Maths Module 4 Powers, Roots and Logarithms
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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. -
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. -
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. -
The Ordered Distribution of Natural Numbers on the Square Root Spiral
The Ordered Distribution of Natural Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 Et Germanytlingen, Germany ------------------------------ mathematical analysis by - Kay Schoenberger - Humboldt-University Berlin ----------------------------- 20. June 2007 Abstract : Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( also named as “Spiral of Theodorus” or “Wurzel Spirale“ or “Einstein Spiral” ). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of three spiral graphs, which divide the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. The Square Root Spiral is a geometrical structure which is based on the three basic constants: 1, sqrt2 and π (pi) , and the continuous application of the Pythagorean Theorem of the right angled triangle. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self-avoiding-walk- constant in the lattice-like structure of the square root spiral. Contents of the general section Page 1 Introduction to the Square Root Spiral 2 2 Mathematical description of the Square Root Spiral 4 3 The distribution -
Microfilmed 1996 Information to Users
UMI MICROFILMED 1996 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 NonhZeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Interpoint Distance Methods for the Analysis of High Dimensional Data DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By John Lawrence, B. -
On Terms of Generalized Fibonacci Sequences Which Are Powers of Their Indexes
mathematics Article On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic, [email protected]; Tel.: +42-049-333-2860 Received: 29 June 2019; Accepted: 31 July 2019; Published: 3 August 2019 (k) Abstract: The k-generalized Fibonacci sequence (Fn )n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2, is defined by the values 0, 0, ... , 0, 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the (k) t (2) 2 (3) 2 Diophantine equation Fm = m , with t > 1 and m > k + 1, has only solutions F12 = 12 and F9 = 9 . Keywords: k-generalized Fibonacci sequence; Diophantine equation; linear form in logarithms; continued fraction MSC: Primary 11J86; Secondary 11B39. 1. Introduction The well-known Fibonacci sequence (Fn)n≥0 is given by the following recurrence of the second order Fn+2 = Fn+1 + Fn, for n ≥ 0, with the initial terms F0 = 0 and F1 = 1. Fibonacci numbers have a lot of very interesting properties (see e.g., book of Koshy [1]). One of the famous classical problems, which has attracted a attention of many mathematicians during the last thirty years of the twenty century, was the problem of finding perfect powers in the sequence of Fibonacci numbers. Finally in 2006 Bugeaud et al. [2] (Theorem 1), confirmed these expectations, as they showed that 0, 1, 8 and 144 are the only perfect powers in the sequence of Fibonacci numbers. -
Number Systems and Radix Conversion
Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will be required for PA1. We humans are comfortable with numbers in the decimal system where each po- sition has a weight which is a power of 10: units have a weight of 1 (100), ten’s have 10 (101), etc. Even the fractional apart after the decimal point have a weight that is a (negative) power of ten. So the number 143.25 has a value that is 1 ∗ 102 + 4 ∗ 101 + 3 ∗ 0 −1 −2 2 5 10 + 2 ∗ 10 + 5 ∗ 10 , i.e., 100 + 40 + 3 + 10 + 100 . You can think of this as a polyno- mial with coefficients 1, 4, 3, 2, and 5 (i.e., the polynomial 1x2 + 4x + 3 + 2x−1 + 5x−2+ evaluated at x = 10. There is nothing special about 10 (just that humans evolved with ten fingers). We can use any radix, r, and write a number system with digits that range from 0 to r − 1. If our radix is larger than 10, we will need to invent “new digits”. We will use the letters of the alphabet: the digit A represents 10, B is 11, K is 20, etc. 2 What is the value of a radix-r number? Mathematically, a sequence of digits (the dot to the right of d0 is called the radix point rather than the decimal point), : : : d2d1d0:d−1d−2 ::: represents a number x defined as follows X i x = dir i 2 1 0 −1 −2 = ::: + d2r + d1r + d0r + d−1r + d−2r + ::: 1 Example What is 3 in radix 3? Answer: 0.1. -
Cheat Sheet of SSE/AVX Intrinsics, for Doing Arithmetic Ll F(Int Ind, Int K) { Return Dp[Ind][K]; } on Several Numbers at Once
