1.1 Place Value, Rounding, Comparing Whole Numbers
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Decimal Rounding
Mathematics Instructional Plan – Grade 5 Decimal Rounding Strand: Number and Number Sense Topic: Rounding decimals, through the thousandths, to the nearest whole number, tenth, or hundredth. Primary SOL: 5.1 The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth. Materials Base-10 blocks Decimal Rounding activity sheet (attached) “Highest or Lowest” Game (attached) Open Number Lines activity sheet (attached) Ten-sided number generators with digits 0–9, or decks of cards Vocabulary approximate, between, closer to, decimal number, decimal point, hundredth, rounding, tenth, thousandth, whole Student/Teacher Actions: What should students be doing? What should teachers be doing? 1. Begin with a review of decimal place value: Display a decimal in the thousandths place, such as 34.726, using base-10 blocks. In pairs, have students discuss how to read the number using place-value names, and review the decimal place each digit holds. Briefly have students share their thoughts with the class by asking: What digit is in the tenths place? The hundredths place? The thousandths place? Who would like to share how to say this decimal using place value? 2. Ask, “If this were a monetary amount, $34.726, how would we determine the amount to the nearest cent?” Allow partners or small groups to discuss this question. It may be helpful if students underlined the place that indicates cents (the hundredths place) in order to keep track of the rounding place. Distribute the Open Number Lines activity sheet and display this number line on the board. Guide students to recall that the nearest cent is the same as the nearest hundredth of a dollar. -
IEEE Standard 754 for Binary Floating-Point Arithmetic
Work in Progress: Lecture Notes on the Status of IEEE 754 October 1, 1997 3:36 am Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic Prof. W. Kahan Elect. Eng. & Computer Science University of California Berkeley CA 94720-1776 Introduction: Twenty years ago anarchy threatened floating-point arithmetic. Over a dozen commercially significant arithmetics boasted diverse wordsizes, precisions, rounding procedures and over/underflow behaviors, and more were in the works. “Portable” software intended to reconcile that numerical diversity had become unbearably costly to develop. Thirteen years ago, when IEEE 754 became official, major microprocessor manufacturers had already adopted it despite the challenge it posed to implementors. With unprecedented altruism, hardware designers had risen to its challenge in the belief that they would ease and encourage a vast burgeoning of numerical software. They did succeed to a considerable extent. Anyway, rounding anomalies that preoccupied all of us in the 1970s afflict only CRAY X-MPs — J90s now. Now atrophy threatens features of IEEE 754 caught in a vicious circle: Those features lack support in programming languages and compilers, so those features are mishandled and/or practically unusable, so those features are little known and less in demand, and so those features lack support in programming languages and compilers. To help break that circle, those features are discussed in these notes under the following headings: Representable Numbers, Normal and Subnormal, Infinite -
Variables and Calculations
¡ ¢ £ ¤ ¥ ¢ ¤ ¦ § ¨ © © § ¦ © § © ¦ £ £ © § ! 3 VARIABLES AND CALCULATIONS Now you’re ready to learn your first ele- ments of Python and start learning how to solve programming problems. Although programming languages have myriad features, the core parts of any programming language are the instructions that perform numerical calculations. In this chapter, we’ll explore how math is performed in Python programs and learn how to solve some prob- lems using only mathematical operations. ¡ ¢ £ ¤ ¥ ¢ ¤ ¦ § ¨ © © § ¦ © § © ¦ £ £ © § ! Sample Program Let’s start by looking at a very simple problem and its Python solution. PROBLEM: THE TOTAL COST OF TOOTHPASTE A store sells toothpaste at $1.76 per tube. Sales tax is 8 percent. For a user-specified number of tubes, display the cost of the toothpaste, showing the subtotal, sales tax, and total, including tax. First I’ll show you a program that solves this problem: toothpaste.py tube_count = int(input("How many tubes to buy: ")) toothpaste_cost = 1.76 subtotal = toothpaste_cost * tube_count sales_tax_rate = 0.08 sales_tax = subtotal * sales_tax_rate total = subtotal + sales_tax print("Toothpaste subtotal: $", subtotal, sep = "") print("Tax: $", sales_tax, sep = "") print("Total is $", total, " including tax.", sep = ") Parts of this program may make intuitive sense to you already; you know how you would answer the question using a calculator and a scratch pad, so you know that the program must be doing something similar. Over the next few pages, you’ll learn exactly what’s going on in these lines of code. For now, enter this program into your Python editor exactly as shown and save it with the required .py extension. Run the program several times with different responses to the question to verify that the program works. -
