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Topic: Integer square root

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	Square root Summary
		Per the fundamental theorem of algebra, there are two solutions to the square root of any number (although these roots may not be distinct, as in the square root of zero).
		Square roots of positive integers are often irrational numbers, i.e., numbers not expressible as a ratio of two integers.
		In geometrical terms, the square root function maps the area of a square to its side length.
	www.bookrags.com /Square_root (1805 words)

	Square root - Wikipedia, the free encyclopedia
		Square roots of positive integers are often irrational numbers, i.e., numbers not expressible as a ratio of two integers.
		In geometrical terms, the square root function maps the area of a square to its side length.
		Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous).
	en.wikipedia.org /wiki/Square_root (1274 words)

	Integer square root - TheBestLinks.com - Integers, Mathematics, Number theory, Newton's method, ...
		In mathematics and Number theory, the integer square root (isqrt) of a number n in the reals which is non-negative and non-zero is the number m which is the greatest integer less than or equal to the square root of n.
		The first and most obvious, but least satisfactory method of calculating the integer square root of a number n is to simply take its square root, and round down.
		The beauty of Newton's Iteration for finding the integer square root of a number n is that it can use solely integers, which is essential for accurate Number theory.
	www.thebestlinks.com /Integer_square_root.html (358 words)

	PlanetMath: square root
		The square root operation is left distributive over multiplication and division, but not over addition or subtraction.
		It is possible to consider square roots in rings other than the integers or the rationals.
		This is version 20 of square root, born on 2001-11-10, modified 2005-08-24.
	planetmath.org /encyclopedia/SquareRoot.html (697 words)

	Encyclopedia :: Square root (Site not responding. Last check: 2007-10-20)
		For a positive real number, the two square roots are the principle square root and the negative square root.
		Square roots of positive integers are often irrational numbers, i.e., numbers not expressible as a quotient of two integers.
		If there is a chance of ambiguity, prefer constructions like a square root or a complex square root to indicate the strict definition, or the positive square root or similar to indicate the looser sense.
	www.hallencyclopedia.com /Square_root (1804 words)

	Integer square root: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-20)
		Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers....
		An integer is often one of the primitive datatypes in computer languages....
		In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b,...
	www.absoluteastronomy.com /encyclopedia/i/in/integer_square_root.htm (476 words)

	Ancient Square Roots
		The basic "ladder rule" for generating a sequence of integers to yield the square root of a number N is the recurrence formula s[j] = 2k s[j-1] + (N-k^2) s[j-2] where k is the largest integer such that k^2 is less than N. The ratio s[j+1]/s[j] approaches sqrt(A) + N as j goes to infinity.
		For example, with j=2 (for square roots) we have y^2 - 2k y - (M - k^2) = 0 The corresponding recurrence is s[n] = 2k s[n-1] + (M-k^2) s[n-2] which is the same as the "ladder arithmetic" square root method given earlier.
		By simply shifting this circle of roots in the positive real direction we can make the magnitude of the positive real root greater than the magnitude of any of the other roots (because the radius of the original circle of roots is smaller than the new positive real root).
	www.mathpages.com /home/kmath190.htm (1412 words)

	The Grinnell Scheme Web: Integer square roots (Site not responding. Last check: 2007-10-20)
		You could do that, but when the square root is irrational the computer generally represents it internally using an approximation to the true value.
		At all times, the recursion keeps track of two numbers, one of which (the ``lower bound'') is known to be less than or equal to the correct integer square root, while the other (the ``upper bound'') is known to be strictly greater than the integer square root.
		An integer square root can't be less than 0, and the integer square root of n is always less than n + 1.
	www.math.grin.edu /~stone/scheme-web/integer-square-root.html (372 words)

	Math Forum: Ask Dr. Math FAQ: Integers, Rational Numbers, Irrational Numbers
		The square root of 2 is an irrational number because it can't be written as a ratio of two integers.
		Other irrational numbers include the square root of 3, the square root of 5, pi, e, and the golden ratio.
		The square root of 2 is another irrational number that cannot be written as a fraction.
	www.mathforum.org /dr.math/faq/faq.integers.html (736 words)