University of Bergen Garbage Collectors Davide Pallotti, Jan Soukup, Olav Røthe Bakken NWERC 2017 Nov 8, 2017 UiB template .bashrc .vimrc troubleshoot 1 tan v + tan w Contest (1) Any possible infinite recursion? tan(v + w) = Invalidated pointers or iterators? 1 − tan v tan w template.cpp Are you using too much memory? v + w v − w 15 lines Debug with resubmits (e.g. remapped signals, see Various). sin v + sin w = 2 sin cos #include <bits/stdc++.h> 2 2 using namespace std; Time limit exceeded: v + w v − w Do you have any possible infinite loops? cos v + cos w = 2 cos cos #define rep(i, a, b) for(int i = a; i < (b); ++i) What is the complexity of your algorithm? 2 2 #define trav(a, x) for(auto& a : x) Are you copying a lot of unnecessary data? (References) #define all(x) x.begin(), x.end() How big is the input and output? (consider scanf) (V + W ) tan(v − w)=2 = (V − W ) tan(v + w)=2 #define sz(x) (int)(x).size() Avoid vector, map. (use arrays/unordered_map) typedef long long ll; What do your team mates think about your algorithm? where V; W are lengths of sides opposite angles v; w. typedef pair<int, int> pii; typedef vector<int> vi; Memory limit exceeded: a cos x + b sin x = r cos(x − φ) What is the max amount of memory your algorithm should need? int main() { Are you clearing all datastructures between test cases? a sin x + b cos x = r sin(x + φ) cin.sync_with_stdio(0); cin.tie(0); cin.exceptions(cin.failbit); p 2 2 } Mathematics (2) where r = a + b ; φ = atan2(b; a). -
Composite Numbers That Give Valid RSA Key Pairs for Any Coprime P
information Article Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p Barry Fagin ID Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-339-4514 Received: 13 August 2018; Accepted: 25 August 2018; Published: 28 August 2018 Abstract: RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific (s, r) pairs can be found for which encryption and decryption will be correct for any (e, d) exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime r - s. We give necessary and sufficient conditions for s with this property. Keywords: cryptography; abstract algebra; RSA; computer science education; cryptography education MSC: [2010] 11Axx 11T71 1. Notation and Background Consider the RSA public-key cryptosystem and its operations of encryption and decryption [1]. Let (p, q) be primes, n = p ∗ q, f(n) = (p − 1)(q − 1) denote Euler’s totient function and (e, d) the ∗ Z encryption/decryption exponent pair chosen such that ed ≡ 1. Let n = Un be the group of units f(n) Z mod n, and let a 2 Un. Encryption and decryption operations are given by: (ae)d ≡ (aed) ≡ (a1) ≡ a mod n We consider the case of RSA encryption and decryption where at least one of (p, q) is a composite number s. -
On Moduli for Which the Fibonacci Sequence Contains a Complete System of Residues S
ON MODULI FOR WHICH THE FIBONACCI SEQUENCE CONTAINS A COMPLETE SYSTEM OF RESIDUES S. A. BURR Belt Telephone Laboratories, Inc., Whippany, New Jersey Shah [1] and Bruckner [2] have considered the problem of determining which moduli m have the property that the Fibonacci sequence {u }, de- fined in the usual way9 contains a complete system of residues modulo m, Following Shah we say that m is defective if m does not have this property, The results proved in [1] includes (I) If m is defective, so is any multiple of m; in particular, 8n is always defective. (II) if p is a prime not 2 or 5, p is defective unless p = 3 or 7 (mod 20). (in) If p is a prime = 3 or 7 (mod 20) and is not defective^ thenthe set {0, ±1, =tu3, ±u49 ±u5, • • • , ±u }9 where h = (p + l)/2 ? is a complete system of residues modulo p„ In [2], Bruckner settles the case of prime moduli by showing that all primes are defective except 2, 3S 5, and 7. In this paper we complete the work of Shah and Bruckner by proving the following re suit f which completely characterizes all defective and nondefective moduli. Theorem. A number m is not defective if and only if m has one of the following forms: k k k 5 , 2 * 5 f 4»5 , 3 k k 3 *5 5 6«5 , k k 7-5 , 14* 5* , where k ^ 0S j — 1. Thus almost all numbers are defective. We will prove a series of lem- mas, from which the theorem will follow directly, We first make some use- ful definitions. -
Grade 7 Mathematics Strand 6 Pattern and Algebra
GR 7 MATHEMATICS S6 1 TITLE PAGE GRADE 7 MATHEMATICS STRAND 6 PATTERN AND ALGEBRA SUB-STRAND 1: NUMBER PATTERNS SUB-STRAND 2: DIRECTED NUMBERS SUB-STRAND 3: INDICES SUB-STARND 4: ALGEBRA GR 7 MATHEMATICS S6 2 ACKNOWLEDGEMENT Acknowledgements We acknowledge the contributions of all Secondary and Upper Primary Teachers who in one way or another helped to develop this Course. Special thanks to the Staff of the mathematics Department of FODE who played active role in coordinating writing workshops, outsourcing lesson writing and editing processes, involving selected teachers of Madang, Central Province and NCD. We also acknowledge the professional guidance provided by the Curriculum Development and Assessment Division throughout the processes of writing and, the services given by the members of the Mathematics Review and Academic Committees. The development of this book was co-funded by GoPNG and World Bank. MR. DEMAS TONGOGO Principal- FODE . Written by: Luzviminda B. Fernandez SCO-Mathematics Department Flexible Open and Distance Education Papua New Guinea Published in 2016 @ Copyright 2016, Department of Education Papua New Guinea All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic, mechanical, photocopying, recording or any other form of reproduction by any process is allowed without the prior permission of the publisher. ISBN: 978 - 9980 - 87 - 250 - 0 National Library Services of Papua New Guinea Printed by the Flexible, Open and Distance Education GR 7 MATHEMATICS S6 3 CONTENTS CONTENTS Page Secretary‟s Message…………………………………….…………………………………......... 4 Strand Introduction…………………………………….…………………………………………. 5 Study Guide………………………………………………….……………………………………. 6 SUB-STRAND 1: NUMBER PATTERNS ……………...….……….……………..………..