Variables and Calculations
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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. -
C Constants and Literals Integer Literals Floating-Point Literals
C Constants and Literals The constants refer to fixed values that the program may not alter during its execution. These fixed values are also called literals. Constants can be of any of the basic data types like an integer constant, a floating constant, a character constant, or a string literal. There are also enumeration constants as well. The constants are treated just like regular variables except that their values cannot be modified after their definition. Integer literals An integer literal can be a decimal, octal, or hexadecimal constant. A prefix specifies the base or radix: 0x or 0X for hexadecimal, 0 for octal, and nothing for decimal. An integer literal can also have a suffix that is a combination of U and L, for unsigned and long, respectively. The suffix can be uppercase or lowercase and can be in any order. Here are some examples of integer literals: Floating-point literals A floating-point literal has an integer part, a decimal point, a fractional part, and an exponent part. You can represent floating point literals either in decimal form or exponential form. While representing using decimal form, you must include the decimal point, the exponent, or both and while representing using exponential form, you must include the integer part, the fractional part, or both. The signed exponent is introduced by e or E. Here are some examples of floating-point literals: 1 | P a g e Character constants Character literals are enclosed in single quotes, e.g., 'x' and can be stored in a simple variable of char type. A character literal can be a plain character (e.g., 'x'), an escape sequence (e.g., '\t'), or a universal character (e.g., '\u02C0'). -
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 -
PYTHON NOTES What Is Python?
PYTHON NOTES What is Python? Python is a popular programming language. It was created in 1991 by Guido van Rossum. It is used for: web development (server-side), software development, mathematics, system scripting. What can Python do? Python can be used on a server to create web applications. Python can connect to database systems. It can also read and modify files. Python can be used to handle big data and perform complex mathematics. Python can be used for production-ready software development. Why Python? Python works on different platforms (Windows, Mac, Linux, Raspberry Pi, etc). Python has a simple syntax similar to the English language. Python has syntax that allows developers to write programs with fewer lines than some other programming languages. Python runs on an interpreter system, meaning that code can be executed as soon as it is written. This means that prototyping can be very quick. Good to know The most recent major version of Python is Python 3, which we shall be using in this tutorial. However, Python 2, although not being updated with anything other than security updates, is still quite popular. Python Syntax compared to other programming languages Python was designed to for readability, and has some similarities to the English language with influence from mathematics. Python uses new lines to complete a command, as opposed to other programming languages which often use semicolons or parentheses. Python relies on indentation, using whitespace, to define scope; such as the scope of loops, functions and classes. Other programming languages often use curly-brackets for this purpose. Python Install Many PCs and Macs will have python already installed. -
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. -
Regular Expressions the Following Are Sections of Different Programming Language Specifications
Regular Expressions The following are sections of different programming language specifications. Write the corresponding regular expressions for each of the described lexical elements. 1) C Identifiers [Kernighan] An identifier is a sequence of letters and digits. The first character must be a letter; the underscore _ is considered a letter. Uppercase and lowercase letters are considered different. 2) Java 6 and previous Integer Literals [Gosling] An integer literal may be expressed in decimal (base 10), hexadecimal (base 16), or octal (base 8). An integer literal is of type long if it is suffixed with an ASCII letter L or l (ell); otherwise it is of type int. The suffix L is preferred, because the letter l (ell) is often hard to distinguish from the digit 1 (one). A decimal numeral is either the single ASCII character 0, representing the integer zero, or consists of an ASCII digit from 1 to 9, optionally followed by one or more ASCII digits from 0 to 9, representing a positive integer. An hexadecimal numeral consists of the leading ASCII characters 0x or 0X followed by one or more ASCII hexadecimal digits and can represent a positive, zero, or negative integer. Hexadecimal digits with values 10 through 15 are represented by the ASCII letters a through f or A through F, respectively; each letter used as a hexadecimal digit may be uppercase or lowercase. An octal numeral consists of an ASCII digit 0 followed by one or more of the ASCII digits 0 through 7 and can represent a positive, zero, or negative integer. Note that octal numerals always consist of two or more digits; 0 is always considered to be a decimal numeral-not that it matters much in practice, for the numerals 0, 00, and 0x0 all represent exactly the same integer value. -
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]. -
1.1 Place Value, Rounding, Comparing Whole Numbers
1.1 Place Value, Rounding, Comparing Whole Numbers Place Value Example: The number 13,652,103 would look like Millions Thousands Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 3 6 5 2 1 0 3 We’d read this in groups of three digits, so this number would be written thirteen million six hundred fifty two thousand one hundred and three Example: What is the place value of 4 in 6,342,105? The 4 is in the ten-thousands place Example: Write the value of two million, five hundred thousand, thirty six 2,500,036 Rounding When we round to a place value, we are looking for the closest number that has zeros in the digits to the right. Example: Round 173 to the nearest ten. Since we are rounding to the nearest ten, we want a 0 in the ones place. The two possible values are 170 or 180. 173 is closer to 170, so we round to 170. Example: Round 97,870 to the nearest thousand. The nearest values are 97,000 and 98,000. The closer value is 98,000. Example: Round 5,950 to the nearest hundred. The nearest values are 5,900 or 6,000. 5,950 is exactly halfway between, so by convention we round up, to 6,000. Comparing To compare to values, we look at which has the largest value in the highest place value. Example: Which is larger: 126 or 132? Both numbers are the same in the hundreds place, so we look in the tens place. -
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.