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Encyclopedia > Square root

In mathematics, a square root (√) of a number x is a number r such that r2 = x, or in words, a number r whose square (the result of multiplying the number by itself) is x. Every non-negative real number x has a unique non-negative square root, called the principal square root and denoted with a radical symbol as $sqrt x$. For example, the principal square root of 9 is 3, denoted $sqrt 9 = 3$, because $3^2 = 3times3 = 9$. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In algebra, the square of a number is that number multiplied by itself. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, an nth root of a number a is a number b, such that bn=a. ...

Square roots often arise when solving quadratic equations, or equations of the form ax2 + bx + c = 0, due to the variable x being squared. In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

Every positive number x has two square roots. One of them is , which is positive, and the other is $-sqrt{x}$, which is negative. Together, these two roots are denoted $pmsqrt{x}$. Square roots of negative numbers can be discussed within the framework of complex numbers. Square roots of objects other than numbers can also be defined. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers. For example, $sqrt 2$ 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. This has been known since ancient times, with the discovery that $sqrt 2$ is irrational attributed to Hipparchus, a disciple of Pythagoras. (See square root of 2 for proofs of the irrationality of this number.) The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... This article is about the mathematical concept. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... For other uses, see Square. ... For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ...

## Properties

The graph of the function $f(x) = sqrt x$, made up of half a parabola with a vertical directrix.

The principal square root function $f(x) = sqrt{x}$ (usually just referred to as the "square root function") is a function which maps the set of non-negative real numbers $mathbb{R}^+ cup {0}$ onto itself, and, like all functions, always returns a unique value. The square root function also maps rational numbers into algebraic numbers (a superset of the rational numbers); $sqrt x$ is rational if and only if x is a rational number which can be represented as a ratio of two perfect squares. In geometrical terms, the square root function maps the area of a square to its side length. Function square root (self made graph) File links The following pages link to this file: Square root Categories: GFDL images ... Function square root (self made graph) File links The following pages link to this file: Square root Categories: GFDL images ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ... For other uses, see Geometry (disambiguation). ... Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ... For other uses, see Square. ...

• For all real numbers x,
$sqrt{x^2} = left|xright| = begin{cases} x, & mbox{if }x ge 0 -x, & mbox{if }x le 0 end{cases}$     (see absolute value)
• For all positive real numbers x and y,
$sqrt{xy} = sqrt x sqrt y$
and
$sqrt x = x^{1/2}.$
• The square root function is continuous for all non-negative x, and differentiable for all positive x. Its derivative is given by
$f'(x) = frac{1}{2sqrt x}.$
• The Taylor series of about $x=0!$ converges for $left| x right| < 1$ and is given by
$sqrt{x+1} = 1 + frac{1}{2}x - frac{1}{8}x^2 + frac{1}{16} x^3 - frac{5}{128} x^4 + dots!$

In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... This article is about derivatives and differentiation in mathematical calculus. ... Series expansion redirects here. ...

## Computation

Many methods of calculating square roots exist today, some meant to be done by hand and some meant to be done by machine. This article presents and explains several methods which can be used to calculate square roots. ...

Many, but not all pocket calculators have a square root key. Computer spreadsheets and other software are also frequently used to calculate square roots. Computer software programs typically implement good routines to compute the exponential function and the natural logarithm or logarithm, and then compute the square root of x using the identity A basic arithmetic calculator. ... Screenshot of a spreadsheet under OpenOffice A spreadsheet is a rectangular table (or grid) of information, often financial information. ... Computer software (or simply software) refers to one or more computer programs and data held in the storage of a computer for some purpose. ... The exponential function is one of the most important functions in mathematics. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... Look up logarithm in Wiktionary, the free dictionary. ...

$sqrt{x} = e^{frac{1}{2}ln x}$ or $sqrt{x} = 10^{frac{1}{2}log x}$

The same identity is exploited when computing square roots with logarithm tables or slide rules. The common logarithm is the logarithm with base 10. ... A typical 10 inch student slide rule (Pickett N902-T simplex trig). ...

