DOCSLIB.ORG

Natural Numbers: N= {1,2,3,· · ·} Integers: Z

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Natural Numbers: N= {1,2,3,· · ·} Integers: Z Real Numbers: Natural Numbers: N= f1; 2; 3; · · ·} Integers: Z= f0; −1; 1; −2; 2; −3; 3; · · ·} Note that every natural number is an integer. There are integers (negative numbers) that are not natural num- bers. 2 −1 13 Rational Numbers: Q=f ; ; 0; 3; − ; · · ·} 5 3 4 A rational number is a ratio of two integers. Note that 8 every integer is a rational number (e.g. 8 = ), but 1 there are rational numbers that are not integers (frac- 2 tions whose denominator is not 1. E.g. ). 3 When expressed as a decimal, every rational number ei- ther terminate after some point, or continue indefinitely in a repeated pattern. E.g. 3 = 0:6 5 79 = 2:393939 ··· = 2:39 33 1 Irrational numbers are numbers thatp cannot be ex- pressed as ratios of two integers. e.g. 2, π. When ex- pressed as decimals, irrational numbers don't terminate and don't repeat. E.g. π = 3:14159 ··· Note that rational numbers are never irrational and vice versa. Putting together rational and irrational numbers give us the set of Real Numbers, R. I.e., real numbers are the collection of all the rational and irrational numbers. Please note that the terms natural, rational, irra- tional, when used in the above case, are not adjectives, but nouns. The set of Real Numbers is a specific collec- tion of numbers that have particular properties in math- ematics. The word real is not used to describe the nature of the numbers and does not imply that the real numbers are any more or less real (used as a description) than any other numbers. 2 An inequality involves one of four inequality symbols: < less than > greater than ≤ less than or equal to ≥ greater than or equal to 3 < 5 is a true statement 5 < 5 is a false statement 5 ≤ 5 is a true statement We can graph an inequality statement like x < 10 on a number line. We use open parentheses ( ) for < or >, and use bracket [ ] for ≤ or ≥. 3 The absolute value of a number is the distance the number is from zero. Algebraically, 8 < x if x ≥ 0 jxj = : −x if x < 0 To add two negative numbers, add the absolute value of the two numbers, and turn the result into negative. E.g. −4 + −3 = −(4 + 3) = −7 To add a positive number with a negative number. First determine which number has a larger absolute value. If the negative number has a larger absolute value, the re- sult is negative, otherwise the result is positive. Subtract the smaller absolute value from the larger absolute value and assign a sign as given above. E.g. −5 + 3 = −(5 − 3) = −2 To subtract a negative number is to add the positive: E.g. 3 − −4 = 3 + 4 = 7 To add a negative is to subtract the positive: 4 + −6 = 4 − 6 = −2 The product of two negative numbers is positive. The product of a negative number with a positive number is negative. Note: j6 − 3j 6= 6 + 3 The absolute value does not change any operation inside it, the absolute value only changes a negative number to its opposite when the final result inside the absolute value, after all the operations are done, is a negative 4 number. 5 Exponents: Any number to a positive integer power means that num- ber times itself that many times. xn = x · x · x ··· x | {z } n times E.g. x3 = x · x · x E.g. 34 = 3 · 3 · 3 · 3 = 81 E.g. 25 = 2 · 2 · 2 · 2 · 2 = 32. In evaluating an expression, we need to always follow the Order of Operations: Parenthesis, acting as a grouping symbol, always has the highest priority. Exponents, or raising to a power, is the next highest priority. Multiplication and division should be done after expo- nents, they have the same priority. If an expression has both multiplication and division written, it should be evaluated from left to right. Addition and subtraction has the same, lowest priority. E.g. 4 + 32 − (2 + 3) + 12 − 2 · 5 = 4 + 32 − 5 + 12 − 2 · 5 = 4 + 9 − 5 + 12 − 2 · 5 = 4 + 9 − 5 + 12 − 10 = 13 − 5 + 12 − 10 6 = 8 + 12 − 10 = 20 − 10 = 10 7 Simplifying Algebraic Expressions: A variable is a letter used to represent a number. A variable is used in one of two purposes: 1. It acts as a pronoun for an unknown quantity, for ex- ample, we may not know the age of John, but we know that John is 5 years older than Mary, we can let x rep- resent the age of John, and y represent the age of Mary, we can then write x = y + 5. 