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Topic: Square root of two

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	Square root of 2 - Wikipedia, the free encyclopedia
		The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1.
		Geometrically, it is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem.
		The square root of two is the aspect ratio of paper sizes under ISO 216.
	en.wikipedia.org /wiki/Square_root_of_two (1116 words)

	A geometric view of the square root algorithm (Site not responding. Last check: 2007-10-31)
		There are two facts concerning the square of an integer that are useful in the inverse process of finding the square root.
		Place 3 in the units position of the square root calculation and subtract the sum of the areas of regions 5 and 6, which is 2 x 250 x 3 = 1 500.
		Thus the square root of 64 009 is 253.
	mathcentral.uregina.ca /RR/database/RR.09.95/grzesina1.html (744 words)

	square root of 2
		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.
	www.daviddarling.info /encyclopedia/S/square_root_of_2.html (334 words)

	nature's word \| musings on sacred geometry
		The square root of three is generated by this three-dimensional object in much the same way that the square root of two is generated by the two-dimensional square.
		In the instance of the square root of two, the line in question is generated by connecting opposite corners, and so it is with the square root of three.
		The fundamental difference between the square roots of two and three is that with the example of the square root of two we were making a relation on a two-dimensional plane - a flat, pre-manifestation level, with the term "manifestation" here referring to actual creation in the three-dimensional physical world.
	www.unitone.org /naturesword/sacred_geometry/square_three/in_geometry (2223 words)

	anda pengunjung ke
		Square root of a number is a second number whose product with itself gives the original number.
		Another convenient way of finding the square root of a number is to use a table of square roots, a table of squares, or a table of logarithms (see LOGARITHMS).
		If the square root of a number that is not in the 1 to 100 range must be found, first divide or multiply the number by 100 to bring it within this range.
	www.geocities.com /siswaonline/squareroot.html (597 words)

	The Square Root of Two Is Not Rational (Site not responding. Last check: 2007-10-31)
		The Square Root Of Two Is Not The Ratio Of Two Integers
		That is, if x is a ratio of two integers, then there are some integers a and b such that b is not 0, x=a/b, and the greatest common divisor of a and b is 1 (1 and -1 both divide a and b, but no other integers do that).
		Suppose that x is the square root of 2 and that x is also the ratio of two integers.
	www.math.hawaii.edu /~ramsey/Root2.html (357 words)

	Proportions: Golden Section or Golden Mean, Modulor, Square Root of Two, Theorie and Construction (Site not responding. Last check: 2007-10-31)
		The simplicity of the derivation (square root of 2 is the diagonal through a square of side length 1) is paralleled by the ease of maintaining the proportion through division or multiplication of the proportioned rectangles.
		The sum of two rectangles of proportion (Square root of 2) : 1 long side by long side is (Square root of 2) : 2.
		Divided by the square root of two we arrive at 1 : (Square root of 2), the same ratio as the two rectangles that were added together, only with a change of orientation.
	photoinf.com /Golden_Mean/Volker_Muller/Proportions_Golden_Section_or_Golden_Mean,_Modulor,_Square_Root_of_Two,_Theorie_and_Construction.htm (1047 words)

	Perfect square - Wikipedia, the free encyclopedia
		For examples, for 4624, the leftmost pair is 46, which is not a perfect square; but the largest perfect square less than 46 is 36 which has a square root of 6.
		Thus, we know the last digit of the square root of 4624 is either 2 or 8; for 729 it is either 3 or 7; for 1600 it is 0.
		The square root of 4624 is either 62 or 68; the square root of 729 is either 23 or 27; the square root of 1600 is 40.
	en.wikipedia.org /wiki/Perfect_square (786 words)

	nature's word \| musings on sacred geometry
		This is a highly effective symbol which conveys not only the circle of unity divided into two equal yet opposing parts (fl and white), but also shows that there can never be a state wherein on force totally dominates the other (note the fl and white dots floating within their opposing color).
		That the two Taoist and Mayan symbols represent the same juxtaposition of dualistic forces is made obvious by the similarity of their designs.
		When we superimpose a matrix based on the square root of two upon these marks (the marks consist of the darker lines in the images), it becomes obvious that these symbols were much more than asthetic designs.
	www.unitone.org /naturesword/sacred_geometry/square_two/in_culture (2252 words)

	Math Lair - Square Root of 2 (Site not responding. Last check: 2007-10-31)
		The square root of 2 to 100 decimals is: 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727.
		The square root of 2 is the solution to the equation x² = 2, which results when using the Pythagorean Theorem to find the length of the hypotenuse of a right triangle where both of the legs are of length 1.
		Therefore, the square root of 2 is irrational.
	www.stormloader.com /ajy/root2.html (377 words)