Introduction to the Python Language
Introduction to the Python language CS111 Computer Programming Department of Computer Science Wellesley College Python Intro Overview o Values: 10 (integer), 3.1415 (decimal number or float), 'wellesley' (text or string) o Types: numbers and text: int, float, str type(10) Knowing the type of a type('wellesley') value allows us to choose the right operator when o Operators: + - * / % = creating expressions. o Expressions: (they always produce a value as a result) len('abc') * 'abc' + 'def' o Built-in functions: max, min, len, int, float, str, round, print, input Python Intro 2 Concepts in this slide: Simple Expressions: numerical values, math operators, Python as calculator expressions. Input Output Expressions Values In [...] Out […] 1+2 3 3*4 12 3 * 4 12 # Spaces don't matter 3.4 * 5.67 19.278 # Floating point (decimal) operations 2 + 3 * 4 14 # Precedence: * binds more tightly than + (2 + 3) * 4 20 # Overriding precedence with parentheses 11 / 4 2.75 # Floating point (decimal) division 11 // 4 2 # Integer division 11 % 4 3 # Remainder 5 - 3.4 1.6 3.25 * 4 13.0 11.0 // 2 5.0 # output is float if at least one input is float 5 // 2.25 2.0 5 % 2.25 0.5 Python Intro 3 Concepts in this slide: Strings and concatenation string values, string operators, TypeError A string is just a sequence of characters that we write between a pair of double quotes or a pair of single quotes. Strings are usually displayed with single quotes. The same string value is created regardless of which quotes are used. -
Programming for Computations – Python
15 Svein Linge · Hans Petter Langtangen Programming for Computations – Python Editorial Board T. J.Barth M.Griebel D.E.Keyes R.M.Nieminen D.Roose T.Schlick Texts in Computational 15 Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick More information about this series at http://www.springer.com/series/5151 Svein Linge Hans Petter Langtangen Programming for Computations – Python A Gentle Introduction to Numerical Simulations with Python Svein Linge Hans Petter Langtangen Department of Process, Energy and Simula Research Laboratory Environmental Technology Lysaker, Norway University College of Southeast Norway Porsgrunn, Norway On leave from: Department of Informatics University of Oslo Oslo, Norway ISSN 1611-0994 Texts in Computational Science and Engineering ISBN 978-3-319-32427-2 ISBN 978-3-319-32428-9 (eBook) DOI 10.1007/978-3-319-32428-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2016945368 Mathematic Subject Classification (2010): 26-01, 34A05, 34A30, 34A34, 39-01, 40-01, 65D15, 65D25, 65D30, 68-01, 68N01, 68N19, 68N30, 70-01, 92D25, 97-04, 97U50 © The Editor(s) (if applicable) and the Author(s) 2016 This book is published open access. Open Access This book is distributed under the terms of the Creative Commons Attribution-Non- Commercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated. -
Efficient, Arbitrarily High Precision Hardware Logarithmic Arithmetic For
Efficient, arbitrarily high precision hardware logarithmic arithmetic for linear algebra Jeff Johnson Facebook AI Research Los Angeles, CA, United States [email protected] Abstract—The logarithmic number system (LNS) is arguably cannot easily apply precision reduction, such as hyperbolic not broadly used due to exponential circuit overheads for embedding generation [5] or structure from motion via matrix summation tables relative to arithmetic precision. Methods to factorization [6], yet provide high local data reuse potential. reduce this overhead have been proposed, yet still yield designs with high chip area and power requirements. Use remains limited The logarithmic number system (LNS) [7] can provide to lower precision or high multiply/add ratio cases, while much energy efficiency by eliminating hardware multipliers and of linear algebra (near 1:1 multiply/add ratio) does not qualify. dividers, yet maintains significant computational overhead We present a dual-base approximate logarithmic arithmetic with Gaussian logarithm functions needed for addition and comparable to floating point in use, yet unlike LNS it is easily subtraction. While reduced precision cases can limit them- fully pipelined, extendable to arbitrary precision with O(n2) overhead, and energy efficient at a 1:1 multiply/add ratio. selves to relatively small LUTs/ROMs, high precision LNS Compared to float32 or float64 vector inner product with FMA, require massive ROMs, linear interpolators and substantial our design is respectively 2.3× and 4.6× more energy efficient MUXes. Pipelining is difficult, requiring resource duplication in 7 nm CMOS. It depends on exp and log evaluation 5.4× and or handling variable latency corner cases as seen in [8]. -
