Notation for the (principal) square root ofx.For example,√25 = 5, since25 = 5 ⋅ 5, or52 (5 squared).
Inmathematics, asquare root of a numberx is a numbery such that; in other words, a numbery whosesquare (the result of multiplying the number by itself, or) isx.[1] For example, 4 and −4 are square roots of 16 because.
Everynonnegativereal numberx has a unique nonnegative square root, called theprincipal square root or simplythe square root (with a definite article, see below), which is denoted by where the symbol "" is called theradical sign[2] orradix. For example, to express the fact that the principal square root of 9 is 3, we write. The term (or number) whose square root is being considered is known as theradicand. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negativex, the principal square root can also be written inexponent notation, as.
Everypositive numberx has two square roots: (which is positive) and (which is negative). The two roots can be written more concisely using the± sign as. Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.[3][4]
Square roots of negative numbers can be discussed within the framework ofcomplex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These includefunction spaces andsquare matrices, among othermathematical structures.
History
YBC 7289 clay tablet
TheYale Babylonian Collection clay tabletYBC 7289 was created between 1800 BC and 1600 BC, showing and respectively as 1;24,51,10 and 0;42,25,35base 60 numbers on a square crossed by two diagonals.[5] (1;24,51,10) base 60 corresponds to 1.41421296, which is correct to 5 decimal places (1.41421356...).
InAncient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as theSulba Sutras, dated around 800–500 BC (possibly much earlier).[7] A method for finding very good approximations to the square roots of 2 and 3 are given in theBaudhayana Sulba Sutra.[8]Apastamba who was dated around 600 BCE has given a strikingly accurate value for which is correct up to five decimal places as.[9][10][11]Aryabhata, in theAryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.
It was known to the ancient Greeks that square roots ofpositive integers that are notperfect squares are alwaysirrational numbers: numbers not expressible as aratio of two integers (that is, they cannot be written exactly as, wherem andn are integers). This is the theoremEuclid X, 9, almost certainly due toTheaetetus dating back toc. 380 BC.[12]The discovery of irrational numbers, including the particular case of thesquare root of 2, is widely associated with the Pythagorean school.[13][14] Although some accounts attribute the discovery toHippasus, the specific contributor remains uncertain due to the scarcity of primary sources and the secretive nature of the brotherhood.[15][16] It is exactly the length of thediagonal of asquare with side length 1.
In the Chinese mathematical workWritings on Reckoning, written between 202 BC and 186 BC during the earlyHan dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[17]
A symbol for square roots, written as an elaborate R, was invented byRegiomontanus (1436–1476). An R was also used for radix to indicate square roots inGerolamo Cardano'sArs Magna.[18]
According to historian of mathematicsD.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe byCataneo—in 1546.
According to Jeffrey A. Oaks, Arabs used the letterjīm/ĝīm (ج), the first letter of the word "جذر" (variously transliterated asjaḏr,jiḏr,ǧaḏr orǧiḏr, "root"), placed in its initial form (ﺟ) over a number to indicate its square root. The letterjīm resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematicianIbn al-Yasamin.[19]
The symbol "√" for the square root was first used in print in 1525, inChristoph Rudolff'sCoss.[20]
Properties and uses
The graph of the functionf(x) = √x, made up of half aparabola with a verticaldirectrix
The principal square root function (usually just referred to as the "square root function") is afunction that maps theset of nonnegative real numbers onto itself. Ingeometrical terms, the square root function maps thearea of a square to its side length.
The square root ofx is rational if and only ifx is arational number that can be represented as a ratio of two perfect squares. (Seesquare root of 2 for proofs that this is an irrational number, andquadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers intoalgebraic numbers, the latter being asuperset of the rational numbers).
The square root function iscontinuous for all nonnegativex, anddifferentiable for all positivex. Iff denotes the square root function, whose derivative is given by:
TheTaylor series of aboutx = 0 converges for|x| ≤ 1, and is given by
A positive number has two square roots, one positive, and one negative, which areopposite to each other. When talking ofthe square root of a positive integer, it is usually the positive square root that is meant.
The square root of a positive integer is the product of the roots of itsprime factors, because the square root of a product is the product of the square roots of the factors. Since only roots of those primes having an odd power in thefactorization are necessary. More precisely, the square root of a prime factorization is
As decimal expansions
The square roots of theperfect squares (e.g., 0, 1, 4, 9, 16) areintegers. In all other cases, the square roots of positive integers areirrational numbers, and hence have non-repeating decimals in theirdecimal representations. Decimal approximations of the square roots of the first few natural numbers are given in the following table.
As with before, the square roots of theperfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers areirrational numbers, and therefore have non-repeating digits in any standardpositional notation system.