The most common method of square root calculation by hand is known as the "Babylonian method". It involves a simple algorithm, which results in a number closer to the actual square root each time it is repeated. To find r, the square root of a real number x: This article presents and explains several methods which can be used to calculate square roots. ...

1. Start with an arbitrary positive start value r (the closer to the square root of x, the better).
2. Replace r by the average between r and x / r. (It is sufficient to take an approximate value of the average, not too close to the previous value of r and x / r in order to ensure convergence.)
3. Repeat step 2 until r and x / r are as close as desired.

The best known time complexity for computing a square root with n digits of precision is the same as that for multiplying two n-digit numbers. In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. ...

## Square roots of negative and complex numbers

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

$sqrt{-x} = i sqrt x$

because

$(isqrt x)^2 = i^2(sqrt x)^2 = (-1)x = -x.$

By the argument given above, i can be neither positive nor negative. This creates a problem: for the complex number z, we cannot define $sqrt z$ to be the "positive" square root of z.

For every non-zero complex number z there exist precisely two numbers w such that w2 = z. For example, the square roots of i are:

$sqrt{i} = frac{1}{sqrt{2}}(1+i)$

and

$- sqrt{i} = - frac{1}{sqrt{2}}(1+i).$

The usual definition of √z is by introducing the following branch cut: if z = r eiφ is represented in polar coordinates with −π < φ ≤ π, then we set the principal value to In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... See also Cauchy principal value for its use in describing improper integrals In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued. ...

$sqrt{z} = sqrt{r} , e^{iphi over 2}.$

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 $sqrt{1+x}$ remains valid for complex numbers x with |x| < 1. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

When the number is in rectangular form the following formula can be used for the principal value: Fig. ...

$sqrt{x+iy} = sqrt{frac{r + x}{2}} + i frac{y}{sqrt{2 (r + x)}}$

where $r = left|x+iyright| = sqrt{x^2+y^2}$ (the absolute value or modulus of the complex number), unless x=-r and y=0. Notice that the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number. The real part of the principal value is always non-negative. In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

Note that because of the discontinuous nature of the square root function in the complex plane, the law $sqrt{zw} = sqrt z cdot sqrt w$ is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of $sqrt z cdot sqrt w$ at all, leading to equality's failure. A similar problem appears with the complex logarithm and the relation log(z)+log(w)=log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that −1 = 1: The natural logarithm is the logarithm to the base e, where e is equal to 2. ...

$-1 = i cdot i = sqrt{-1} cdot sqrt{-1} = sqrt{-1 cdot -1} = sqrt{1} = 1$

The third equality cannot be justified (see invalid proof), however, it can be adjusted to be true if we (1) permit freedom in the choice of branch by no longer requiring the principal square root (defined in the beginning of the article) implicit in the √ notation and (2) choose the square root's branch so as to exclude the value 1. The left hand side becomes either $sqrt{-1} cdot sqrt{-1}=i*i=-1$ if the branch includes + i or $sqrt{-1} cdot sqrt{-1}=(-i)*(-i)=-1$ if the branch includes i while the right hand side becomes $sqrt{-1 cdot -1}=sqrt{1}=-1$, again by the choice of branch. In mathematics, there are a variety of spurious proofs of obvious contradictions. ...

## 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 B2 = A; we then define √A = B. The square root of a matrix A is a matrix B such that the matrix product B B is equal to A. Numerical method Given a square matrix A, one way to find its square root is the Denman-Beavers square root iteration, described below: Given matrix A, and the... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...

More generally, to every normal matrix or operator A there exist normal operators B such that B2 = 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 real numbers, and normal operators are akin to complex numbers. In functional analysis, a normal operator on a Hilbert space is a continuous linear operator that commutes with its hermitian adjoint : The main importance of this concept is that the spectral theorem applies to normal operators. ...

