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Topic: Constructible number

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	What's Special About This Number?
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.stetson.edu /~efriedma/numbers.html (7257 words)

	Constructible number - Wikipedia, the free encyclopedia
		A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual x- and y-coordinate axes.
		Note that this is quite a distinct notion from Gödel's constructible universe, L; though every number that is constructible in the sense of this article is in L, the converse fails badly.
		The algebraic characterization of constructible numbers provides an important necessary condition for constructibility: if z is constructible, then it is algebraic, and its minimal irreducible polynomial has degree a power of 2, or equivalently, the field extension Q(z)/Q has dimension a power of 2.
	en.wikipedia.org /wiki/Constructible_number (892 words)

	Rational number
		In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero.
		The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part).
		The rational numbers are a (dense) subset of the real numbers, and as such they also carry a subspace topology.
	www.askfactmaster.com /Rational_number (712 words)

	[No title]
		Sometimes the negatives of constructible numbers are also called constructible.
		All rational numbers are constructible, and all constructible numbers are algebraic numbers.
		The cube root of a general constructible number
	en-cyclopedia.com /wiki/Constructible_number (72 words)

	Constructions in Advanced Euclidean Geometry (Site not responding. Last check: 2007-09-07)
		Construct a triangle, given the length of one side and lengths of the altitude and median to that side.
		Construct a triangle, given one angle, the length of the opposite side, and the length of the altitude to that side.
		Construct a triangle, given the measure of two angles and the length of the bisector of the third angle.
	pegasus.cc.ucf.edu /~xli/construction2.htm (158 words)

	Constructible Numbers
		If we call numbers that express lengths of segments constructed with a straightedge and a compass constructible, then we find that all (positive) integers and rational numbers are constructible as are the square roots of such numbers.
		This simply means that the number was determined through a sequence of geometric constructions with a straightedge and a compass.
		Since the set of all algebraic numbers is countable, we get, as an immediate consequence of this result, that the set of all constructible numbers is also countable.
	www.cut-the-knot.org /arithmetic/rational.shtml (1273 words)

	Fermat number - Wikipedia, the free encyclopedia
		Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime.
		Because of the size of Fermat numbers, it is difficult to factorize or to prove primality of those.
		Elliptic curve method is a fast method for finding small prime divisors of numbers, and at least GIMPS is trying to find prime divisors of Fermat numbers by elliptic curve method.
	en.wikipedia.org /wiki/Fermat_prime (1063 words)

	Ruler-and-compass construction - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-07)
		A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass.
		This shows that the constructible points form a field, which one treats as a subfield of the complex numbers.
		Therefore the degree for the minimal polynomial for cos 20° is of degree three, so cos 20° is not constructible and 60° cannot be trisected.
	xahlee.org /_p/wiki/Trisecting_the_angle.html (1442 words)

	Squaring the circle - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-07)
		The problem is to construct, using only ruler-and-compass constructions, a square with the same area as a given circle.
		it is non-algebraic, and therefore a non-constructible number.
		This does not imply that it is impossible to construct a square with an area very close to that of a given circle.
	xahlee.org /_p/wiki/Squaring_the_circle.html (308 words)

	text
		But this is not all constructible numbers allow us to do -- we can also take the square root of a constructible number and the result will, again, be a constructible number.
		Also, the construction will apply, as in Lemma II, to drop a perpendicular on a given line, where the perpendicular is an integral multiple of sqrt(3)/2 --- that is, this solves the only problem we had in Lemma II in constructing a perpendicular from a given point to a line.
		In the match-stick model, we can construct the points that correspond to the intersections of a line, defined by two points, and a circle, defined by a point C (centre) and a point D on the circumference (CD is the radius).
	www.cim.mcgill.ca /~pdimit/cs507/webpage/text.html (2724 words)

	Construction Workers (Site not responding. Last check: 2007-09-07)
		Although this may be thought of as a single activity, in fact construction is a feat of multitasking.
		The vast majority of building construction projects are small renovations, such as addition of a room, or renovation of a bathroom.
		Constructiveness is the seat of inititative, creativity and originality.
	www.wwwtln.com /finance/52/construction-workers.html (552 words)