2. A variable also can be used as a generic expression to represent a collection of numbers (or other objects) that share a particular characteristic when we need to write formulas about the numbers: E.g. For any real number x, y, we have: x + y = y + x. A constant is a number whose value is fixed. The num- ber 3, for example, is a constant. A constant can also be represented by a letter (or any other symbol) as long as it is understood that the number represented by the letter has a fixed value. An expression is a collection of variables and constants with mathematical operations: E.g. 3x + 5 p E.g. x + 4y − x Each expression can have a value if each variable of the expression is assigned a value. For example, in the ex- pression: 5x + 3y − 4z, if x = 4, y = 1, and z = 2, then the expression has the value: 8 5(4) + 3(1) − 4(2) = 20 + 3 − 8 = 15. To find the value of the expression given the values of the variables is to evaluate the expression. 9 If we have the equal sign (=) between two expressions, we have an equation. An equation can be true or false. E.g. 3 + 2 = 9 E.g. 4 − 2 = 2 E.g. x + 3 = 5 The first equation is always false, the second equation is always true (called an identity), and the third equation is sometimes true (if x = 2). To find the value(s) of the variable that would make the equation a true equation is to solve the equation. Please understand the difference between an expression and an equation. An expression does not need to be solved. An expression may be evaluated if all the values of the variables are known or assigned. An equation, on the other hand, can be solved. An equation is either true or false. 10 A term in an expression is separated by addition or sub- traction from other terms. E.g. 4x − 3xy + 2z has three terms, they are 4x, −3xy, and 2z. Note that we say the first term is 4x, not 4 and x, and we say that the second term is −3xy, because the sign in front of a term is part of the term. The constant (the number) in a term is called the coefficient of the term. In the example, the coefficient of the first term is 4, the coefficient of the second term is −3, and the coefficient of the third term is 2. If a term does not have a number explicitely written, the coefficient is 1. E.g. x + 3y. In this expression with two terms, the coefficient of the first term is 1. E.g. 4x2 −y+12xz2. In this expression with three terms, the coefficient of the second term is −1. Two terms are like terms if they involve the same vari- ables to the same power: E.g. 4x2y, 6x2y are like terms, they both have variables x and y, and the power of x are both 2, and the power of y are both 1. E.g. 3x, 4xy are unlike terms, since the first term does not have y. E.g. 4x2, −8x are unlike terms, because the first term has x to the 2 power, and second term has x to the first power. If the terms are alike, we can combine like terms by 11 adding their coefficients. E.g. 3x + 5x = 8x E.g. 4x2 + 3x − 2x2 = 2x2 + 3x E.g. x + 4x = 5x. Notice in the third example that there is the invisible coefficient of 1 in the first term, and that's why the sim- plified expression is 5x. When simplifying an expression, we are trying to write an expression to an equivalent expression that is easier to read or evaluate. Two expressions are equivalent if they would have the same value for the same values of the variables. In the above 3 examples, the expression on the right is equiva- lent to the expression on the left. 12 Properties of the Real Numbers: For any real numbers x, y, z, Commutative property of Addition: x + y = y + x Commutative property of Multiplication: xy = yx Associative property of Addition: (x + y) + z = x + (y + z) Associative property of Multiplication: (xy)z = x(yz) Distributive property x(y + z) = xy + xz (y + z)x = yx + zx = xy + xz x(y − z) = xy − xz (y − z)x = yx − zx = xy − xz Notice that when we use the distributive property with a negative number outside of the parentheses, the negative sign will also need to be distributed. E.g. 4x − 3(x − 5) = 4x − 3x + 15 = x + 15. Notice that the negative sign in front of 3 is also dis- tributed to the x (making it −3x) and to the −5, making it +15.
