 2 on page 2 [21]. However, Descartesmade no claim to originality and his audience would have been quite familiar with Euclid.Euclid's second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length a+ b with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections ofthe perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales' theorem and as in theproof of Pythagoras' theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are totriangle ACB, though we don't need that but it is the essence of the proof of Pythagoras' theorem) so that AH:CH isas HC:HB i.e. from which we conclude by cross-multiplication that and finally that . Notefurther that if you were to mark the midpoint O of the line segment AB and draw the radius OC of length then clearly OC > CH i.e. (with equality when and only when a = b), which is thearithmeticgeometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greekunderstanding of "Heron's method".Another method of geometric construction uses right triangles and induction: can, of course, be constructed, andonce has been constructed, the right triangle with 1 and for its legs has a hypotenuse of . The Spiralof Theodorus is constructed using successive square roots in this manner.

HistoryThe Yale Babylonian Collection YBC 7289 clay tablet was created between 1800 BC and 1600 BC, showing and as 1;24,51,10 and 42;25,35 base 60 numbers on a square crossed by two diagonals.[22]

The Rhind Mathematical Papyrus is a copy from 1650 BC of an even earlier work and shows how the Egyptiansextracted square roots.[23]

In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as theSulba Sutras, dated around 800500 BC (possibly much earlier). A method for finding very good approximations tothe square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.[24] Aryabhata in the Aryabhatiya (section 2.4),has given a method for finding the square root of numbers having many digits.In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early HanDynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine theexcess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and theexcess numerator times the deficiency denominator, combine them as the dividend."[25]

A symbol for square roots, written as an elaborate R, was invented by Regiomontanus (1436 - 1476). An R was alsoused for Radix to indicate square roots in Giralamo Cardano's Ars Magna. [26]

Square root 10

According to historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was firstintroduced in Europe by Cataneo in 1546.[27]

The symbol for the square root was first used in print in 1525 in Christoph Rudolff's Coss, which was also the firstto use the then-new signs '+' and ''.[28]

Notes[1] Gel'fand, p. 120 (http:/ / books. google. com/ books?id=Z9z7iliyFD0C& pg=PA120)[2] Zill, Dennis G.; Shanahan, Patrick (2008). A First Course in Complex Analysis With Applications (http:/ / books. google. com/

books?id=YKZqY8PCNo0C) (2nd ed.). Jones & Bartlett Learning. p.78. ISBN0-7637-5772-1. ., Extract of page 78 (http:/ / books. google.com/ books?id=YKZqY8PCNo0C& pg=PA78)

[3] Heath, Sir Thomas L. (1908). The Thirteen Books of The Elements, Vol. 3 (http:/ / www. archive. org/ stream/thirteenbookseu03heibgoog#page/ n14/ mode/ 1up). Cambridge University Press. p.3. .

[4] Heath, Sir Thomas L. (1921). A History of Greek Mathematics, Vol. 2 (http:/ / books. google. com/ ?id=LOA5AAAAMAAJ& pg=PR323).Oxford: Clarendon Press. pp.323324. .

[5] Muller, Jean-Mic (2006). Elementary functions: algorithms and implementation (http:/ / books. google. com/ ?id=g3AlWip4R38C). Springer.pp.9293. ISBN0-8176-4372-9. , Chapter 5, p 92 (http:/ / books. google. com/ books?id=g3AlWip4R38C& pg=PA92)

[6] Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables (http:/ /books. google. com/ books?id=MtU8uP7XMvoC). Courier Dover Publications. p.17. ISBN0-486-61272-4. ., Section 3.7.27, p. 17 (http:/ /www. math. sfu. ca/ ~cbm/ aands/ page_17. htm)

[7] Cooke, Roger (2008). Classical algebra: its nature, origins, and uses (http:/ / books. google. com/ books?id=lUcTsYopfhkC). John Wileyand Sons. p.59. ISBN0-470-25952-3. ., Extract: page 59 (http:/ / books. google. com/ books?id=lUcTsYopfhkC& pg=PA59)

[8] Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", Mathematical Gazette 87, November 2003, 499500.[9] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt2. 1mil[10] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt2. 2mil[11] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt2. 5mil[12] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt2. 10mil[13] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt3. 1mil[14] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt5. 1mil[15] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt6. 1mil[16] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt7. 1mil[17] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt8. 1mil[18] http:/ / antwrp. gsfc. nasa. gov/ htmltest/ gifcity/ sqrt10. 1mil[19] http:/ / aleph0. clarku. edu/ ~djoyce/ java/ elements/ bookII/ propII14. html[20] http:/ / aleph0. clarku. edu/ ~djoyce/ java/ elements/ bookVI/ propVI13. html[21] http:/ / historical. library. cornell. edu/ cgi-bin/ cul. math/ docviewer?did=00570001& seq=12& frames=0& view=50[22] Math.ubc.ca (http:/ / www. math. ubc. ca/ ~cass/ Euclid/ ybc/ analysis. html)[23] Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.[24][24] Joseph, ch.8.[25][25] Dauben (2007), p. 210.[26] Nrich (http:/ / nrich. maths. org/ 6546)[27][27] Smith, p. 148.[28] Manguel, Alberto (2006). "Done on paper: the dual nature of numbers and the page". The Life of Numbers. ISBN84-86882-14-1.