	Square Root Table - Your ultimate square root table resource. (Site not responding. Last check: 2007-10-20)
		Square Root Calculator (Allows you to find the value of a number squared or that number's square root...
		The name comes from the fact that it is the square root of the mean of the squares of the values.
		...2". The length of the leg is the square root of (A squared + B squared).
	tables.finance-articles.com /index.php?k=square-root-table (518 words)

	Integer Square Roots - GameDev.Net Discussion Forums
		I would say, go and do your normal binary square root funtion, but go and rightshift the output number (because shifting the input number isn't too good for this), to go and get the 16.16 notation.
		For example, SSE has a 1 cycle reciprocal square root that is accurate to 12 bits.
		This could probably be used as a good guess for the square root of a number.
	www.gamedev.net /community/forums/topic.asp?topic_id=297059 (3605 words)

	Square Root Algorithm Derivation (Site not responding. Last check: 2007-10-20)
		Find the greatest square less than or equal to that group of digits and its square root will be your first approximation of the entire square root.
		Step 1: The square root of a number between 1 and 100 is a number between 1 and 10.
		Furthermore, the square root of a number between 100 and 10000 is a number between 10 and 100.
	jwilson.coe.uga.edu /EMT668/EMAT6680.F99/Challen/squareroot/sqrt.html (690 words)

	The square root of 2 (Site not responding. Last check: 2007-10-20)
		The length of the diagonal is the square root of 2.
		In the case of the square root of 2 there is no repeating pattern.
		If it is not a perfect square, like 2, 3 or 12 then its square root does not have a finite or repeating decimal expansion.
	mathcentral.uregina.ca /qq/database/QQ.09.01/roger1.html (217 words)

	Square root algorithms
		It is known that the square root of an integer that is itself not the square of another integer will be irrational.
		For instance, 5 is not the square of an integer.
		Finding the square root by a procedure similar to long division was popular prior to the advent of electronic calculators.
	www.mathpath.org /Algor/squareroot/algor.square.root.htm (1017 words)

	CISC 280 Lab Module B (Site not responding. Last check: 2007-10-20)
		Most basically, the number n = kk is a "square"* because n dots could be arranged in a k by k square.
		We won't explore triangular numbers here, but rather will consider numbers which can be written as a sum of squares (you can think of stacking the squares of dots like a child playing with blocks of various sizes).
		Some famous open questions in number theory concern how each positive integer may be written as a sum of squares.
	www.eecis.udel.edu /~saunders/courses/280/01s/lab/labC.html (394 words)

	Paul Hsieh's Square Root page
		Any composite positive integer has a positive divisor that is greater than 1 and less or equal to its square root.
		The square root of a positive irrational is always irrational and the square root of a positive algebraic (which includes rationals) is always algebraic.
		Square roots continued to be important to Pythagoras, of course, as a method for calculating right triangle edge lengths from Pythagoras' theorem.
	www.azillionmonkeys.com /qed/sqroot.html (4748 words)

	[No title]
		Square roots of positive integers are often irrational numbers, i.e., numbers not expressible as a quotient of two integers.
		The discovery that \sqrt 2 is irrational is attributed to Hippasus, a disciple of Pythagoras.
		Many of the methods used for finding the square root of a number requires an initial seed value which is close to the actual value of the square root.
	www.hermes-press.com /square_root.htm (3720 words)

	integer - Perl pragma to use integer arithmetic instead of floating point (Site not responding. Last check: 2007-10-20)
		In addition, the range of operands and results is restricted to that of familiar two's complement integers, i.e., -(2**31)..
		is also not affected, so that 2 .5 is always the square root** of 2.
		Internally, native integer arithmetic (as provided by your C compiler) is used.
	theory.uwinnipeg.ca /CPAN/perl/lib/integer.html (442 words)

	Decomposing a positive integer into a sum of four squares
		The simplest solution one can think of works by subtracting a trial square from the number to be decomposed and then trying to decompose the resulting number into a sum of three squares.
		Note that one could also try to subtract the largest possible square from n that makes sure that n is not of the forbidden in order to have a smaller number to be represented as a sum of three squares.
		Indeed, numerical experiments suggest that 9634 is the largest integer which cannot be represented as a square plus a prime congruent 1 mod 4 or as a square plus twice a prime congruent 1 mod 4.
	www.schorn.ch /howto.html (1122 words)

