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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, the square of a number is that number multiplied by itself 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 √x. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, an n th root of a Number a is a number b such that bn = a. For example, the principal square root of 9 is 3, denoted √9 = 3, because 3² = 3 × 3 = 9. If otherwise unqualified, "the square root" of a number refers to the principal square root: the square root of 2 is approximately 1. 4142.

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 √x, which is positive, and the other −√x, which is negative. Together, these two roots are denoted ±√x. Square roots of negative numbers can be discussed within the framework of complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Square roots of objects other than numbers can also be defined.

Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a square number, sometimes also called a Perfect square, is an Integer that can be written as the square of some other In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction A ratio is an expression which compares quantities relative to each other For example, √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. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides This has been known since ancient times, with the discovery that √2 is irrational attributed to Hipparchus, a disciple of Pythagoras. Hipparchus ( Greek; ca 190 BC &ndash ca 120 BC was a Greek Astronomer, Geographer, and Mathematician of the Hellenistic "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. (See square root of 2 for proofs of the irrationality of this number. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2 )

1 Properties
2 Computation
3 Square roots of negative and complex numbers
4 Square roots of matrices and operators
5 Principal square roots of the first 20 positive integers
- 5.1 As decimal expansions
- 5.2 As periodic continued fractions
6 Geometric construction of the square root
7 History
8 Notes
9 References
10 See also
11 External links

Properties

The graph of the function f(x) = √x, made up of half a parabola with a vertical directrix. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

The principal square root function f(x) = √x (usually just referred to as the "square root function") is a function which maps the set of non-negative real numbers R⁺ ∪ {0} onto itself, and, like all functions, always returns a unique value. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The square root function also maps rational numbers into algebraic numbers (a superset of the rational numbers); √x is rational if and only if x is a rational number which can be represented as a ratio of two perfect squares. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or This article refers to the REM live recording For the mathematical term see Perfect square. In geometrical terms, the square root function maps the area of a square to its side length. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides

For all real numbers x,

$\sqrt{x^2} = \left|x\right| = \begin{cases} x, & \mbox{if }x \ge 0 \\ -x, & \mbox{if }x \le 0 \end{cases}$ (see absolute value)

For all non-negative real numbers x and y,

$\sqrt{xy} = \sqrt x \sqrt y$

and

$\sqrt x = x^{\frac{1}{2}}.$

The square root function is continuous for all non-negative x and differentiable for all positive x. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Its derivative is given by

$f'(x) = \frac{1}{2\sqrt x}.$

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

$\sqrt{1 + x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + \dots\!$

Computation

Main article: Methods of computing square roots

Many methods of calculating square roots exist today, some meant to be done by hand and some meant to be done by machine. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives This article presents and explains several methods which can be used to calculate Square roots Exponential identity Pocket calculators typically implement good

Many, but not all pocket calculators have a square root key. A calculator is device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive Computer spreadsheets and other software are also frequently used to calculate square roots. A spreadsheet is a Computer application that simulates a paper worksheet 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

$\sqrt{x} = e^{\frac{1}{2}\ln x}$ 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 exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base The slide rule, also known as a slipstick, is a mechanical Analog computer.

The most common method of square root calculation by hand is known as the "Babylonian method". This article presents and explains several methods which can be used to calculate Square roots Exponential identity Pocket calculators typically implement good 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:

Start with an arbitrary positive start value r (the closer to the square root of x, the better).
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. 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 )
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. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources

Square roots of negative and complex numbers

See also: Complex number

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

The square of any positive or negative number is positive, and the square of 0 is 0. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Therefore, no negative number can have a real square root. However, it is possible to work with a larger set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity) and called the imaginary unit, which is defined such that i² = −1. Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation Using this notation, we can think of i as the square root of −1, but notice that we also have (−i)² = i² = −1 and so −i is also a square root of −1. Similarly to the real numbers, we say the principal square root of −1 is i, or more generally, if x is any positive number, then the principal square root of −x is

$\sqrt{-x} = i \sqrt x$

because

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

By the argument given above, i can be neither positive nor negative. This creates a problem: for the complex number z, we cannot define √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 e^iφ is represented in polar coordinates with −π < φ ≤ π, then we set the principal value to

$\sqrt{z} = \sqrt{r} \, e^{i\phi \over 2}.$

Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous). In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In considering complex Multiple-valued functions in Complex analysis, the principal values of a function are the values along one chosen branch of that Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The above Taylor series for √1 + x remains valid for complex numbers x with | x | < 1.

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

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

where

$r = |x + iy| = \sqrt{x^2+ y^2}$

is the absolute value or modulus of the complex number, unless x = −r and y = 0. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. 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.

Note that because of the discontinuous nature of the square root function in the complex plane, the law √zw = √z√w is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis 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 √z√w at all, leading to the equality's failure. A similar problem appears with the complex logarithm and the relation log z + log w = log(zw). In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers ) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that −1 = 1:

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

The third equality cannot be justified (see invalid proof). In Mathematics, there are a variety of spurious proofs of obvious Contradictions Although the proofs are flawed the errors usually by design are comparatively subtle It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root but selects a branch for the square root that contains $\sqrt{-1} \cdot \sqrt{-1}$ . The left hand side becomes either

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

if the branch includes +i or

$\sqrt{-1} \cdot \sqrt{-1}=(-i) \cdot (-i)=-1$

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

$\sqrt{-1 \cdot -1}=\sqrt{1}=-1,$

where the last equality, $\sqrt{1}=-1$ , is a consequence of the choice of branch in the redefinition of √.