Thesquare bracket notation used above is a short form for a continued fraction. Written in the more suggestive algebraic form, the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this:
where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since11 = 32 + 2, the above is also identical to the followinggeneralized continued fractions:
Square roots of positive numbers are not in generalrational numbers, and so cannot be written as a terminating or recurring decimal expression. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained.
Mostpocket calculators have a square root key. Computerspreadsheets and othersoftware are also frequently used to calculate square roots. Pocket calculators typically implement efficient routines, such as theNewton's method (frequently with an initial guess of 1), to compute the square root of a positive real number.[21][22] When computing square roots withlogarithm tables orslide rules, one can exploit the identitieswhereln andlog10 are thenatural andbase-10 logarithms.
By trial-and-error,[23] one can square an estimate for and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it is prudent to use the identityas it allows one to adjust the estimatex by some amountc and measure the square of the adjustment in terms of the original estimate and its square.
The most commoniterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopherHeron of Alexandria, who first described it.[24] The method uses the same iterative scheme as theNewton–Raphson method yields when applied to the functiony =f(x) =x2 −a, using the fact that its slope at any point isdy/dx =f′(x) = 2x, but predates it by many centuries.[25] The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that ifx is an overestimate to the square root of a nonnegative real numbera thena/x will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, theinequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as notedbelow), and so it can serve as a new overestimate with which to repeat the process, whichconverges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To findx:
Start with an arbitrary positive start valuex. The closer to the square root ofa, the fewer the iterations that will be needed to achieve the desired precision.
Replacex by the average(x +a/x) / 2 betweenx anda/x.
Repeat from step 2, using this average as the new value ofx.
That is, if an arbitrary guess for isx0, andxn + 1 = (xn +a/xn) / 2, then eachxn is an approximation of which is better for largen than for smalln. Ifa is positive, the convergence isquadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. Ifa = 0, the convergence is only linear; however, so in this case no iteration is needed.
Using the identitythe computation of the square root of a positive number can be reduced to that of a number in the range[1, 4). This simplifies finding a start value for the iterative method that is close to the square root, for which apolynomial orpiecewise-linearapproximation can be used.
Thetime complexity for computing a square root withn 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 forn = 2.
Using theRiemann surface of the square root, it is shown 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 number can have areal square root. However, it is possible to work with a more inclusive set of numbers, called thecomplex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted byi (sometimes byj, especially in the context ofelectricity wherei traditionally represents electric current) and called theimaginary unit, which isdefined such thati2 = −1. Using this notation, we can think ofi as the square root of −1, but 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 isi, or more generally, ifx is any nonnegative number, then the principal square root of−x is
The right side (as well as its negative) is indeed a square root of−x, since
For every non-zero complex numberz there exist precisely two numbersw such thatw2 =z: the principal square root ofz (defined below), and its negative.
Principal square root of a complex number
Geometric representation of the 2nd to 6th roots of a complex numberz, in polar formreiφ wherer = |z | andφ = argz. Ifz is real,φ = 0 orπ. Principal roots are shown in black.
To find a definition for the square root that allows us to consistently choose a single value, called theprincipal value, we start by observing that any complex number can be viewed as a point in the plane, expressed usingCartesian coordinates. The same point may be reinterpreted usingpolar coordinates as the pair where is the distance of the point from the origin, and is the angle that the line from the origin to the point makes with the positive real () axis. In complex analysis, the location of this point is conventionally written Ifthen theprincipal square root of is defined to be the following: The principal square root function is thus defined using the non-positive real axis as abranch cut. If is a non-negative real number (which happens if and only if) then the principal square root of is in other words, the principal square root of a non-negative real number is just the usual non-negative square root. It is important that because if, for example, (so) then the principal square root isbut using would instead produce the other square root
The principal square root function isholomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not evencontinuous). The above Taylor series for remains valid for complex numbers with
When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root:[28][29]
wheresgn(y) = 1 ify ≥ 0 andsgn(y) = −1 otherwise.[30] In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative.
For example, the principal square roots of±i are given by:
Notes
In the following, the complexz andw may be expressed as:
where and.
Because of the discontinuous nature of the square root function in the complex plane, the following laws arenot true in general.
Counterexample for the principal square root:z = −1 andw = −1 This equality is valid only when
Counterexample for the principal square root:w = 1 andz = −1 This equality is valid only when
Counterexample for the principal square root:z = −1) This equality is valid only when
A similar problem appears with other complex functions with branch cuts, e.g., thecomplex logarithm and the relationslogz + logw = log(zw) orlog(z*) = log(z)* which are not true in general.
Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that−1 = 1:
The third equality cannot be justified (seeinvalid proof).[31]: Chapter VI, Section I, Subsection 2The fallacy that +1 = −1 It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains The left-hand side becomes eitherif the branch includes+i orif the branch includes−i, while the right-hand side becomeswhere the last equality, is a consequence of the choice of branch in the redefinition of√.
nth roots and polynomial roots
The definition of a square root of as a number such that has been generalized in the following way.