Load more

Recommended publications
  • The Orthogonal Planes Split of Quaternions and Its Relation to Quaternion Geometry of Rotations
    Home Search Collections Journals About Contact us My IOPscience The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Conf. Ser. 597 012042 (http://iopscience.iop.org/1742-6596/597/1/012042) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 131.169.4.70 This content was downloaded on 17/02/2016 at 22:46 Please note that terms and conditions apply. 30th International Colloquium on Group Theoretical Methods in Physics (Group30) IOP Publishing Journal of Physics: Conference Series 597 (2015) 012042 doi:10.1088/1742-6596/597/1/012042 The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations1 Eckhard Hitzer Osawa 3-10-2, Mitaka 181-8585, International Christian University, Japan E-mail: [email protected] Abstract. Recently the general orthogonal planes split with respect to any two pure unit 2 2 quaternions f; g 2 H, f = g = −1, including the case f = g, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [7]. Applications include color image processing, where the orthogonal planes split with f = g = the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations [3], that the pure quaternion units f; g and the analysis planes, which they define, play a key role in the geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit quaternions, as relevant for texture analysis in crystallography.
    [Show full text]
  • The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
    To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition.
    [Show full text]
  • Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS
    Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS Students will be able to: Understand the meanings of Irrational Numbers Key Vocabulary: • Irrational Numbers • Examples of Rational Numbers and Irrational Numbers • Decimal expansion of Irrational Numbers • Steps for representing Irrational Numbers on number line IRRATIONAL NUMBERS A rational number is a number that can be expressed as a ratio or we can say that written as a fraction. Every whole number is a rational number, because any whole number can be written as a fraction. Numbers that are not rational are called irrational numbers. An Irrational Number is a real number that cannot be written as a simple fraction or we can say cannot be written as a ratio of two integers. The set of real numbers consists of the union of the rational and irrational numbers. If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square, and √2 is irrational. EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Rational Number The number 7 is a rational number because it can be written as the 7 fraction . 1 The number 0.1111111….(1 is repeating) is also rational number 1 because it can be written as fraction . 9 EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Irrational Numbers The square root of 2 is an irrational number because it cannot be written as a fraction √2 = 1.4142135…… Pi(휋) is also an irrational number. π = 3.1415926535897932384626433832795 (and more...) 22 The approx. value of = 3.1428571428571..
    [Show full text]
  • 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.
    [Show full text]
  • 81684-1B SP.Pdf
    INTERMEDIATE ALGEBRA The Why and the How First Edition Mathew Baxter Florida Gulf Coast University Bassim Hamadeh, CEO and Publisher Jess Estrella, Senior Graphic Designer Jennifer McCarthy, Acquisitions Editor Gem Rabanera, Project Editor Abbey Hastings, Associate Production Editor Chris Snipes, Interior Designer Natalie Piccotti, Senior Marketing Manager Kassie Graves, Director of Acquisitions Jamie Giganti, Senior Managing Editor Copyright © 2018 by Cognella, Inc. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of Cognella, Inc. For inquiries regarding permissions, translations, foreign rights, audio rights, and any other forms of reproduction, please contact the Cognella Licensing Department at [email protected]. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Cover image copyright © Depositphotos/agsandrew. Printed in the United States of America ISBN: 978-1-5165-0303-2 (pbk) / 978-1-5165-0304-9 (br) iii |  CONTENTS 1 The Real Number System 1 RealNumbersandTheirProperties 2:1.1 RealNumbers 2 PropertiesofRealNumbers 6 Section1.1Exercises 9 TheFourBasicOperations 11:1.2 AdditionandSubtraction 11 MultiplicationandDivision