References Dauben, Joseph W. (2007). "Chinese Mathematics I" (http:/ / books. google. com/ books?id=3ullzl036UEC). In

Katz, Victor J.. The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: PrincetonUniversity Press. ISBN0-691-11485-4.

Gel'fand, Izrael M.; Shen, Alexander (1993). Algebra (http:/ / books. google. com/ books?id=Z9z7iliyFD0C) (3rded.). Birkhuser. p.120. ISBN0-8176-3677-3.

Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN0-691-00659-8. Smith, David (1958). History of Mathematics. 2. New York: Dover Publications. ISBN978-0-486-20430-7.

Square root 11

External links Algorithms, implementations, and more (http:/ / www. azillionmonkeys. com/ qed/ sqroot. html) - Paul Hsieh's

square roots webpage How to manually find a square root (http:/ / johnkerl. org/ doc/ square-root. html)

Article Sources and Contributors 12

Article Sources and ContributorsSquare root Source: http://en.wikipedia.org/w/index.php?oldid=528866168 Contributors: (jarbarf), 200.45.42.xxx, 2001:558:6026:81:7560:20C2:D985:2A70,2601:9:4180:2A:F488:FCDD:1615:729F, 4.46.3.xxx, ABCD, ABF, AK456, Aaron north, AceMyth, Adam majewski, Adj08, Afa86, Ahoerstemeier, Ajmas, Alansohn, Aliekens, Amcnea, AnasSalloum, AndreaPersephone, Andrewmc123, Andrewrost3241981, Animum, Ann Stouter, Anonymous Dissident, Antandrus, Antilived, Aristotle2600, Arithmonic, Arthur Rubin, Asclepius,AugPi, AxelBoldt, Aziraphael, B, Badger151, Beland, Bharatveer, BillFlis, Bk1 168, Bkell, Black Kite, Blotwell, Blue william, Bob K31416, Bobdoleisevil, Bongwarrior, Bryan Derksen,Bsadowski1, Bsimmons666, Bsmith671, Burn, CPMcE, CWY2190, CWii, Calebbbbb, Caltrop, CameronPG, Capitalistroadster, Capricorn42, Catgut, Catherineyronwode, Cdc, Cesiumfrog,Chetvorno, Chuunen Baka, CiteCop, Clarince63, Coffee, Complex (de), Constructive editor, Corpx, Cp111, Cyp, Czenek, DARTH SIDIOUS 2, DVdm, DaHuzyBru, Daniel,levine, Daphne A,Daqu, Dataryder, Daven200520, David Shay, DavidCBryant, DavidMCEddy, Dcljr, Dcoetzee, Deathflyer, Deeptrivia, Demmy100, DerHexer, Derekleungtszhei, Discospinster, Divide, Dmcq,Domthedude001, Doshell, Ductapedaredevil, Duoduoduo, Duplode, DynamicDonkey, Dysprosia, Eeekster, Eequor, Egg, Ellywa, Emoll, Epbr123, Eric119, Esrob, Exor674, FAThomssen,Favonian, Fgnievinski, FightingMac, Folajimi, Foregone conclusion, Forgottenanimator, Frank, Fredrik, Frencheigh, Fresheneesz, Funkter, Gabetarian, Gail, Galoubet, Gandalf61, Garo,Garykempen, Gauge00, Gesslein, Gfoley4, Ghaly, Giftlite, Gilmer, Gisling, Glenn L, GoGi, Golbez, Gozwei, Graham87, Grunt, Guanaco, Gkhan, H8ter boi, Happy-melon, Harasty, Harry,Hdt83, He Who Is, Heather1144, Henrygb, Herbee, Hildebrandt20031, Iamunknown, Icairns, Idkwattoput, Inferno, Lord of Penguins, Iridescent, Irishguy, Isaac, Isheden, Itsmejudith, J.delanoy,JGrusd, JHunterJ, JRSpriggs, JamesBWatson, Jamesooders, Jaxl, Jeff G., Jeppesn, JeremyA, Jewbacca, Jiddisch, Jim1138, Jim45678, JimVC3, Jimothy 46, Jimp, Jitse Niesen, Jllompi,Jmnbatista, Joe9320, JohnCD, Josh Cherry, Josh Grosse, Jumbuck, JustUser, Justin Eiler, Justine hutchinson, JzWmtr, KDesk, KSmrq, Kaobear, Karl0905, KeyBlade999, Kiersere, Kirbytime,Knowz, Kortaggio, Kukini, Kuru, LFaraone, La goutte de pluie, Lambiam, Langing, Lawnchair, Linas, LinuxJuice, Lkesteloot, Loadmaster, Logan, Logical2u, Lolzqueen, Lord Emsworth, Lotje,Luna Santin, M00npirate, MC10, MFH, MONGO, Madmath789, Malcolm Farmer, Manning Bartlett, ManningBartlett, Marauder40, Mardybum59, Marek69, Mat cross, Materialscientist,MathMan64, Mathtyke, Mathx314, Matt