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Elementary Number Theory and Methods of Proof
CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright © Cengage Learning. All rights reserved. SECTION 4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Copyright © Cengage Learning. All rights reserved. Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem The quotient-remainder theorem says that when any integer n is divided by any positive integer d, the result is a quotient q and a nonnegative remainder r that is smaller than d. 3 Example 1 – The Quotient-Remainder Theorem For each of the following values of n and d, find integers q and r such that and a. n = 54, d = 4 b. n = –54, d = 4 c. n = 54, d = 70 Solution: a. b. c. 4 div and mod 5 div and mod A number of computer languages have built-in functions that enable you to compute many values of q and r for the quotient-remainder theorem. These functions are called div and mod in Pascal, are called / and % in C and C++, are called / and % in Java, and are called / (or \) and mod in .NET. The functions give the values that satisfy the quotient-remainder theorem when a nonnegative integer n is divided by a positive integer d and the result is assigned to an integer variable. 6 div and mod However, they do not give the values that satisfy the quotient-remainder theorem when a negative integer n is divided by a positive integer d. 7 div and mod For instance, to compute n div d for a nonnegative integer n and a positive integer d, you just divide n by d and ignore the part of the answer to the right of the decimal point. -
Introduction to Uncertainties (Prepared for Physics 15 and 17)
Introduction to Uncertainties (prepared for physics 15 and 17) Average deviation. When you have repeated the same measurement several times, common sense suggests that your “best” result is the average value of the numbers. We still need to know how “good” this average value is. One measure is called the average deviation. The average deviation or “RMS deviation” of a data set is the average value of the absolute value of the differences between the individual data numbers and the average of the data set. For example if the average is 23.5cm/s, and the average deviation is 0.7cm/s, then the number can be expressed as (23.5 ± 0.7) cm/sec. Rule 0. Numerical and fractional uncertainties. The uncertainty in a quantity can be expressed in numerical or fractional forms. Thus in the above example, ± 0.7 cm/sec is a numerical uncertainty, but we could also express it as ± 2.98% , which is a fraction or %. (Remember, %’s are hundredths.) Rule 1. Addition and subtraction. If you are adding or subtracting two uncertain numbers, then the numerical uncertainty of the sum or difference is the sum of the numerical uncertainties of the two numbers. For example, if A = 3.4± .5 m and B = 6.3± .2 m, then A+B = 9.7± .7 m , and A- B = - 2.9± .7 m. Notice that the numerical uncertainty is the same in these two cases, but the fractional uncertainty is very different. Rule2. Multiplication and division. If you are multiplying or dividing two uncertain numbers, then the fractional uncertainty of the product or quotient is the sum of the fractional uncertainties of the two numbers. -
Unit 6: Multiply & Divide Fractions Key Words to Know
Unit 6: Multiply & Divide Fractions Learning Targets: LT 1: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b) LT 2: Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g. by using visual fraction models or equations to represent the problem. LT 3: Apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction. LT 4: Interpret the product (a/b) q ÷ b. LT 5: Use a visual fraction model. Conversation Starters: Key Words § How are fractions like division problems? (for to Know example: If 9 people want to shar a 50-lb *Fraction sack of rice equally by *Numerator weight, how many pounds *Denominator of rice should each *Mixed number *Improper fraction person get?) *Product § 3 pizzas at 10 slices each *Equation need to be divided by 14 *Division friends. How many pieces would each friend receive? § How can a model help us make sense of a problem? Fractions as Division Students will interpret a fraction as division of the numerator by the { denominator. } What does a fraction as division look like? How can I support this Important Steps strategy at home? - Frac&ons are another way to Practice show division. https://www.khanacademy.org/math/cc- - Fractions are equal size pieces of a fifth-grade-math/cc-5th-fractions-topic/ whole. tcc-5th-fractions-as-division/v/fractions- - The numerator becomes the as-division dividend and the denominator becomes the divisor. Quotient as a Fraction Students will solve real world problems by dividing whole numbers that have a quotient resulting in a fraction. -
Division Into Cases and the Quotient-Remainder Theorem