	Search Results for root*
		Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction.
		It is also wrong to deny the existence of approximations to irrational square roots, to assume a geometrical basis of the quadratic equations or to deny the existence of texts of this type in the Hellenistic period.
		In the case of a non-square number, subtract the nearest square number, divide the remainder by twice this nearest square; half the square of this is divided by the sum of the approximate root and the fraction.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=root*&CONTEXT=1 (13205 words)

	Math Forum - Geometry Problem of the Week (Site not responding. Last check: 2007-10-31)
		The chord's length is 6 root 2, as is the hypotenuse.
		This states that the sum of the squares of the lengths of the legs of the triangle is equal to the square of the hypotenuse, or c^2=a^2+b^2.
		The square root of 2 is approximately 1.414213562.
	mathforum.org /geopow/fullsolutions/19971031.fullsolution.html (13026 words)

	Question Corner -- A Geometric Proof That The Square Root Of Two Is Irrational
		Therefore, if the square root of 2 were rational, you would be able to write it in the form a/b where a and b are not both even.
		If the square root of two were a rational number, you could write it as a fraction a/b in lowest terms, where a and b were integers, not both even.
		Therefore, because it is impossible to find two integers a and b with the property that a and b are not both even and a/b = sqrt(2), the square root of two cannot be a rational number.
	www.math.toronto.edu /mathnet/plain/questionCorner/rootoftwo.html (827 words)

	Square root of 2 is irrational
		Their squares are still mutually prime for they are built from the same factors.
		The latter proof makes it entirely obvious that unless a square root of an integer is an integer itself, it is bound to be irrational.
		2 is the length of the diagonal of the unit square.
	www.cut-the-knot.org /proofs/sq_root.shtml (1143 words)

	The Square Root of Ten 3.16227766 (Site not responding. Last check: 2007-10-31)
		In relation, then, to all of the computations shown for the square root of two, we must remember that the square root of five could be employed, thereby obtaining the same results simply by changing the sign of the computation from division to multiplication or vice versa.
		The square roots and their reciprocals of certain number would appear to be related directly to the measurements and their fractal expression of specific ancient reckoning counts.
		A study of the square root of ten and its reciprocal would easily teach a student such a possibility in reckoning numbers.
	www.earthmatrix.com /giza/square_roots.html (1545 words)

	[No title]
		By inspection, the "root mean squared" value is the square root of 1/2 (approximately 0.707).
		Finally, the "root mean squared" is the square root of this final result.
		Inexpensive multimeters measure the average positive voltage of a waveform and scale this value using the square root of two to produce a display value.
	www.ee.unb.ca /tervo/ee2791/vrms.htm (519 words)

	square root records
		Square Root will be there, handing out free copies of the Catalog compilation.
		The Catalog compilation, featuring two tracks from each of the seven Square Root artists, is available now directly from us or from various independent record stores throughout Michigan.
		AFNP have been a part of each of the three Square Root releases: two of their songs are showcased on the Catalog compilation, they provided two remixes for the Chauceworth Aif's Impress/Depress EP, and they performed on numerous pieces on the Post Tapes improv album.
	www.squarerootmusic.com (2248 words)

	the square root of two
		Now, the square root of 2 multiplied by the square root of 2 is the same as saying p/q x p/q.
		And the square root of 2 times the square root of 2 is 2—simply because a square root of any one number, n, times the square root of that same number is just n.
		The square root of 3 times the square root of 3 is 3, the square root of 4 times the square root of 4 is 4, and so on.
	www.cgshock.com /forums/showthread.php?t=5038 (962 words)

	Harmony and Proportion: The Square Root of 2
		Because the irrational relationship between the square and its diagonal seems to give a glimpse of a kind of order which is not resultant from (or dependent on) manifest number, but appears to indicate a "meta-physical" cause, it has in the past been held by architects and artists to have special significance.
		Its appearance in the underlying geometry of the Squared Circle echoes or symbolises the "dialogue" between the Heavens and the Earth, the Infinite and the Finite.
		The relationship between the side and diagonal of a square is not of this same order, but nevertheless fulfills the same function as a symbol (analogous to the squared circle) of the harmonious relationship between Form and Essence, Finite and Infinite.
	aboutscotland.co.uk /harmony/prop7.html (431 words)

