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Large but Finite
Department of Mathematics MathClub@WMU Michigan Epsilon Chapter of Pi Mu Epsilon Large but Finite Drake Olejniczak Department of Mathematics, WMU What is the biggest number you can imagine? Did you think of a million? a billion? a septillion? Perhaps you remembered back to the time you heard the term googol or its show-off of an older brother googolplex. Or maybe you thought of infinity. No. Surely, `infinity' is cheating. Large, finite numbers, truly monstrous numbers, arise in several areas of mathematics as well as in astronomy and computer science. For instance, Euclid defined perfect numbers as positive integers that are equal to the sum of their proper divisors. He is also credited with the discovery of the first four perfect numbers: 6, 28, 496, 8128. It may be surprising to hear that the ninth perfect number already has 37 digits, or that the thirteenth perfect number vastly exceeds the number of particles in the observable universe. However, the sequence of perfect numbers pales in comparison to some other sequences in terms of growth rate. In this talk, we will explore examples of large numbers such as Graham's number, Rayo's number, and the limits of the universe. As well, we will encounter some fast-growing sequences and functions such as the TREE sequence, the busy beaver function, and the Ackermann function. Through this, we will uncover a structure on which to compare these concepts and, hopefully, gain a sense of what it means for a number to be truly large. 4 p.m. Friday October 12 6625 Everett Tower, WMU Main Campus All are welcome! http://www.wmich.edu/mathclub/ Questions? Contact Patrick Bennett ([email protected]) . -
Grade 7/8 Math Circles the Scale of Numbers Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero. -
Maths Secrets of Simpsons Revealed in New Book
MONDAY 7 OCTOBER 2013 WWW.THEDAY.CO.UK Maths secrets of Simpsons revealed in new book The most successful TV show of all time is written by a team of brilliant ‘mathletes’, says writer Simon Singh, and full of obscure mathematical jokes. Can numbers really be all that funny? MATHEMATICS Nerd hero: The smartest girl in Springfield was created by a team of maths wizards. he world’s most popular cartoon a perfect number, a narcissistic number insist that their love of maths contrib- family has a secret: their lines are and a Mersenne Prime. utes directly to the more obvious humour written by a team of expert mathema- Another of these maths jokes – a black- that has made the show such a hit. Turn- Tticians – former ‘mathletes’ who are board showing 398712 + 436512 = 447212 ing intuitions about comedy into concrete as happy solving differential equa- – sent shivers down Simon Singh’s spine. jokes is like wrestling mathematical tions as crafting jokes. ‘I was so shocked,’ he writes, ‘I almost hunches into proofs and formulas. Comedy Now, science writer Simon Singh has snapped my slide rule.’ The numbers are and maths, says Cohen, are both explora- revealed The Simpsons’ secret math- a fake exception to a famous mathemati- tions into the unknown. ematical formula in a new book*. He cal rule known as Fermat’s Last Theorem. combed through hundreds of episodes One episode from 1990 features a Mathletes and trawled obscure internet forums to teacher making a maths joke to a class of Can maths really be funny? There are many discover that behind the show’s comic brilliant students in which Bart Simpson who will think comparing jokes to equa- exterior lies a hidden core of advanced has been accidentally included. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n. -
A Child Thinking About Infinity
A Child Thinking About Infinity David Tall Mathematics Education Research Centre University of Warwick COVENTRY CV4 7AL Young children’s thinking about infinity can be fascinating stories of extrapolation and imagination. To capture the development of an individual’s thinking requires being in the right place at the right time. When my youngest son Nic (then aged seven) spoke to me for the first time about infinity, I was fortunate to be able to tape-record the conversation for later reflection on what was happening. It proved to be a fascinating document in which he first treated infinity as a very large number and used his intuitions to think about various arithmetic operations on infinity. He also happened to know about “minus numbers” from earlier experiences with temperatures in centigrade. It was thus possible to ask him not only about arithmetic with infinity, but also about “minus infinity”. The responses were thought-provoking and amazing in their coherent relationships to his other knowledge. My research in studying infinite concepts in older students showed me that their ideas were influenced by their prior experiences. Almost always the notion of “limit” in some dynamic sense was met before the notion of one to one correspondences between infinite sets. Thus notions of “variable size” had become part of their intuition that clashed with the notion of infinite cardinals. For instance, Tall (1980) reported a student who considered that the limit of n2/n! was zero because the top is a “smaller infinity” than the bottom. It suddenly occurred to me that perhaps I could introduce Nic to the concept of cardinal infinity to see what this did to his intuitions. -
Parameter Free Induction and Provably Total Computable Functions
