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Topic: Decimal expansion

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	Decimal - Wikipedia, the free encyclopedia
		Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) to represent numbers.
		Decimal fractions can be expressed without a denominator, the decimal point being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator.
		That a rational must produce a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only (q-1) possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q-1.
	en.wikipedia.org /wiki/Decimal (1120 words)

	Expanding Fractions (Site not responding. Last check: 2007-10-29)
		In this problem you are to print the decimal expansion of a quotient of two integers.
		You will print the decimal expansion of the integer quotient given, stopping just as the expansion terminates or just as the repeating pattern is to repeat itself for the first time.
		If the expansion is infinite, you should print the decimal expansion up to, but not including the digit where the repeated pattern first repeats itself.
	acm.uva.es /p/v2/275.html (357 words)

	Decimal (Site not responding. Last check: 2007-10-29)
		Decimal, also called denary, is the base 10 numeral system, which uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to represent numbers.
		Decimal numerals can be encoded for computers using binary-coded decimal or more efficient schemes.
		Decimal fractions are usually expressed without a denominator, the decimal point being inserted into the numerator at a position corresponding to the power of ten of the denominator.
	www.yotor.com /wiki/en/de/Decimal.htm (690 words)

	PlanetMath: decimal expansion
		If the tails of 0's are not accepted, then the digital expansion of every positive rational is unique (then e.g.
		Also any irrational number has a unique decimal expansion, but it is non-periodic; for example the Liouville's number
		This is version 19 of decimal expansion, born on 2005-02-19, modified 2005-05-04.
	planetmath.org /encyclopedia/DecimalExpansion.html (208 words)

	[No title]
		Decimal Point Binary numbers also have a binary expansion which is almost exactly like the decimal expansion we've just seen.
		Expansion of binary 11011: 4 3 2 1 0 1 x 2 + 1 x 2 + 0 x 2 + 1 x 2 + 1 x 2 Expansion of binary.1101 -1 -2 -3 -4 1 x 2 + 1 x 2 + 0 x 2 + 1 x 2 Where...
		To convert a decimal fraction to hexadecimal: Successively multiply by 16 and use the integer portion of the result in the answer.
	www.unf.edu /public/cot3100/jgiles/lecture1 (1420 words)

	Rational Numbers from Repeating Fractions (Site not responding. Last check: 2007-10-29)
		If we are given the decimal expansion of a rational fraction (with an indication of which digits are repeated, if necessary), we can determine the rational fraction (that is, the integer values of p and q in p/q) using the following algorithm.
		is the decimal expansion of the rational number 315/990.
		There may be as many as nine (9) digits in the decimal expansion (that is, the value of k+j may be as large as 9).
	acm.uva.es /p/v3/332.html (408 words)

	The space of decimal expansions (Site not responding. Last check: 2007-10-29)
		In this chapter we are interested in a topological structure on the set of decimal expansions of real numbers, and its relation to the topological structure on the set of real numbers.
		Notice the distinction; a decimal expansion is a concrete syntactic entity (a sequence of digits), but a real number is an abstract mathematical entity.
		For simplicity, we shall consider only fractional infinite decimal expansions in these notes; we therefore ignore the decimal point and the leading digit zero, so that a decimal expansion is just a sequence of decimal digits.
	www.cs.bham.ac.uk /~mhe/issac/node21.html (581 words)

	Number and Operations Session 7, Part A: Fractions to Decimals
		For decimals of this type, we can examine the period of the decimal, or the number of digits that appear before the digit string begins repeating itself.
		In the decimal expansion of 1/3, only the digit 3 repeats, and so the period is one.
		For example, in the repeating decimal 0.3333..., the repetend is 3 and, as we've just seen, the period is one; in 0.142857142857..., the repetend is 142857, and the period is six.
	www.learner.org /channel/courses/learningmath/number/session7/part_a/repeating.html (454 words)