Square roots of matrices and operators

Main article: square root of a matrix

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. In Mathematics, the square root of a matrix extends the notion of Square root from numbers to matrices. 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 Mathematics, especially Functional analysis, a normal operator on a Hilbert space H (or more generally in a C* algebra) is 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.

Principal square roots of the first 20 positive integers

As decimal expansions

The square roots of the perfect squares (1, 4, 9, and 16) are integers. This article refers to the REM live recording For the mathematical term see Perfect square. In all other cases, the square roots are irrational numbers, and therefore their decimal representations are non-repeating decimals. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction This article gives a mathematical definition For a more accessible article see Decimal. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is

$\sqrt {1}$	$=\,$	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
$\sqrt {11}$	$\approx$	3. 3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609
$\sqrt {12}$	$\approx$	3. 4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818
$\sqrt {13}$	$\approx$	3. 6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293
$\sqrt {14}$	$\approx$	3. 7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307
$\sqrt {15}$	$\approx$	3. 8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937
$\sqrt {16}$	$=\,$	4
$\sqrt {17}$	$\approx$	4. 1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338
$\sqrt {18}$	$\approx$	4. 2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386
$\sqrt {19}$	$\approx$	4. 3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203
$\sqrt {20}$	$\approx$	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. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of digits.

$\sqrt {2}$	$=\,$	[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, . . . ]
$\sqrt {11}$	$=\,$	[3; 3, 6, 3, 6, . . . ]
$\sqrt {12}$	$=\,$	[3; 2, 6, 2, 6, . . . ]
$\sqrt {13}$	$=\,$	[3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, . . . ]
$\sqrt {14}$	$=\,$	[3; 1, 2, 1, 6, 1, 2, 1, 6, . . . ]
$\sqrt {15}$	$=\,$	[3; 1, 6, 1, 6, . . . ]
$\sqrt {16}$	$=\,$	[4]
$\sqrt {17}$	$=\,$	[4; 8, 8, . . . ]
$\sqrt {18}$	$=\,$	[4; 4, 8, 4, 8, . . . ]
$\sqrt {19}$	$=\,$	[4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, . . . ]
$\sqrt {20}$	$=\,$	[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, . In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} . . ] – looks like this:

$\sqrt{11} = 3 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{\ddots}}}}}\,$

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. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers Since the geometric mean of a and b is √ab, one can construct √a simply by taking b = 1.

The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. La Géométrie was published in 1637 as an appendix to Discours de la méthode ( Discourse on Method) written However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Another method of geometric construction uses right triangles and induction: √1 can, of course, be constructed, and once √x has been constructed, the right triangle with 1 and √x for its legs has a hypotenuse of √x + 1. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that A hypotenuse is the longest side of a Right triangle, the side opposite of the Right angle.

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. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish ^[1]

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. This article is about the history of South Asia prior to the Partition of British India in 1947 The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the 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. Baudhāyana, (fl ca 800 BCE was an Indian mathematician whowas most likely also a priest ^[2] Aryabhata in the Aryabhatiya (section 2. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Āryabhatīya, an astronomical treatise is the Magnum opus and only extant work of the 5th century Indian mathematician Aryabhata. 4), has given a method for finding the square root of numbers having many digits.

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". Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics ^[3]

Notes

^ Anglin, W. S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
^ Joseph, ch. 8.
^ 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. Penguin Books is a British Publisher founded in 1935 by Allen Lane. (2000). ISBN 0-691-00659-8.

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. The Penrose method is a method devised in 1946 by Professor Lionel Penrose, for allocating seats or votes in legislatures based on the Square root of the An Integer q is called a quadratic residue modulo n if it is congruent to a perfect square (mod n) i In Mathematics, an n th root of a Number a is a number b such that bn = a. In Mathematics, a quadratic irrational, also known as a quadratic irrationality or quadratic surd, is an Irrational number that is the solution In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 = x In Algebra, nested radicals are radical expressions that contain another radical expression In Number theory, the integer square root (isqrt of a Positive integer n is the positive integer m which is the greatest integer less than or In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power This article presents and explains several methods which can be used to calculate Square roots Exponential identity Pocket calculators typically implement good In Mathematics, the square root of a matrix extends the notion of Square root from numbers to matrices. In Algebra, the square of a number is that number multiplied by itself The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2 Geometry If an equilateral triangle ( Equilateral polygon with three sides with sides of length 1 is cut into two equal halves by bisecting an internal angle across Continued fraction It can be expressed as the Continued fraction 4 4 4 4 4
MATHPATH: Square roots via Mediants
Rational Mean: High.Order square-root methods
How to manually find a square root

Dictionary

-noun

(mathematics) Of a number, another number which, when squared, yields the original number; sometimes constrained to be the positive number where two solutions exist.

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