Parameter free induction and provably total computable functions Lev D. Beklemishev∗ Steklov Mathematical Institute Gubkina 8, 117966 Moscow, Russia e-mail: [email protected] November 20, 1997 Abstract We give a precise characterization of parameter free Σn and Πn in- − − duction schemata, IΣn and IΠn , in terms of reflection principles. This I − I − allows us to show that Πn+1 is conservative over Σn w.r.t. boolean com- binations of Σn+1 sentences, for n ≥ 1. In particular, we give a positive I − answer to a question, whether the provably recursive functions of Π2 are exactly the primitive recursive ones. We also characterize the provably − recursive functions of theories of the form IΣn + IΠn+1 in terms of the fast growing hierarchy. For n = 1 the corresponding class coincides with the doubly-recursive functions of Peter. We also obtain sharp results on the strength of bounded number of instances of parameter free induction in terms of iterated reflection. Keywords: parameter free induction, provably recursive functions, re- flection principles, fast growing hierarchy Mathematics Subject Classification: 03F30, 03D20 1 Introduction In this paper we shall deal with the first order theories containing Kalmar ele- mentary arithmetic EA or, equivalently, I∆0 + Exp (cf. [11]). We are interested in the general question how various ways of formal reasoning correspond to models of computation. This kind of analysis is traditionally based on the con- cept of provably total recursive function (p.t.r.f.) of a theory. Given a theory T containing EA, a function f(~x) is called provably total recursive in T , iff there is a Σ1 formula φ(~x, y), sometimes called specification, that defines the graph of f in the standard model of arithmetic and such that T ⊢ ∀~x∃!y φ(~x, y). -
Simple Statements, Large Numbers
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2007 Simple Statements, Large Numbers Shana Streeks University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Streeks, Shana, "Simple Statements, Large Numbers" (2007). MAT Exam Expository Papers. 41. https://digitalcommons.unl.edu/mathmidexppap/41 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Master of Arts in Teaching (MAT) Masters Exam Shana Streeks In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Gordon Woodward, Advisor July 2007 Simple Statements, Large Numbers Shana Streeks July 2007 Page 1 Streeks Simple Statements, Large Numbers Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, as defined by Wikipedia (2007). Large numbers typically refer to large positive integers, or more generally, large positive real numbers, but may also be used in other contexts. Very large numbers often occur in fields such as mathematics, cosmology, and cryptography. Sometimes people refer to numbers as being “astronomically large”. However, it is easy to mathematically define numbers that are much larger than those even in astronomy. We are familiar with the large magnitudes, such as million or billion. -
Primality Testing for Beginners
STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. -
The Notion Of" Unimaginable Numbers" in Computational Number Theory
Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers. -
Hyperoperations and Nopt Structures
Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles. -
A Family of Sequences Generating Smith Numbers
1 2 Journal of Integer Sequences, Vol. 16 (2013), 3 Article 13.4.6 47 6 23 11 A Family of Sequences Generating Smith Numbers Amin Witno Department of Basic Sciences Philadelphia University 19392 Jordan [email protected] Abstract Infinitely many Smith numbers can be constructed using sequences which involve repunits. We provide an alternate method of construction by introducing a generaliza- tion of repunits, resulting in new Smith numbers whose decimal digits are composed of zeros and nines. 1 Introduction We work with natural numbers in their decimal representation. Let S(N) denote the “digital sum” (the sum of the base-10 digits) of the number N and let Sp(N) denote the sum of all the digital sums of the prime factors of N, counting multiplicity. For instance, S(2013) = 2+0+1+3 = 6 and, since 2013 = 3 × 11 × 61, we have Sp(2013) = 3+1+1+6+1 = 12. The quantity Sp(N) does not seem to have a name in the literature, so let us just call Sp(N) the p-digit sum of N. The natural number N is called a Smith number when N is composite and S(N)= Sp(N). For example, since 22 = 2 × 11, we have both S(22) = 4 and Sp(22) = 4. Hence, 22 is a Smith number—thus named by Wilansky [4] in 1982. Several different ways of constructing Smith numbers are already known. (See our 2010 article [5], for instance, which contains a brief historical account on the subject as well as a list of references. -
Ackermann and the Superpowers
Ackermann and the superpowers Ant´onioPorto and Armando B. Matos Original version 1980, published in \ACM SIGACT News" Modified in October 20, 2012 Modified in January 23, 2016 (working paper) Abstract The Ackermann function a(m; n) is a classical example of a total re- cursive function which is not primitive recursive. It grows faster than any primitive recursive function. It is usually defined by a general recurrence together with two \boundary" conditions. In this paper we obtain a closed form of a(m; n), which involves the Knuth superpower notation, namely m−2 a(m; n) = 2 " (n + 3) − 3. Generalized Ackermann functions, that is functions satisfying only the general recurrence and one of the bound- ary conditions are also studied. In particular, we show that the function m−2 2 " (n + 2) − 2 also belongs to the \Ackermann class". 1 Introduction and definitions The \arrow" or \superpower" notation has been introduced by Knuth [1] as a convenient way of expressing very large numbers. It is based on the infinite sequence of operators: +, ∗, ",... We shall see that the arrow notation is closely related to the Ackermann function (see, for instance, [2]). 1.1 The Superpowers Let us begin with the following sequence of integer operators, where all the operators are right associative. a × n = a + a + ··· + a (n a's) a " n = a × a × · · · × a (n a's) 2 a " n = a " a "···" a (n a's) m In general we define a " n as 1 Definition 1 m m−1 m−1 m−1 a " n = a " a "··· " a | {z } n a's m The operator " is not associative for m ≥ 1.