	Math Forum: Ask Dr. Math FAQ: Regular Polygon Formulas
		It's a VERY famous theorem of Gauss that the only regular polygons with a prime number of sides that can be constructed with straightedge and compass are those for which the prime is one of the Fermat primes 3, 5, 17, 257, 65537,...
		The book also gives similar constructions for the regular polygons with 13 and 17 sides (for the regular 11-gon there's a construction using an angle-quinquesector, but it was too complicated for us to put into the book).
		Construction of a regular pentagon: Let N,S,E,W be the points of a circle C with center O in the four compass directions, and let M be the midpoint of ON.
	mathforum.org /dr.math/faq/formulas/faq.regpoly.html (1064 words)

	geometric problems of antiquity on Encyclopedia.com
		The cube root of 2 is not constructible, since it involves a cube root.
		Finally, the solution of problem (3′) did not come until 1882, when the German Ferdinand Lindemann showed that π is a transcendental number and thus cannot be expressed in terms of any roots of any rational numbers (see number).
		Constructing prehistory: lithic analysis in the Levantine Epipalaeolithic.
	www.encyclopedia.com /html/g1/geom-pr.asp (548 words)

	Talk: Pi - Open Encyclopedia (Site not responding. Last check: 2007-09-07)
		Numbers with a non-recurring decimal expansion are irrational.
		Furthermore, a constant IS a number, a constant does not "have a value which is a number".
		The definition of a constructible real number is a number which lies in a field gotten by taken a finite sequence of quadratic extensions of the rationals, i.e.
	talk.open-encyclopedia.com /Pi (5152 words)

	M3210 Sample Exam II (Site not responding. Last check: 2007-09-07)
		To construct an altitude, we drop a perpendicular from a vertex to the opposite side.
		A particular instance of this problem would be to construct a cube whose volume is twice that of the unit cube.
		This entails constructing a side of the larger cube, and in this case that means constructing a length equal to the cube root of 2.
	www-math.cudenver.edu /~wcherowi/courses/m3210/hgex2sam.html (623 words)

	Archimedes Plutonium (Site not responding. Last check: 2007-09-07)
		Define density of identical circles as to the number of points of tangency (sci.math in 1993 informed me that this is called "kissing points" and they remarked that in higher dimensions the kissing points diverges from density--- and I retort that no dimension higher than 3rd exists because physics proves the case).
		And pi is an evenly divisible irrational number by 2,3,4,5,6,8,10,12,15,16,17,20,.
		For when any number x is evenly divisible in base 10 by an integer y in base 10, that same number x in base z is also evenly divisible by the same integer y after y is converted to base z.
	www.iw.net /~a_plutonium/File112.html (948 words)

	Edge: Stuart Kauffman & Lee Smolin Paper [page 5]
		One model for how to do physics in the absence of a constructible Hilbert space is seen in a recent formulation of the path integral for quantum gravity in terms of spin networks by Markopoulou and Smolin[14] (This followed the development of a Euclidean path integral by Reisenberger[20] and by Reisenberger and Rovelli[21].
		The way in which the amplitudes are constructed in the absence of a specifiable basis or Hilbert structure requires a notion of successor states.
		In this case it is the evolution itself that constructs the subspace of the space of states that is needed to describe the possible futures of any given state.
	www.edge.org /3rd_culture/smolin/smolin_p5.html (1659 words)

	[No title] (Site not responding. Last check: 2007-09-07)
		One last concept, but probably the most important, is that of constructibility: a number is constructible iff it belongs to the smallest class containing the ordinals and closed under the Godel operations, the class L (the constructible universe).
		Now all constructible real numbers are ordinal definable, and the converse is consistent (but it is also consistent that there exists a real number that is ordinal definable but not constructible).
		Finally, it is consistent that all real numbers are constructible, and also it is consistent that some real number is not constructible.
	www.math.niu.edu /~rusin/known-math/98/definable (316 words)

	The construction of arctan(1/2)/Pi
		Note : the construction is done on a plain white paper and could be done on the sand in fact with small precision.
		This includes numbers of the form arctan(A)/Pi where A is algebraic and constructible with a ruler and compass.
		In this context it means that we can't construct an arc length of 1 radian with the ruler and compass.
	www.cs.uwaterloo.ca /journals/JIS/compass.html (1055 words)