4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem 4.4 Quotient-Remainder Theorem 1 / 4 1 n = 34 and d = 6. 2 n = −34 and d = 6. Examples For each of the following values of n and d, find integers q and r such that n = dq + r and 0 ≤ r < d. The Quotient-Remainder Theorem Theorem Given any integer n and positive integer d, there exist unique integers q and r such that n = dq + r and 0 ≤ r < d: 4.4 Quotient-Remainder Theorem 2 / 4 2 n = −34 and d = 6. The Quotient-Remainder Theorem Theorem Given any integer n and positive integer d, there exist unique integers q and r such that n = dq + r and 0 ≤ r < d: Examples For each of the following values of n and d, find integers q and r such that n = dq + r and 0 ≤ r < d. 1 n = 34 and d = 6. 4.4 Quotient-Remainder Theorem 2 / 4 The Quotient-Remainder Theorem Theorem Given any integer n and positive integer d, there exist unique integers q and r such that n = dq + r and 0 ≤ r < d: Examples For each of the following values of n and d, find integers q and r such that n = dq + r and 0 ≤ r < d. 1 n = 34 and d = 6. 2 n = −34 and d = 6. 4.4 Quotient-Remainder Theorem 2 / 4 1 Compute 33 div 9 and 33 mod 9 (by hand and Python). 2 Keeping in mind which years are leap years, what day of the week will be 1 year from today? 3 Suppose that m is an integer. -
4. Reconfigurations of Screen Borders: the New Or Not-So-New Aspect Ratios
4. Reconfigurations of Screen Borders: The New or Not-So-New Aspect Ratios Miriam Ross Abstract The ubiquity of mobile phone cameras has resulted in many videos foregoing the traditional horizontal (landscape) frame in favour of a vertical (portrait) mode. While vertical framing is often derided as amateur practice, these new framing techniques are part of a wider contemporary screen culture in which filmmakers and artists are using unconventional aspect ratios and/or expanding and contracting aspect ratios over the course of their audio-visual work. This chapter briefly outlines historical contexts in which the border of the screen has been more flexible and open to changing configurations than is widely acknowledged. It then uses recent case studies to consider how our understanding of on-screen and off-screen space is determined by these framing configurations. Keywords: Aspect ratios, embodiment, framing, cinema In recent years, the increasing ubiquity of mobile phone videos has drawn attention to a radical challenge to traditional screen culture. It is not just that a wide variety of amateur users now have a filmmaking device at their fingertips—rather, that many of them are foregoing the more than a century- long norm for shooting with a horizontal frame. Appearing on social media sites such as YouTube, Facebook, and Twitter as well as in commercial news broadcasts, their footage stands tall in a vertical format. When replayed on horizontal screens, the startling strangeness of wide black bands on either side of the content focuses attention on the border of the frame as well as seem- ingly absent screen space. -
IMAX Technician Robert Kilburn Uses Aspect Ratios to Make Sure Audiences Enjoy the Movie
THE Scenes from upcoming IMAX releases Star Trek and (below) Monsters vs. Aliens f you've ever seen a movie in an IMAX theater—with a huge screen and excellent sound and picture—you know how realistic it can be. You might have even felt like you were inside the movie. It takes many people to make sure that the film looks the way it's supposed to. One of those people is Robert kilburn, the manager of system installations at IMAX. His job is to IMAX technician supervise the installation of IMAX screens and projection Robert Kilburn and sound systems. Kilburn's favorite films to uses aspect work on might also be your favorites to watch. "I like the ratios to make visually spectacular films that take you places you sure audiences wouldn't normally be able to go, from the deepest depths enjoy the movie. of our oceans to the magical Continued on next page world of Harry Potter," Kilburn OA modem flat-screen television Some IMAX theaters use the told MATH. "The Harry Potter M(LCD or plasma) could have a 416:9 aspect ratio. If the height of films are my favorites." width of 64 inches and a height of an image at this ratio is 51.3 feet... In order to make sure the 36 inches, a. what proportion could you picture looks perfect on an a. To find the aspect ratio for write to find the width? IMAX screen, Kilburn uses flat-screen TVs, write those aspect ratios. A ratio compares dimensions as a ratio in two different numbers of the simplest form: same unit. -
Eureka Math™ Tips for Parents Module 2