	Search Results for decimal*
		For example if asked for the decimal expansion of 1/851 he would think of 851 as 23 cross 37, if asked for the square root of 851 then he thought of it as 292 + 10, if asked for the decimal expansion of 17/851 then he would think of it as almost 0.02.
		The decimal place-value system developed when a decimal scale came to be associated with the value of the places of the numbers arranged left to right or right to left.
		Ten decimal places of pi are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?BIOGS=1&TOPICS=1&CURVES=1&REFS=1&BIBLI=1&SOCIETIES=1"=1&WORD=decimal*&CONTEXT=1 (4616 words)

	Repeating Nines
		That is I have to accept that there might be two distinct decimals a and b, where a and b do not begin to differ until after an infinite number of digits from the decimal point.
		In the last section of this essay, I suggested that two numbers might have the same decimal representation for an infinite number of digits and still not be the same number.
		Certainly, each step in the expansion of the decimal gets me closer to the true value of 1/3, but there is still a remainder.
	descmath.com /diag/nines.html (2341 words)

	Richard Julius Wilhelm Dedekind
		It can be shown that every rational number has a terminating or infinite periodic decimal expansion and every terminate or infinite periodic decimal expansion represents a rational number.
		If we work out the decimal expansion of 2 by the method of finding square roots then this rule would enable us to put any given rational number into one of two classes A or B, depending on whether it is smaller or larger than
		For example, suppose the given rational number is 707/500 whose decimal expansion is 1.414.
	www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/rdedek.htm (843 words)

	Digits in the decimal expansion (Site not responding. Last check: 2007-10-29)
		In the decimal expansion of 1/17 what digit is in the 1997th place?
		into decimal form, and look for a pattern.
		They are 14257 and hence the 1997th digit in the decimal representation of
	mathcentral.uregina.ca /QQ/database/QQ.09.03/leslie1.html (139 words)

	The period length of the decimal expansion of a fraction (Site not responding. Last check: 2007-10-29)
		The period length of the decimal expansion of a fraction
		It is explained how, for a given natural number, the period length of the decimal fraction of the reciprocal of that number can be determined without actually performing the division.
		The article begins with examples in the decimal number system, but the theory is then displayed for arbitrary bases of number systems.
	www.lrz-muenchen.de /~hr/numb/period.html (1176 words)

	[No title]
		Without decimal expansion, 808 over Cutter number, it designates books or sets of works collected from several literatures, which could not be classed in 810-899 since they are not in one original language.
		With decimal expansion, 808.1 - 808.9 is for collections of essays about literary forms, following language table 5: 808.1 on poetry, 808.2 on drama.
		There is a special letter expansion, 809.22B for biographies of actors, directors and so on, with a second line Cutter number for the person written about, not the writer.
	www.columbia.edu /~brennan/library/cc.800.html (948 words)

	Is 0.999... = 1? (Site not responding. Last check: 2007-10-29)
		That means that you can't necessarily compute the decimal expansion of a sum from the decimal expansions of its addends.
		The fact that you can't compute the decimal expansion of a sum from the decimal expansions of its addends is a well known phenomenon that was noticed by Turing.
		Instead of extending the decimal numbers so as to include additive inverses of those decimal numbers that are cancellable under addition, we could extend them so as to include multiplicative inverses of those decimal numbers that are cancellable under multiplication.
	www.math.fau.edu /Richman/HTML/999.htm (2442 words)

	Read This: Ergodic Theory of Numbers
		For an integer n>1, the n-ary expansion of x in [0,1) is its base-n expansion.
		The partition of [0,1) for a Lüroth series expansion is given by the intervals [1/(n+1),1/n), n>0, and the corresponding map T is linear (with positive slope) and bijective with range [0,1) on each atom of the partition.
		The authors go on to use dynamics to provide a new proof of a generalization of Borel's theorem and to prove (not assign!) a result due to Barbolosi and Jager (1994), generalizing a result of Legendre's, that states precisely when a given rational is a continued fraction convergent for the irrational x.
	www.maa.org /reviews/ergodicnt.html (3054 words)