## Principal square roots of the first 20 positive integers

### As non-periodic decimal fractions

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

### As periodic continued fractions

One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange circa 1780. Lagrange found that the square root of any non-square positive integer can be represented by a periodic continued fraction. That is, in which a certain pattern of digits occurs over and over in the denominators (see example below the table). In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of digits. In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... Periodicity is the quality of occurring at regular intervals (e. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... A denominator is a name. ...

 $sqrt {2}$ [1; 2, 2, ...] $sqrt {3}$ [1; 1, 2, 1, 2, ...] $sqrt {4}$ [2] $sqrt {5}$ [2; 4, 4, ...] $sqrt {6}$ [2; 2, 4, 2, 4, ...] $sqrt {7}$ [2; 1, 1, 1, 4, 1, 1, 1, 4, ...] $sqrt {8}$ [2; 1, 4, 1, 4, ...] $sqrt {9}$ [3] $sqrt {10}$ [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 more traditional notation the simple continued fraction for the square root of 11 – [3; 3, 6, 3, 6, ...] – looks like this: In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

where the two-digit pattern {3, 6} repeats over and over and over again in the partial denominators.

## Geometric construction of the square root

A square root can be constructed with a compass and straightedge. 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 , one can construct simply by taking b = 1. The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... For other uses, see Euclid (disambiguation). ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

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. René Descartes René Descartes (IPA: , March 31, 1596 – February 11, 1650), also known as Cartesius, worked as a philosopher and mathematician. ... La GÃ©omÃ©trie was published in 1637 and written by RenÃ© Descartes. ...

Another method of geometric construction uses right triangles and induction: can, of course, be constructed, and once has been constructed, the right triangle with 1 and for its legs has a hypotenuse of . This article is about angles in geometry. ... A triangle. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ...

## History

The Rhind Mathematical Papyrus is a copy from 1650 BC of an even earlier work and shows us how the Egyptians extracted square roots.[1] The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ...

In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800-500 B.C. (possibly much earlier). A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.[2] Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits. The History of India begins with the Indus Valley Civilization, which flourished in the north-western part of the Indian subcontinent from 3300 to 1700 BCE. This Bronze Age civilization was followed by the Iron Age Vedic period, which witnessed the rise of major kingdoms known as the Mahajanapadas. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... BaudhÄyana, (fl. ... For other uses, see Aryabhata (disambiguation). ... Ä€ryabhatÄ«ya, an astronomical treastise, is the Magnum Opus and only extant work of the 5th century Indian Mathematician, Aryabhatta. ...

D.E. Smith in History of Mathematics, says, about the existing situation in Europe: "In Europe these methods (for finding out the square and square root) did not appear before Cataneo (1546). He gave the method of Aryabhata for determining the square root".[3] For other uses, see Aryabhata (disambiguation). ...

## Notes

1. ^ Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
2. ^ Joseph, ch.8.
3. ^ Smith, p. 148.

## References

• Smith D.E., History of Mathematics (book 2)
• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.

It has been suggested that Penguin Modern Poets, Penguin Great Ideas be merged into this article or section. ...

Results from FactBites:

 Square root - Wikipedia, the free encyclopedia (1682 words) In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. Square roots of non-perfect squares are always irrational numbers, i.e., numbers not expressible as a ratio of two integers. Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous).
 square root of 2 (334 words) It was the first number shown to be what is now known as an irrational number (a number that can't be written in the form a/b, where both a and b are integers.) This discovery was made by Pythagoras or, at any rate, by the Pythagorean group that he founded. The square root of 2 is the length of the hypotenuse (longest side) of a right triangle whose other two sides are each one unit long. Squaring each of these numbers gives 1, 36, 1225, 41616, 1413721, 48024900, 1631432881,..., each of which is also a triangular number.
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