The decimal expansion of a number

2020 ◽  
pp. 167-207
Keyword(s):  
Decimal Expansion

scholarly journals Repdigits as products of consecutive Padovan or Perrin numbers

2021 ◽  
Author(s):  
Salah Eddine Rihane ◽  
Alain Togbé
Keyword(s):  
Positive Integer ◽  
Decimal Expansion

AbstractA repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number of the form $$a(10^m-1)/9$$ a ( 10 m - 1 ) / 9 , for some $$m\ge 1$$ m ≥ 1 and $$1 \le a \le 9$$ 1 ≤ a ≤ 9 . Let $$\left( P_n\right) _{n\ge 0}$$ P n n ≥ 0 and $$\left( E_n\right) _{n\ge 0}$$ E n n ≥ 0 be the sequence of Padovan and Perrin numbers, respectively. This paper deals with repdigits that can be written as the products of consecutive Padovan or/and Perrin numbers.

Keep it simple whenever possible, since 0.999…=1

2020 ◽  
pp. 151-152
Author(s):  
Susan D'Agostino
Keyword(s):  
Mathematical Proof ◽  
Mathematics Students ◽  
Decimal Expansion ◽  
Loss Of Information

“Keep it simple whenever possible, since 0.999…=1” presents and discusses a very short mathematical proof demonstrating the long-known result that 0.999…=1. The ellipsis in the number 0.999… indicates that this number repeats in an infinite decimal expansion. As such, this number is unwieldy to lug around, insert into equations, and even describe. However, the number 1 is not simply a good approximation for 0.999…., but rather the number 1 may be used in place of 0.999… without loss of information. Mathematics students and enthusiasts are encouraged to keep their mathematical and life pursuits simple whenever possible. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

89.61 Rapid decimal expansion of rational fractions

2005 ◽  
Vol 89 (516) ◽  
pp. 458-460
Author(s):  
P. MacGregor
Keyword(s):  
Decimal Expansion

scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

Expanding the Limits of the Calculator Display

1986 ◽  
Vol 79 (1) ◽  
pp. 20-21
Author(s):  
Laurence Sherzer
Keyword(s):  
Outer Space ◽  
Rational Numbers ◽  
Decimal Expansion ◽  
Electronic Calculator

The decimal expansions of 1/7 or 5/13 can be found immediately on the average electronic calculator. These decimal expansions are limited to six digits before they begin to repeat. But what do we say to students who want to explore the repeating properties of rational numbers? What would we answer if they asked for the decimal expansion of 1/17 or 3/23? Do we just say, “Keep dividing and you'll find the answer”? Hardly. Long decimal expansions are usually left to actuaries who don't want to mismanage thousands of dollars over the life of an annuity or mortgage table, or to spacecraft engineers who don't want to miss their targets in outer space.

A secure cryptosystem using the decimal expansion of an irrational number

2015 ◽  
Vol 9 ◽  
pp. 5293-5303
Author(s):  
M. K. Viswanath ◽  
M. Ranjith Kumar
Keyword(s):  
Irrational Number ◽  
Decimal Expansion

A Generalized Tauberian Theorem

1958 ◽  
Vol 10 ◽  
pp. 111-114
Author(s):  
F. R. Keogh ◽  
G. M. Petersen
Keyword(s):  
Tauberian Theorem ◽  
Decimal Place ◽  
Real Sequence ◽  
Decimal Expansion ◽  
First Category ◽  
Dense Set

Let {s(n)} be a real sequence and let x be any number in the interval 0 < x ⩽ 1. Representing x by a non-terminating binary decimal expansion we shall denote by {s(n,x)} the subsequence of {s(n)} obtained by omitting s(k) if and only if there is a 0 in the decimal place in the expansion of x. With this correspondence it is then possible to speak of “a set of subsequences of the first category,” “an everywhere dense set of subsequences,” and so on.

scholarly journals Extensional constructive real analysis via locators

2020 ◽  
pp. 1-25
Author(s):  
Auke B. Booij
Keyword(s):  
Cauchy Sequence ◽  
Computable Analysis ◽  
Real Analysis ◽  
Additional Structure ◽  
Real Numbers ◽  
Decimal Expansion ◽  
Discrete Information

Abstract Real numbers do not admit an extensional procedure for observing discrete information, such as the first digit of its decimal expansion, because every extensional, computable map from the reals to the integers is constant, as is well known. We overcome this by considering real numbers equipped with additional structure, which we call a locator. With this structure, it is possible, for instance, to construct a signed-digit representation or a Cauchy sequence, and conversely, these intensional representations give rise to a locator. Although the constructions are reminiscent of computable analysis, instead of working with a notion of computability, we simply work constructively to extract observable information, and instead of working with representations, we consider a certain locatedness structure on real numbers.

Intrasystem research for elementary school teachers

1965 ◽  
Vol 12 (1) ◽  
pp. 5-8
Author(s):  
Leonard Pikaart ◽  
Charles Berryman
Keyword(s):  
Elementary School ◽  
Elementary School Teachers ◽  
School Teachers ◽  
Decimal Expansion ◽  
Simple Observation ◽  
New Information ◽  
The World ◽  
Chance Occurrence

Pi was once assumed by ancient Hebrew “experts” to equal three. More careful observers subsequently added1/7, and by 1874 a decimal expansion to 707 decimal places was computed. Research by mathematicians proved that pi is an inational number, or, more precisely, transcendental. Research was slow in coming. The world was oriented to “experts,” and their opinions were often so sacrosanct that simple observation was virtually taboo. Occasionally, observations demonstrated such obvious merit that new information was forced upon an unwilling world. As the advantages of observation became more apparent, it was realized that casual observation must rely upon chance occurrence. Some unknown genius deliberately created an event to be observed, and research was born.

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