	Section 13.1: Approximation with Continued Fractions
		Suppose we had a decimal approximation of an unknown rational number, and we wanted to recover the original rational number that produced the decimal approximation.
		One approach would be to use the fact that the digits of the decimal approximation of a rational number are ultimately periodic.
		For example, 0.46017699115044247787 is the first 20 decimal places of a rational number with a moderately small denominator.
	www.math.mtu.edu /mathlab/COURSES/holt/dnt/pell1.html (598 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-29)
		What matters is that the decimal expansion is infinite.
		The Nth term in the decimal expansion of pi is d_N/10^N where d_N is the Nth digit (0 <= d_N < 10).
		Infinity, infinite series, and infinite decimal expansions are hard to think about.
	mathforum.org /library/drmath/view/57091.html (659 words)

	Exam 1 (Site not responding. Last check: 2007-10-29)
		Prove that the decimal expansion of a rational number is either terminating or repeating.
		Note that the decimal expansion of a rational number can be found using long division.
		When that is done, either the remainder is 0 at some stage (in which case the expansion is terminating) or the remainder must repeat.
	www.math.colostate.edu /~manvel/M425sp02/Exam1Sol2002.html (539 words)

	POLYA013 (Site not responding. Last check: 2007-10-29)
		The decimal expansion of 1/N is periodic after a certain number of digits beyond the decimal point.
		Since 1/N has a periodic part of its decimal expansion, there is a smallest number M such that (10^M - 1)/N has terminating decimal expansion.
		The number of decimals of (10^M - 1)/N is equal to the length L of N. Hence, for an integer P, we have (10^(M+L) - 10^L)/N = P where P is not a positive multiple of 10, or 10^(M+L) - NP = 10^L. So 10^L'=10^L. Bibliography.
	forumgeom.fau.edu /POLYA/ProblemCenter/POLYA013.html (198 words)

	Problem 2 (Site not responding. Last check: 2007-10-29)
		It is important to note that the expression in the numerator and the denominator of this expression will always yield integer values, and these represent the numerator and denominator of the __ rational number.
		Thus the repeated fraction.318 is the decimal expansion of the rational number 315/990.
		The input data for this problem will be a sequence of test cases, each test case appearing on a line by itself, followed by a -1.
	www.mathcs.carleton.edu /acm/archives/prob2.html (325 words)

	Puzzle 95. K consecutive primes p whose decimal expansion has a (p-1) period
		K consecutive primes p such that the decimal expansion of the reciprocal 1/p, for each p, has exactly a period of length (p-1) (
		As you know the decimal expansion of 1/p for p=7 has a period = 6 equal to (p-1)
		It can be shown that the period x of the decimal expansion of the reciprocal (1/p) of any prime p, is the least x value (1<=x<=p-1) that satisfies 10^x = 1 (mod p).
	www.primepuzzles.net /puzzles/puzz_095.htm (601 words)

	Number and Operations Session 7: Homework
		Shigeto and Consuela were computing the decimal expansion of 1/19.
		Teague asked the class to compute a decimal expansion with period 42.
		Is it possible to provide a convincing argument to prove that the decimal expansion of 1/n has a period that is less than n?
	www.learner.org /channel/courses/learningmath/number/session7/part_h/homework.html (385 words)

	[No title]
		Use this ;; to read off the decimal expansion.
		(field [ht (make-hash-table 'equal)] [expansions 0]) ;; this field holds the state of the current computation ;; of the numbers digits.
		void ;; iterates until the numbers decimal expansion is completely computed, ;; or the number's decimal expansion terminates.
	www.bath.ac.uk /~masjap/plt/collects/drscheme/private/number-snip.ss (590 words)

	Decimal Representations
		Conversely, a terminating or repeating decimal is equal to a rational.
		If the decimal repeats, such as 4.40909090909..., write it as the sum 4.4 + 0.00909090909..., which is 44/10 + one tenth of 9/99.
		The correspondence between rationals and decimals is almost 1-1, but not quite.
	www.mathreference.com /top-ms,dec.html (508 words)

	Synopses of Topics - Decimal Expansion
		Some numbers have an infinite number of decimal digits, however it is possible for us to work with such numbers.
		Note that if we start with a decimal expression, then there is always a real number that is represented by that decimal expression.
		Also, any decimal representation of a number with a repeating component can be changed to a rational number.
	math.usask.ca /emr/dec.html (499 words)

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