    [Show full text]
  • Is Liscussed and a Class of Spaces in Yhich the Potian of Dittance Is Defin Time Sd-Called Metric Spaces, Is Introduced
    DOCUMENT RESEW Sit 030 465 - ED in ) AUTH3R Shreider, Yu A. TITLE What Is Distance? Populatt Lectu'res in Mathematios. INSTITUTI3N :".hicago Univ., Ill: Dept. of Mathematics. SPONS A3ENCY National Science Foundation, Washington, D.C. P.11B DATE 74 GRANT NSF-G-834 7 (M Al NOTE Blip.; For.relateE documents,see SE- 030 460-464: Not availabie in hard copy due :to copyright restrActi. cals. Translated and adapted from the° Russian edition. AVAILABLE FROM!he gniversity ofwChicago Press, Chicago, IL 60637 (Order No. 754987; $4.501. EDRS- PRICL: MF0-1,Plus-Postage. PC Not Available from ELMS. DESCRIPTORS College Mathemitties; Geometric Concepts; ,Higker Education; Lecture Method; Mathematics; Mathematics Curriculum; *Mathematics Education; *tlatheatatic ,Instruction; *Measurment; Secondary Education; *Secondary echool Mathematics IDENKFIERp *Distance (Mathematics); *Metric Spaces ABSTRACT Rresened is at elaboration of a course given at Moscow University for-pupils in nifith and tenthgrades/eTh development through ibstraction of the'general definiton of,distance is liscussed and a class of spaces in yhich the potian of dittance is defin time sd-called metric spaces, is introduced. The gener.al ..oon,ce V of diztance is related to a large number of mathematical phenomena. (Author/MO 14. R producttions supplied by EDRS are the best that can be made from the original document. ************************t**************:********.********************** , . THIStiocuMENT HAS BEEN REgoltb., OOCEO EXACTve AVECCEIVEO fROM- THE 'PERSON OR.OROANIZATIONC41OIN- AIR** IT PONTS Of Vie*41010114IONS4- ! STATED 00 NOT NECESSAACT REPRE SEATOFFICIAL NATIONAL INSTITUTIEOF IMOUCATIOH P9SITION OR POLKY ;.- r ilk 4 #. .f.) ;4',C; . fr; AL"' ' . , ... , , AV Popular Lectures in Matheniatics Survey of Recent East European Mathernatical- Literature A project conducted by Izaak Wirszup, .
    [Show full text]
  • The Unique Natural Number Set and Distributed Prime Numbers
    mp Co uta & tio d n e a li Jabur and Kushnaw, J Appl Computat Math 2017, 6:4 l p M p a Journal of A t h DOI: 10.4172/2168-9679.1000368 f e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access The Unique Natural Number Set and Distributed Prime Numbers Alabed TH1* and Bashir MB2 1Computer Science, Shendi University, Shendi, Sudan 2Shendi University, Shendi, Sudan Abstract The Natural number set has a new number set behind three main subset: odd, even and prime which are unique number set. It comes from a mathematical relationship between the two main types even and odd number set. It consists of prime and T-semi-prime numbers set. However, finding ways to understand how prime number was distributed was ambiguity and it is impossible, this new natural set show how the prime number was distributed and highlight focusing on T-semi-prime number as a new sub natural set. Keywords: Prime number; T-semi-prime number; Unique number purely mathematical sciences and it has been used prime number set within theoretical physics to try and produce a theory that ties together prime numbers with quantum chaos theory [3]. However, researchers Introduction were made a big effort to find algorithm which can describe how prime The Unique natural number set is sub natural number set which number are distributed, still distribution of prime numbers throughout consist of prime and T-semi-prime numbers. T-semi-prime is natural a list of integers leads to the emergence of many unanswered and number divided by one or more prime numbers.
    [Show full text]
  • Hypercomplex Algebras and Their Application to the Mathematical
    Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H.
