A Short History of Mathematical Symbolism
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Euler's Square Root Laws for Negative Numbers
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Complex Numbers Mathematics via Primary Historical Sources (TRIUMPHS) Winter 2020 Euler's Square Root Laws for Negative Numbers Dave Ruch Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_complex Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Euler’sSquare Root Laws for Negative Numbers David Ruch December 17, 2019 1 Introduction We learn in elementary algebra that the square root product law pa pb = pab (1) · is valid for any positive real numbers a, b. For example, p2 p3 = p6. An important question · for the study of complex variables is this: will this product law be valid when a and b are complex numbers? The great Leonard Euler discussed some aspects of this question in his 1770 book Elements of Algebra, which was written as a textbook [Euler, 1770]. However, some of his statements drew criticism [Martinez, 2007], as we shall see in the next section. 2 Euler’sIntroduction to Imaginary Numbers In the following passage with excerpts from Sections 139—148of Chapter XIII, entitled Of Impossible or Imaginary Quantities, Euler meant the quantity a to be a positive number. 1111111111111111111111111111111111111111 The squares of numbers, negative as well as positive, are always positive. ...To extract the root of a negative number, a great diffi culty arises; since there is no assignable number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of 4; we here require such as number as, when multiplied by itself, would produce 4; now, this number is neither +2 nor 2, because the square both of 2 and of 2 is +4, and not 4. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
The Geometry of René Descartes
BOOK FIRST The Geometry of René Descartes BOOK I PROBLEMS THE CONSTRUCTION OF WHICH REQUIRES ONLY STRAIGHT LINES AND CIRCLES ANY problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.1 Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to add or subtract ther lines; or else, taking one line which I shall call unity in order to relate it as closely as possible to numbers,2 and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as multiplication) ; or, again, to find a fourth line which is to one of the given lines as unity is to the other (which is equivalent to division) ; or, finally, to find one, two, or several mean proportionals between unity and some other line (which is the same as extracting the square root, cube root, etc., of the given line.3 And I shall not hesitate to introduce these arithmetical terms into geometry, for the sake of greater clearness. For example, let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA ; then BE is the product of BD and BC. -
Number Theory
“mcs-ftl” — 2010/9/8 — 0:40 — page 81 — #87 4 Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . Which one don’t you understand? Sec- ond, what practical value is there in it? The mathematician G. H. Hardy expressed pleasure in its impracticality when he wrote: [Number theorists] may be justified in rejoicing that there is one sci- ence, at any rate, and that their own, whose very remoteness from or- dinary human activities should keep it gentle and clean. Hardy was specially concerned that number theory not be used in warfare; he was a pacifist. You may applaud his sentiments, but he got it wrong: Number Theory underlies modern cryptography, which is what makes secure online communication possible. Secure communication is of course crucial in war—which may leave poor Hardy spinning in his grave. It’s also central to online commerce. Every time you buy a book from Amazon, check your grades on WebSIS, or use a PayPal account, you are relying on number theoretic algorithms. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in Chapters 2 and 3. Since we’ll be focusing on properties of the integers, we’ll adopt the default convention in this chapter that variables range over the set of integers, Z. 4.1 Divisibility The nature of number theory emerges as soon as we consider the divides relation a divides b iff ak b for some k: D The notation, a b, is an abbreviation for “a divides b.” If a b, then we also j j say that b is a multiple of a. -
A Quartically Convergent Square Root Algorithm: an Exercise in Forensic Paleo-Mathematics
A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics David H Bailey, Lawrence Berkeley National Lab, USA DHB’s website: http://crd.lbl.gov/~dhbailey! Collaborator: Jonathan M. Borwein, University of Newcastle, Australia 1 A quartically convergent algorithm for Pi: Jon and Peter Borwein’s first “big” result In 1985, Jonathan and Peter Borwein published a “quartically convergent” algorithm for π. “Quartically convergent” means that each iteration approximately quadruples the number of correct digits (provided all iterations are performed with full precision): Set a0 = 6 - sqrt[2], and y0 = sqrt[2] - 1. Then iterate: 1 (1 y4)1/4 y = − − k k+1 1+(1 y4)1/4 − k a = a (1 + y )4 22k+3y (1 + y + y2 ) k+1 k k+1 − k+1 k+1 k+1 Then ak, converge quartically to 1/π. This algorithm, together with the Salamin-Brent scheme, has been employed in numerous computations of π. Both this and the Salamin-Brent scheme are based on the arithmetic-geometric mean and some ideas due to Gauss, but evidently he (nor anyone else until 1976) ever saw the connection to computation. Perhaps no one in the pre-computer age was accustomed to an “iterative” algorithm? Ref: J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity}, John Wiley, New York, 1987. 2 A quartically convergent algorithm for square roots I have found a quartically convergent algorithm for square roots in a little-known manuscript: · To compute the square root of q, let x0 be the initial approximation. -
Unit 2. Powers, Roots and Logarithms
English Maths 4th Year. European Section at Modesto Navarro Secondary School UNIT 2. POWERS, ROOTS AND LOGARITHMS. 1. POWERS. 1.1. DEFINITION. When you multiply two or more numbers, each number is called a factor of the product. When the same factor is repeated, you can use an exponent to simplify your writing. An exponent tells you how many times a number, called the base, is used as a factor. A power is a number that is expressed using exponents. In English: base ………………………………. Exponente ………………………… Other examples: . 52 = 5 al cuadrado = five to the second power or five squared . 53 = 5 al cubo = five to the third power or five cubed . 45 = 4 elevado a la quinta potencia = four (raised) to the fifth power . 1521 = fifteen to the twenty-first . 3322 = thirty-three to the power of twenty-two Exercise 1. Calculate: a) (–2)3 = f) 23 = b) (–3)3 = g) (–1)4 = c) (–5)4 = h) (–5)3 = d) (–10)3 = i) (–10)6 = 3 3 e) (7) = j) (–7) = Exercise: Calculate with the calculator: a) (–6)2 = b) 53 = c) (2)20 = d) (10)8 = e) (–6)12 = For more information, you can visit http://en.wikibooks.org/wiki/Basic_Algebra UNIT 2. Powers, roots and logarithms. 1 English Maths 4th Year. European Section at Modesto Navarro Secondary School 1.2. PROPERTIES OF POWERS. Here are the properties of powers. Pay attention to the last one (section vii, powers with negative exponent) because it is something new for you: i) Multiplication of powers with the same base: E.g.: ii) Division of powers with the same base : E.g.: E.g.: 35 : 34 = 31 = 3 iii) Power of a power: 2 E.g. -
Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J
Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J. Keely. All Rights Reserved. Mathematical expressions can be typed online in a number of ways including plain text, ASCII codes, HTML tags, or using an equation editor (see Writing Mathematical Notation Online for overview). If the application in which you are working does not have an equation editor built in, then a common option is to write expressions horizontally in plain text. In doing so you have to format the expressions very carefully using appropriately placed parentheses and accurate notation. This document provides examples and important cautions for writing mathematical expressions in plain text. Section 1. How to Write Exponents Just as on a graphing calculator, when writing in plain text the caret key ^ (above the 6 on a qwerty keyboard) means that an exponent follows. For example x2 would be written as x^2. Example 1a. 4xy23 would be written as 4 x^2 y^3 or with the multiplication mark as 4*x^2*y^3. Example 1b. With more than one item in the exponent you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x2n must be written as x^(2n) and NOT as x^2n. Writing x^2n means xn2 . Example 1c. When using the quotient rule of exponents you often have to perform subtraction within an exponent. In such cases you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x5 The middle step of ==xx52− 3 must be written as x^(5-2) and NOT as x^5-2 which means x5 − 2 . -
Schaum's Outline of Linear Algebra (4Th Edition)
SCHAUM’S SCHAUM’S outlines outlines Linear Algebra Fourth Edition Seymour Lipschutz, Ph.D. Temple University Marc Lars Lipson, Ph.D. University of Virginia Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 2001, 1991, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ- ten permission of the publisher. ISBN: 978-0-07-154353-8 MHID: 0-07-154353-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154352-1, MHID: 0-07-154352-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected]. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. -
FROM HARMONIC ANALYSIS to ARITHMETIC COMBINATORICS: a BRIEF SURVEY the Purpose of This Note Is to Showcase a Certain Line Of
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS: A BRIEF SURVEY IZABELLA ÃLABA The purpose of this note is to showcase a certain line of research that connects harmonic analysis, speci¯cally restriction theory, to other areas of mathematics such as PDE, geometric measure theory, combinatorics, and number theory. There are many excellent in-depth presentations of the vari- ous areas of research that we will discuss; see e.g., the references below. The emphasis here will be on highlighting the connections between these areas. Our starting point will be restriction theory in harmonic analysis on Eu- clidean spaces. The main theme of restriction theory, in this context, is the connection between the decay at in¯nity of the Fourier transforms of singu- lar measures and the geometric properties of their support, including (but not necessarily limited to) curvature and dimensionality. For example, the Fourier transform of a measure supported on a hypersurface in Rd need not, in general, belong to any Lp with p < 1, but there are positive results if the hypersurface in question is curved. A classic example is the restriction theory for the sphere, where a conjecture due to E. M. Stein asserts that the Fourier transform maps L1(Sd¡1) to Lq(Rd) for all q > 2d=(d¡1). This has been proved in dimension 2 (Fe®erman-Stein, 1970), but remains open oth- erwise, despite the impressive and often groundbreaking work of Bourgain, Wol®, Tao, Christ, and others. We recommend [8] for a thorough survey of restriction theory for the sphere and other curved hypersurfaces. Restriction-type estimates have been immensely useful in PDE theory; in fact, much of the interest in the subject stems from PDE applications. -
Problems in Abstract Algebra
STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 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 select pages 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. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder . -
Measuring Fractals by Infinite and Infinitesimal Numbers
MEASURING FRACTALS BY INFINITE AND INFINITESIMAL NUMBERS Yaroslav D. Sergeyev DEIS, University of Calabria, Via P. Bucci, Cubo 42-C, 87036 Rende (CS), Italy, N.I. Lobachevsky State University, Nizhni Novgorod, Russia, and Institute of High Performance Computing and Networking of the National Research Council of Italy http://wwwinfo.deis.unical.it/∼yaro e-mail: [email protected] Abstract. Traditional mathematical tools used for analysis of fractals allow one to distinguish results of self-similarity processes after a finite number of iterations. For example, the result of procedure of construction of Cantor’s set after two steps is different from that obtained after three steps. However, we are not able to make such a distinction at infinity. It is shown in this paper that infinite and infinitesimal numbers proposed recently allow one to measure results of fractal processes at different iterations at infinity too. First, the new technique is used to measure at infinity sets being results of Cantor’s proce- dure. Second, it is applied to calculate the lengths of polygonal geometric spirals at different points of infinity. 1. INTRODUCTION During last decades fractals have been intensively studied and applied in various fields (see, for instance, [4, 11, 5, 7, 12, 20]). However, their mathematical analysis (except, of course, a very well developed theory of fractal dimensions) very often continues to have mainly a qualitative character and there are no many tools for a quantitative analysis of their behavior after execution of infinitely many steps of a self-similarity process of construction. Usually, we can measure fractals in a way and can give certain numerical answers to questions regarding fractals only if a finite number of steps in the procedure of their construction has been executed.