	Proportions: Golden Section or Golden Mean, Modulor, Square Root of Two, Theorie and Construction (Site not responding. Last check: 2007-10-31)
		The Golden Section or Golden Mean is derived with simple geometric constructions, its ratio expressed in numbers is, however, irrational ((Square root of five) - 1) : 2 =.618034 : 1 (= 1 : 1.618034).
		That creates a certain tension between golden section and square, which may contribute to the interest that this proportioning scheme can maintain (see Corbu's Modulor), especially when compared to schemes that use the square as only proportioning scheme (see O.M.U.).
		The prevalent paper formats in Germany are defined by the DIN 476 (DIN is the German Insitute for Standardization, comparable to ANSI in the U.S.A, ISO internationally, etc.).
	home.att.net /~vmueller/prop/theo.html (1076 words)

	The Grinnell Scheme Web: Integer square roots
		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)

	Reductio ad absurdum
		A proof of the irrationality of the square root of two.
		But the square of an even number is even, [footnote 1] and the square of an odd number is odd.
		Therefore, the square root of two cannot be rational; it is irrational, which cannot be expressed as the ratio of two integers, no matter how big those integers are.
	www.alcyone.com /max/writing/essays/reductio-ad-absurdum.html (429 words)

	square root
		Calculation of a square root by hand is a little like long-hand division.
		That is, we have improved our knowledge of the square root by one digit, B. We take whatever remains, scale again by 100, by bringing down two more digits, and repeat the process.
		Note: In computers and hand-held calculators, square root, sine, cosine, and other transcendental functions are calculated with sophisticated functions based on the Newton-Raphson method, sometimes called Newton's method.
	www.nist.gov /dads/HTML/squareRoot.html (575 words)

	The square root of two is never supposed to end (Site not responding. Last check: 2007-10-31)
		if you have a square with sides of length one then the diagonal of the square is the square root of 2.
		The correct statement is that if you represent the square root of two in decimal form then the non-zero digits after the decimal never end.
		In the case of the square root of two there is the additional fact that the decimal part has no repeating pattern as it does for 1/3.
	mathcentral.uregina.ca /QQ/database/QQ.09.98/wonderingstudent1.html (377 words)

	Square root of 2 is irrational
		No two integers are equidistant from the square root of 2.
		No two rational numbers are equidistant from the square root of 2.
		Second, if a real number is equidistant from two distinct rational points, the number is necessarily the average of the two points, and is, therefore, rational.
	www.cut-the-knot.org /do_you_know/diffDist.shtml (772 words)

	Dissertations, Essays on Essay on the square root of 2
		Dissertations, Essays on Essay on the square root of 2
		and#8730;2 Although it wasn't Pythagoras himself who discovered the square root of two and the changes it caused to Ancient Greek mathematics as well as the future of mathematics, his follower did and because of this he is mainly accredited.
		The discovery of the square root of two was a frightening thing to the Pythagoreans however we can now thank them for our wonderful math classes today.
	www.essayboom.com /essay/Essay_on_the_square_root_of_2-169390.html (187 words)

	MID-ATLANTIC GEOMANCY: Sacred Geometry 4 (Site not responding. Last check: 2007-10-31)
		The diagonal of a square whose side is 1 is √2, or 1.4142135624∞.
		So while the square represents the physical, it has within it, a connection to the spiritual.
		The square was found was in the Holy of Holies (the back room) of Solomon's Temple (G,H,F,E).
	www.geomancy.org /sacred_geometry/sacgeo-4.html (194 words)

	Square Root of Two (Site not responding. Last check: 2007-10-31)
		A variant of this piece was inadvertently made when I realized that instead of threading the first "1" after the decimal I had instead threaded "0".
		After contemplating the result, I decided that that piece would be called "The Square Root of Two Minus 0.01".
		Current status: 2 copies of "The Square Root of Two Minus 0.01" remain.
	home.comcast.net /~rexcreations/square_root_of_two.htm (100 words)

	Key Curriculum Press - The Square Root of Two: A Mathematical Dialog and Excursion
		The square root of 2 is a very important number, even if it's often overshadowed by more famous numbers such as pi.
		Through an imaginary dialog between a teacher and a student, readers see why the square root of 2 is so important, and how, in puzzling out its special qualities, mathematicians came to their insights about the nature of irrational numbers.
		The Square Root of Two: A Mathematical Dialog and Excursion is available from our Partner in Publishing or Amazon.com.
	www.keypress.com /x17517.xml (108 words)

	Irrationality of Sqrt(2) (PRIME)
		Therefore the first equation is false, and we have our result: the assumption that the square root of two is a rational number must be false, so the contrary statement is true – the square root of two is an irrational number.
		A slightly more general argument along the same lines shows that all square roots of integers are irrational, except when the integers are perfect squares (1, 4, 9, 16, 25, etc.).
		For example, since the square root of 5 is approximately 2.236, the integer part is 2.
	www.mathacademy.com /pr/prime/articles/irr2 (721 words)

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