	There are trisectable angles that are not constructible
		As usual, a number is constructible if it is constructible with a compass and a straightedge.
		That the angle is not constructible has been shown in the last century by Gauss and Wantzel.
		As a corollary, the sum of a constructible but non-trisectable angle and a trisectable but non-constructible angle is neither constructible, nor trisectable.
	www.cut-the-knot.org /do_you_know/trisect.shtml (317 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-09-07)
		A "perfect" number is equal to the sum of its factors excluding itself; 6 = 1+2+3.
		In fact if the numbers got real large, say if they were about a hundred digits long, Ron Rivest (at MIT) conjectured it would take 4 BILLION years to factor the numbers into primes, while finding the gcd this way would only take a couple of months (or less).
		An example of this is Euler's proof that any divisor of a number of the form 2^2^n - 1 is of the form k*2^n - 1.
	mathforum.org /library/drmath/view/51447.html (789 words)

	Three Problems Of Antiquity
		The number of prime factors on the left side of the latter equation is divisible by 3.
		The number of prime factors on the right side of the equation, when divided by three, leaves a remainder of 1.
		Another important observation is that the problem of constructing an acute angle is equivalent to that of constructing a right triangle with a given angle.
	www.cut-the-knot.org /arithmetic/cubic.shtml (769 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 6.3 (Site not responding. Last check: 2007-09-07)
		The real number a is said to be a constructible number if it is possible to construct a line segment of length a
		If u is a constructible real number, then u is algebraic over Q, and the degree of its minimal polynomial over Q is a power of 2.
		It is impossible to find a general construction for trisecting an angle, duplicating a cube, or squaring a circle.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/63.html (246 words)

	Term Tests 3 Sample Questions (Site not responding. Last check: 2007-09-07)
		Prove (you can quote without proof theorems we've proven in class) that the cube root of a natural number is not constructible unless it is an integer.
		Say that the complex number at a +bi is constructible if the point (a,b) is constructible (equivalently, if a and b are both constructible real numbers).
		Say that the complex number is algebraic if it is the root of a polynomial with integer coefficients.
	www.math.toronto.edu /jkorman/Math246Y/test3.htm (660 words)

	MAT 246Y1 EXAM REVIEW SHEET (Site not responding. Last check: 2007-09-07)
		Is the number of natural numbers m {1,…., m – 1} that are relatively prime to m.
		Definition: A real number is algebraic if there exists a polynomial with integer coefficients that has it as a root (not counting 0 polynomial).
		Theorem: S is infinite iff there is a proper subset of S that has the same cardinality of S. The number that can be marked on the line starting from 0 and 1 being given and using straight edge and compass are the Constructible numbers.
	www.math.toronto.edu /rosent/246/review.htm (1243 words)

	The book of Threes - Mathematics (Site not responding. Last check: 2007-09-07)
		The discriminant in the quadratic equation has three possibilities: The discriminant is the part inside the radical and shows where the roots of a particular quadratic equation exists (where the curve crosses X axis), and is defined by the trichotomy property as follows; 1.
		The number of terms in a mathematical equation is the called the degree of the equation.
		The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself.
	www.threes.com /math_book.html (7174 words)

	cubic.html
		(Diophantus 200AD) To find two numbers, one a square and one a cube, so that the sum of their squares is a square.
		if it can be arrived at from the rationals using a finite number of applications of the four arithmetic operations together with the operation of taking the positive square root of a positive number.
		Now construct a sequence of 21 plots of the roots as a moves from 2 to 0 like this.
	www.ms.uky.edu /~carl/ma330/html/cubic1.html (884 words)

	Constructible Numbers
		operations” that can be constructed: x + y, x – y, x*y, y/x, and Öx or Öy.
		Therefore, the constructible numbers (numbers that can be constructed) are those that can be found using a finite number of applications of the constructible operations.
		If a number is transcendental, then it is not the root of a rational equation.
	jwilson.coe.uga.edu /EMT668/EMAT6680.2001/Hays/GeometryProject/constructible.html (99 words)

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