    [Show full text]
  • New Formulas for Semi-Primes. Testing, Counting and Identification
    New Formulas for Semi-Primes. Testing, Counting and Identification of the nth and next Semi-Primes Issam Kaddouraa, Samih Abdul-Nabib, Khadija Al-Akhrassa aDepartment of Mathematics, school of arts and sciences bDepartment of computers and communications engineering, Lebanese International University, Beirut, Lebanon Abstract In this paper we give a new semiprimality test and we construct a new formula for π(2)(N), the function that counts the number of semiprimes not exceeding a given number N. We also present new formulas to identify the nth semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots 3 of N : P , P ....P 3 √N. 1 2 π( √N) ≤ Keywords: prime, semiprime, nth semiprime, next semiprime 1. Introduction Securing data remains a concern for every individual and every organiza- tion on the globe. In telecommunication, cryptography is one of the studies that permits the secure transfer of information [1] over the Internet. Prime numbers have special properties that make them of fundamental importance in cryptography. The core of the Internet security is based on protocols, such arXiv:1608.05405v1 [math.NT] 17 Aug 2016 as SSL and TSL [2] released in 1994 and persist as the basis for securing dif- ferent aspects of today’s Internet [3]. The Rivest-Shamir-Adleman encryption method [4], released in 1978, uses asymmetric keys for exchanging data. A secret key Sk and a public key Pk are generated by the recipient with the following property: A message enciphered Email addresses: [email protected] (Issam Kaddoura), [email protected] (Samih Abdul-Nabi) 1 by Pk can only be deciphered by Sk and vice versa.
    [Show full text]
  • From Counting to Quaternions -- the Agonies and Ecstasies of the Student Repeat Those of D'alembert and Hamilton
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Keck Graduate Institute Journal of Humanistic Mathematics Volume 1 | Issue 1 January 2011 From Counting to Quaternions -- The Agonies and Ecstasies of the Student Repeat Those of D'Alembert and Hamilton Reuben Hersh University of New Mexico Follow this and additional works at: https://scholarship.claremont.edu/jhm Recommended Citation Hersh, R. "From Counting to Quaternions -- The Agonies and Ecstasies of the Student Repeat Those of D'Alembert and Hamilton," Journal of Humanistic Mathematics, Volume 1 Issue 1 (January 2011), pages 65-93. DOI: 10.5642/jhummath.201101.06 . Available at: https://scholarship.claremont.edu/jhm/vol1/ iss1/6 ©2011 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. From Counting to Quaternions { The Agonies and Ecstasies of the Student Repeat Those of D'Alembert and Hamilton Reuben Hersh Department of Mathematics and Statistics, The University of New Mexico [email protected] Synopsis Young learners of mathematics share a common experience with the greatest creators of mathematics: \hitting a wall," meaning, first frustration, then strug- gle, and finally, enlightenment and elation.
    [Show full text]
  • The Evolution of Numbers
    The Evolution of Numbers Counting Numbers Counting Numbers: {1, 2, 3, …} We use numbers to count: 1, 2, 3, 4, etc You can have "3 friends" A field can have "6 cows" Whole Numbers Whole numbers are the counting numbers plus zero. Whole Numbers: {0, 1, 2, 3, …} Negative Numbers We can count forward: 1, 2, 3, 4, ...... but what if we count backward: 3, 2, 1, 0, ... what happens next? The answer is: negative numbers: {…, -3, -2, -1} A negative number is any number less than zero. Integers If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers: {…, -3, -2, -1, 0, 1, 2, 3, …} The Integers include zero, the counting numbers, and the negative counting numbers, to make a list of numbers that stretch in either direction indefinitely. Rational Numbers A rational number is a number that can be written as a simple fraction (i.e. as a ratio). 2.5 is rational, because it can be written as the ratio 5/2 7 is rational, because it can be written as the ratio 7/1 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3 More formally we say: A rational number is a number that can be written in the form p/q where p and q are integers and q is not equal to zero. Example: If p is 3 and q is 2, then: p/q = 3/2 = 1.5 is a rational number Rational Numbers include: all integers all fractions Irrational Numbers An irrational number is a number that cannot be written as a simple fraction.
    [Show full text]
  • 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.
    [Show full text]