Deres, Mayooranathan, McAusten, Meaghan, Meaning of Fife, Meni Rosenfeld, Mental Blank, Mentisock, Mets501, Michael Courtney, Michael Hardy,Michael Slone, Mifter, Mild Bill Hiccup, Milogardner, Mind21 98, Modster, Monedula, Mr. Manwhich, MrFish, MrOllie, Mrdempsey, Mshonle, Mukmantheorigina, Musiphil, MylesMc, N8chz,NHRHS2010, Natman 2000, NawlinWiki, Nerdson123, Nonagonal Spider, NotAnonymous0, Ocean12, Oda Mari, Ohanian, Oleg Alexandrov, Oleg Viro, Onore Baka Sama, Onorem, OreoPriest, Orulo, Ost000101, Paddles, Paepaok, Paolo.dL, Papito123, Parins, Paste, Patrick, Paul August, Pdcook, Pen1515, Pepito123, Pepper, PericlesofAthens, PerryTachett, Peter Gawtry,Philg88, Piano non troppo, PierreAbbat, Pinethicket, Pizza Puzzle, Plugwash, Potatoswatter, PranksterTurtle, ProfBradley, Psy guy, Pt, Pyrospirit, Qef, Qniemiec, Qooth, Quondum, RDBury,RMFan1, RadioFan, Radu242, Raljah, Random user 8384993, Rashayw, Raulshc, Razor2988, RedWolf, Redaktor, Renzeescoto, RexNL, Reywas92, Rich Farmbrough, Rick Norwood, Rinpoche,Rnelsonee, Robertvan1, Rohan Ghatak, Roland Kaufmann, Rothgo, Ruakh, RyanGerbil10, S h i v a (Visnu), SGBailey, SQL, Salix alba, Sam Korn, Sceptre, Scetoaux, SchfiftyThree, SchreyP,Seav, Sergei, Seschubert, Setokaiba, Shenweiwei, Showgun45, Shreevatsa, Sidhekin, SimonP, SimpsonDG, Sjakkalle, Skullfission, Sleigh, Sligocki, Smaug123, Sopoforic, Sparthorse, Spoon!,Stephenb, Stevey7788, Stj6, Sudhirb, SvNH, SvavarL, Symane, T1gerch3n, TUF-KAT, TakuyaMurata, Tarquin, Tavilis, Taxman, The Anome, The Man in Question, The Thing That Should NotBe, TheGerm, Thecheesykid, Theryguy512, Thewikicontributor, Tombomp, Tparameter, Trappist the monk, Treisijs, Tresiden, Trevor Marron, Troll breath, Trusilver, Twin Bird, Typuifre,Uaxuctum, Vaccuumsauce, Valermos, Vanished User 0001, Versus22, Vipinhari, Vsmith, WeijiBaikeBianji, Wereon, Westc55998, Wik, Wiki alf, Wiki-vr, WikiLaurent, Wolfrock, Wolvereness,Wragge, XJamRastafire, Xantharius, Xchrisblackx, Yamamoto Ichiro, Yk Yk Yk, Zhorken, Zundark, var Arnfjr Bjarmason, , , 749 anonymous edits

Image Sources, Licenses and ContributorsImage:Drop time.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Drop_time.jpg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Michael Courtney aten.wikipediaImage:Square root 0 25.svg Source: http://en.wikipedia.org/w/index.php?title=File:Square_root_0_25.svg License: Public Domain Contributors: Qef wasssuupp!!?Image:Complex sqrt leaf1.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_sqrt_leaf1.jpg License: Public Domain Contributors: Jan HomannImage:Complex sqrt leaf2.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_sqrt_leaf2.jpg License: Public Domain Contributors: Jan HomannImage:Riemann surface sqrt.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Riemann_surface_sqrt.jpg License: Public Domain Contributors: Jan HomannImage:Imaginary2Root.svg Source: http://en.wikipedia.org/w/index.php?title=File:Imaginary2Root.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Loadmaster(David R. Tribble)

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/

Square rootPropertiesComputationSquare roots of negative and complex numbersSquare root of an imaginary numberPrincipal square root of a complex numberAlgebraic formulaNotes

Square roots of matrices and operatorsUniqueness of square roots in general ringsPrincipal square roots of the positive integersAs decimal expansionsAs expansions in other numeral systemsAs periodic continued fractions

Geometric construction of the square rootHistoryNotesReferencesExternal links

License