Acube root of is a number such that; it is denoted
Ifn is an integer greater than two, an-th root of is a number such that; it is denoted
Given anypolynomialp, aroot ofp is a numbery such thatp(y) = 0. For example, thenth roots ofx are the roots of the polynomial (iny)
Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms ofnth roots.
IfA is apositive-definite matrix or operator, then there exists precisely one positive definite matrix or operatorB withB2 =A; we then defineA1/2 =B. In general matrices may have multiple square roots or even an infinitude of them. For example, the2 × 2identity matrix has an infinity of square roots,[32] though only one of them is positive definite.
In integral domains, including fields
Each element of anintegral domain has no more than 2 square roots. Thedifference of two squares identityu2 −v2 = (u −v)(u +v) is proved using thecommutativity of multiplication. Ifu andv are square roots of the same element, thenu2 −v2 = 0. Because there are nozero divisors this impliesu =v oru +v = 0, where the latter means that two roots areadditive inverses of each other. In other words if an element a square rootu of an elementa exists, then the only square roots ofa areu and−u. The only square root of 0 in an integral domain is 0 itself.
In a field ofcharacteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that−u =u. If the field isfinite of characteristic 2 then every element has a unique square root. In afield of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.
Given an oddprime numberp, letq =pe for some positive integere. A non-zero element of the fieldFq withq elements is aquadratic residue if it has a square root inFq. Otherwise, it is a quadratic non-residue. There are(q − 1)/2 quadratic residues and(q − 1)/2 quadratic non-residues; zero is not counted in either class. The quadratic residues form agroup under multiplication. The properties of quadratic residues are widely used innumber theory.
In rings in general
Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring of integersmodulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3.
Another example is provided by the ring ofquaternions which has no zero divisors, but is not commutative. Here, the element −1 hasinfinitely many square roots, including±i,±j, and±k. In fact, the set of square roots of−1 is exactly
A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in any multiple ofn is a square root of 0.
The square root of a positive number is usually defined as the side length of asquare with thearea equal to the given number. But the square shape is not necessary for it: if one of twosimilarplanar Euclidean objects has the areaa times greater than another, then the ratio of their linear sizes is.
A square root can be constructed with a compass and straightedge. In hisElements,Euclid (fl. 300 BC) gave the construction of thegeometric mean of two quantities in two different places:Proposition II.14 andProposition VI.13. Since the geometric mean ofa andb is, one can construct simply by takingb = 1.
The construction is also given byDescartes in hisLa Géométrie, see figure 2 onpage 2. However, Descartes made 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 ofsimilar triangles. Let AHB be a line segment of lengtha +b withAH =a andHB =b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH ash. Then, usingThales' theorem and, as in theproof of Pythagoras' theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that, but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB, i.e.a/h =h/b, from which we conclude by cross-multiplication thath2 =ab, and finally that. When marking the midpoint O of the line segment AB and drawing the radius OC of length(a +b)/2, then clearly OC > CH, i.e. (with equality if and only ifa =b), which is thearithmetic–geometric mean inequality for two variables and, as notedabove, is the basis of theAncient Greek understanding of "Heron's method".
Another method of geometric construction usesright triangles andinduction: can be constructed, and once has been constructed, the right triangle with legs 1 and has ahypotenuse of. Constructing successive square roots in this manner yields theSpiral of Theodorus depicted above.
^Anglin, W.S. (1994).Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
^Seidenberg, A. (1961)."The ritual origin of geometry".Archive for History of Exact Sciences.1 (5):488–527.doi:10.1007/bf00327767.ISSN0003-9519.S2CID119992603.Seidenberg (pp. 501-505) proposes: "It is the distinction between use and origin." [By analogy] "KEPLER needed the ellipse to describe the paths of the planets around the sun; he did not, however invent the ellipse, but made use of a curve that had been lying around for nearly 2000 years". In this manner Seidenberg argues: "Although the date of a manuscript or text cannot give us the age of the practices it discloses, nonetheless the evidence is contained in manuscripts." Seidenberg quotes Thibaut from 1875: "Regarding the time in which the Sulvasutras may have been composed, it is impossible to give more accurate information than we are able to give about the date of the Kalpasutras. But whatever the period may have been during which Kalpasutras and Sulvasutras were composed in the form now before us, we must keep in view that they only give a systematically arranged description of sacrificial rites, which had been practiced during long preceding ages." Lastly, Seidenberg summarizes: "In 1899, THIBAUT ventured to assign the fourth or the third centuries B.C. as the latest possible date for the composition of the Sulvasutras (it being understood that this refers to a codification of far older material)."
^J J O'Connor; E F Robertson (November 2020)."Apastamba".www.mathshistory.st-andrews.ac.uk. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved30 March 2024.