	GNU Emacs Calc 2.02 Manual - Integer Truncation
		There are four commands for truncating a real number to an integer, differing mainly in their treatment of negative numbers.
		All of these commands have the property that if the argument is an integer, the result is the same integer.
		(integer square root) commands, which are analogous to
	theory.uwinnipeg.ca /localfiles/infofiles/calc/calc_195.html (422 words)

	Non integer square roots and pi = irrational?
		The OP's question seems to be, are all irrationals comparable, so that irrational pi being taken as a unit, irrational square root 5 comes out as, I don't know what he wants, maybe a linear combination with integer coefficients.
		A non-integer square root is a number that is not an integer that when multiplied by itself equals an integer.
		If you were to draw a triangle like the one I said above (legs with a length of 2 and hypotenuse with a length of the square root of 8) on a sheet of graph paper, you have just drawn a line with an irrational length.
	www.physicsforums.com /showthread.php?threadid=104832 (2676 words)

	Paul Hsieh's Square Root page (Site not responding. Last check: 2007-10-20)
		Integer squareroot approximation which executes in 16-27 cycles through effective bitsearch and 256 byte LUT table.
		Unfortunately, at about the time this was discovered, it was also discovered that the Athlon's hardware square root is very impressive (it beats it by almost 2x.) Nevertheless for other platforms this does appears to be the algorithm of choice.
		This leads to (a,b) <= ((a+yb)/2,b(2-ab)), where a is supposed to converge to the square* root and b to the reciprocal of the square root.
	www.jbenjamin.org /docs/asm-sqroot.html (3140 words)

	[No title]
		Square root is about as complex, and takes about as much time, as a divide.
		A square root is like a division in which the quotient and divisor are equal.
		However, I knew that square root could be done as quickly as division at least as far back as high school, before I had ever heard of Dijkstra (or seen a computer for that matter).
	hyperarchive.lcs.mit.edu /HyperArchive/Archive/_Periodical/csmp/csmp-digest-v3-036.txt (15962 words)

	Integer square root Summary
		Roundoff error is a form of noise that pervades all calculations performed on computers (other than those dealing strictly with integers).
		Roundoff error occurs because computers can only use finite strings of digits to represent any given number, while many numbers are often not representable by finite strings.
		In number theory (a branch of mathematics), the integer square root (isqrt) of a positive integer n is the positive integer m which is the greatest integer less than or equal to the square root of n,
	www.bookrags.com /Integer_square_root (876 words)

	Module Integer
		The value of isqrt n is the floor of the square root of n.
		the floor of the square root of n.
		Since integers have unlimited precision, only the 32 least significant bits are computed.
	www.informatik.uni-kiel.de /~pakcs/lib/CDOC/Integer.html (397 words)

	integer - perldoc.perl.org
		In addition, the range of operands and results is restricted to that of familiar two's complement integers, i.e., -(2**31)..
		is also not affected, so that 2 .5 is always the square root** of 2.
		Internally, native integer arithmetic (as provided by your C compiler) is used.
	perldoc.perl.org /integer.html (402 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		function [ q, r ] = i_sqrt (n) %% I_SQRT finds the integer square root of N by solving N = Q2 + R. % Discussion: % % The integer square root of N is an integer Q such that % Q2 <= N but N < (Q+1)**2.
		% % Parameters: % % Input, integer N, the number whose integer square root is desired.
		% % Output, integer Q, R, the integer square root, and positive remainder, % of N. n_abs = abs (n); q = n_abs; if (0 < n_abs) while (floor (n_abs / q) < q) q = floor ((q + floor (n_abs / q)) / 2); end end r = n_abs - q * q;
	www.csit.fsu.edu /~burkardt/m_src/subset/i_sqrt.m (214 words)

	Irrational number - All About All (Site not responding. Last check: 2007-10-20)
		Legendre (1794) completed Lambert's proof, and showed that π is not the square root of a rational number.
		One proof of the irrationality of the square root of 2 is the following reductio ad absurdum.
		However, this metric space is homeomorphic to the complete metric space of all sequences of positive integers; the homeomorphism is given by the infinite continued fraction expansion.
	www.answers-zone.com /article/Irrational_number (1852 words)

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