Interval Arithmetic with Fixed Rounding Mode
NOLTA, IEICE Paper Interval arithmetic with fixed rounding mode Siegfried M. Rump 1 ,3 a), Takeshi Ogita 2 b), Yusuke Morikura 3 , and Shin’ichi Oishi 3 1 Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstraße 95, Hamburg 21071, Germany 2 Division of Mathematical Sciences, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan 3 Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169–8555, Japan a) [email protected] b) [email protected] Received December 10, 2015; Revised March 25, 2016; Published July 1, 2016 Abstract: We discuss several methods to simulate interval arithmetic operations using floating- point operations with fixed rounding mode. In particular we present formulas using only rounding to nearest and using only chop rounding (towards zero). The latter was the default and only rounding on GPU (Graphics Processing Unit) and cell processors, which in turn are very fast and therefore attractive in scientific computations. Key Words: rounding mode, chop rounding, interval arithmetic, predecessor, successor, IEEE 754 1. Notation There is a growing interest in so-called verification methods, i.e. algorithms with rigorous verification of the correctness of the result. Such algorithms require error estimations of the basic floating-point operations. A specifically convenient way to do this is using rounding modes as defined in the IEEE 754 floating- point standard [1]. In particular performing an operation in rounding downwards and rounding upwards computes the narrowest interval with floating-point bounds including the correct value of the operation. However, such an approach requires frequent changing of the rounding mode. -
SIGNIFICANT FIGURES, EXPONENTS, and SCIENTIFIC NOTATION ©2004, 1990 by David A
SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION ©2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES Significant figures are those digits necessary to express the results of a measurement to the precision with which it was made. No measurement is ever absolutely correct since every measurement is limited by the accuracy or reliability of the measuring instrument used. For example, if a thermometer is graduated in one degree intervals and the temperature indicated by the mercury column is between 55°C and 56°C, then the temperature can be read precisely only to the nearest degree (55°C or 56°C, whichever is closer). If the graduations are sufficiently spaced, the fractional degrees between 55°C and 56°C can be estimated to the nearest tenth of a degree. If a more precise measurement is required, then a more precise measuring instrument (e.g., a thermometer graduated in one-tenth degree intervals) can be used. This will increase the number of significant figures in the reported measurement. (See Figure 1) Figure 1. A typical Laboratory Thermometer graduated in °C. In dealing with measurements and significant figures the following terms must be understood: Precision tells the reproducibility of a particular measurement or how often a particular measurement will repeat itself in a series of measurements. Accuracy tells how close the measured value is to a known or standard accepted value of the same measurement. Measurements showing a high degree of precision do not always reflect a high degree of accuracy nor does a high degree of accuracy mean that a high degree of precision has been obtained. -
Numbers in Javascript
NUMBERS IN JAVASCRIPT Math constants JavaScript understands numbers in several formats, allowing you to specify numbers in hex, decimal, and octal. If a 0 precedes a number and there is no number higher than 7, the number is considered to be in octal (base 8) and if the number is preceded by 0x, then it is considered to be in hex (base 16) and can also contain characters A, B, C, D, E, F. Neither may contain decimal points. With decimal numbers, 12e+4 may be used to replace 12x104 and 12e-4 may be used to replace 12x10-4 etc. There are a few maths constants (or properties) that are held by the Math object and can be used as variables, These values are read only, you cannot assign values to them. The available Math object constants are: Math object constants Value Mathematical Math object property (approx) equivalent Math.E 2.718 e Math.LN2 0.693 ln(2) Math.LN10 2.303 ln(10) Math.LOG2E 1.442 log2(e) Math.LOG10E 0.434 log10(e) Math.PI 3.142 Pi Math.SQRT1_2 0.707 (sqrt(2))-1 or sqrt(1/2) Math.SQRT2 1.414 sqrt(2) Precision - javascript Math.round function The concepts of accuracy and precision are often confused. The accuracy of a number x is given by the number of significant decimal (or other) digits to the right of the decimal point in x. the precision of x is the total number of significant decimal (or other) digits. The Math.round() function will round a number (up or down) to the nearest integer. -
Pascal Page 1
P A S C A L Mini text by Paul Stewart Ver 3.2 January 93 reprint 2000 PASCAL PAGE 1 TABLE OF CONTENTS LESSON 1 ...........................4 A BRIEF HISTORY........................4 PROGRAM............................5 VAR..............................5 Rules for Variable Names ...................6 BEGIN and END.........................6 arithmetic operators .....................7 Order of operation ......................8 WRITELN............................8 LESSON 2 ...........................9 WHILE-DO .......................... 10 relational operators .................... 11 READLN ........................... 12 ASSIGNMENT LESSON 2..................... 19 LESSON 3 .......................... 20 Mixed Mode Arithmetic.................... 20 REALS & INTEGERS ...................... 20 IF THEN ELSE statement................... 22 Finished .......................... 24 IF statement........................ 25 P. G. STEWART January 93 PAGE 2 PASCAL ASSIGNMENT LESSON 3..................... 26 LESSON 4 .......................... 27 REPEAT UNTIL loop...................... 27 FOR DO loop......................... 28 ASSIGNMENT LESSON 4..................... 31 LESSON 5 .......................... 32 WRITE statement....................... 32 ORMATTING OF OUTPUT..................... 33 CHARACTERS ......................... 34 CASE statement ....................... 36 ASSIGNMENT LESSON 5..................... 40 LESSON 6 .......................... 41 READ statement ....................... 41 EOLN ............................ 41 ASSIGNMENT LESSON 6.................... -
Learn Pascal.Pdf
Index • Introduction • History of Pascal • Pascal Compilers • Hello, world. • Basics o Program Structure o Identifiers o Constants o Variables and Data Types o Assignment and Operations o Standard Functions o Punctuation and Indentation o Programming Assignment o Solution • Input/Output o Input o Output o Formatting output o Files o EOLN and EOF o Programming Assignment o Solution • Program Flow o Sequential control o Boolean Expressions o Branching . IF . CASE o Looping . FOR..DO . WHILE..DO . REPEAT..UNTIL o Programming Assignments: Fibonacci Sequence and Powers of Two o Solutions • Subprograms o Procedures o Parameters o Functions o Scope o Recursion o Forward Referencing o Programming Assignment: the Towers of Hanoi o Solution • Data types o Enumerated types o Subranges o 1-dimensional arrays o Multidimensional arrays o Records o Pointers • Final words 2 Introduction Welcome to Learn Pascal! This tutorial is an introduction to the Pascal simple, yet complete, introduction to the Pascal programming language. It covers all of the syntax of standard Pascal, including pointers. I have tried to make things are clear as possible. If you don't understand anything, try it in your Pascal compiler and tweak things a bit. Pascal was designed for teaching purposes, and is a very structured and syntactically-strict language. This means the compiler will catch more beginner errors and yield more beginner-friendly error messages than with a shorthand-laden language such as C or PERL. This tutorial was written for beginner programmers, so assumes no knowledge. At the same time, a surprising number of experienced programmers have found the tutorial a useful reference source for picking up Pascal. -
Complete Interval Arithmetic and Its Implementation on the Computer
Complete Interval Arithmetic and its Implementation on the Computer Ulrich W. Kulisch Institut f¨ur Angewandte und Numerische Mathematik Universit¨at Karlsruhe Abstract: Let IIR be the set of closed and bounded intervals of real numbers. Arithmetic in IIR can be defined via the power set IPIR of real numbers. If divisors containing zero are excluded, arithmetic in IIR is an algebraically closed subset of the arithmetic in IPIR, i.e., an operation in IIR performed in IPIR gives a result that is in IIR. Arithmetic in IPIR also allows division by an interval that contains zero. Such division results in closed intervals of real numbers which, however, are no longer bounded. The union of the set IIR with these new intervals is denoted by (IIR). This paper shows that arithmetic operations can be extended to all elements of the set (IIR). Let F ⊂ IR denote the set of floating-point numbers. On the computer, arithmetic in (IIR) is approximated by arithmetic in the subset (IF ) of closed intervals with floating- point bounds. The usual exceptions of floating-point arithmetic like underflow, overflow, division by zero, or invalid operation do not occur in (IF ). Keywords: computer arithmetic, floating-point arithmetic, interval arithmetic, arith- metic standards. 1 Introduction or a Vision of Future Computing Computers are getting ever faster. On all relevant processors floating-point arith- metic is provided by fast hardware. The time can already be foreseen when the P C will be a teraflops computer. With this tremendous computing power scientific computing will experience a significant shift from floating-point arithmetic toward increased use of interval arithmetic.