Grade 6 Eureka Math™ Tips for Parents Module 2 The chart below shows the relationships between various fractions and may be a great Key Words tool for your child throughout this module. Greatest Common Factor In this 19-lesson module, students complete The greatest common factor of two whole numbers (not both zero) is the their understanding of the four operations as they study division of whole numbers, division greatest whole number that is a by a fraction, division of decimals and factor of each number. For operations on multi-digit decimals. This example, the GCF of 24 and 36 is 12 expanded understanding serves to complete because when all of the factors of their study of the four operations with positive 24 and 36 are listed, the largest rational numbers, preparing students for factor they share is 12. understanding, locating, and ordering negative rational numbers and working with algebraic Least Common Multiple expressions. The least common multiple of two whole numbers is the least whole number greater than zero that is a What Came Before this Module: multiple of each number. For Below is an example of how a fraction bar model can be used to represent the Students added, subtracted, and example, the LCM of 4 and 6 is 12 quotient in a division problem. multiplied fractions and decimals (to because when the multiples of 4 and the hundredths place). They divided a 6 are listed, the smallest or first unit fraction by a non-zero whole multiple they share is 12. number as well as divided a whole number by a unit fraction. -
Digital Image Prep for Displays
Digital Image Prep for Displays Jim Roselli ISF-C, ISF-R, HAA Tonight's Agenda • A Few Basics • Aspect Ratios • LCD Display Technologies • Image Reviews Light Basics We view the world in 3 colors Red, Green and Blue We are Trichromatic beings We are more sensitive to Luminance than to Color Our eyes contain: • Light 4.5 million cones (color receptors) 90 million rods (lumen receptors) Analog vs. Digital Light Sample • Digitize the Photon Light Sample Light Sample DSLR’s Canon 5D mkII and Nikon D3x 14 bits per color channel (16,384) Color channels (R,G & B) • Digital Capture Digital Backs Phase One 16 bits per channel (65,536) Color channels (R,G & B) Square Pixels Big pixels have less noise … Spatial Resolution Round spot vs. Square pixel • Size Matters Square Pixels • What is this a picture of ? Aspect Ratio Using Excel’s “Greatest Common Divisor” function • The ratio of the width to =GCD(x,y) xRatio = (x/GCD) the height of the image yRatio = (y/GCD) on a display screen Early Television • How do I calculate this ? Aspect Ratio 4 x 3 Aspect Ratio • SAR - Storage Aspect Ratio • DAR – Display Aspect Ratio • PAR – Pixel Aspect Ratio Aspect Ratio Device Width(x) Heigth(y) GCD xRa5o yRa5o Display 4096 2160 16 256 135 Display 1920 1080 120 16 9 • Examples Display 1280 720 80 16 9 Display 1024 768 256 4 3 Display 640 480 160 4 3 Aspect Ratio Device Width(x) Heigth(y) GCD xRa5o yRa5o Nikon D3x 6048 4032 2016 3 2 • Examples Canon 5d mkII 4368 2912 1456 3 2 Aspect Ratio Device Width(x) Heigth(y) GCD xRa5o yRa5o iPhone 1 480 320 160 3 2 HTC Inspire 800 480 160 5 3 • Examples Samsung InFuse 800 480 160 5 3 iPhone 4 960 640 320 3 2 iPad III 2048 1536 512 4 3 Motorola Xoom Tablet 1280 800 160 8 5 Future Aspect Ratio Device Width(x) Heigth(y) GCD xRa5o yRa5o Display 2048 1536 512 4 3 Display 3840 2160 240 16 9 Display 7680 4320 480 16 9 Aspect Ratio 4K Format Aspect Ratio • Don’t be scared … Pan D3x Frame . -
Facultad De Matemáticas, Univ. De Sevilla, Sevilla, Spain Arias@ Us. Es
#A51 INTEGERS 18 (2018) ARITHMETIC OF THE FABIUS FUNCTION J. Arias de Reyna1 Facultad de Matem´aticas, Univ. de Sevilla, Sevilla, Spain [email protected] Received: 5/29/17, Revised: 2/27/18, Accepted: 5/31/18, Published: 6/5/18 Abstract The Fabius function was defined in 1935 by Jessen and Wintner and has been independently defined at least six times since. We attempt to unify notations re- lated to the Fabius function. The Fabius function F (x) takes rational values at dyadic points. We study the arithmetic of these rational numbers. In partic- ular, we define two sequences of natural numbers that determine these rational numbers. Using these sequences we solve a conjecture raised in MathOverflow by n Vladimir Reshetnikov. We determine the dyadic valuation of F (2− ), showing that n n ⌫2(F (2− )) = 2 ⌫2(n!) 1. We give the proof of a formula that allows an efficient computation− −of exact−or approximate values of F (x). 1. Introduction The Fabius function is a natural object. It is not surprising that it has been inde- pendently defined several times. As far as we know it was defined independently in the following cases: 1. In 1935 by B. Jessen and A. Wintner [7, Ex. 5, p. 62] (English). They showed in five lines that it is infinitely di↵erentiable. 2. In 1966 by J. Fabius [4] (English); considered as the distribution function of a random variable. 3. In 1971 by V. A. Rvach¨ev [12] (Ukrainian); defined as a solution to the equa- tion y0(x) = 2(y(2x + 1) y(2x 1)). -
Math 1312 Section 5.1 Ratios, Rates, and Proportions Definition
Math 1312 Section 5.1 Ratios, Rates, and Proportions Definition : a A ratio is the quotient , where b ≠ 0 that provides comparison between the numbers a b and b . Units of measure found in a ratio must be convertible to the same unit of measure. Example 1 : The ratio of two numbers (a and b) may be written in a variety of ways. a a ÷ b a to b a : b b In writing the ratio of two numbers, it is usually helpful to express the ratio (fraction) in simplest form. 50 1 Example 2: = 100 2 Example 3: Find the best form of each ratio: 8 a) 4 8 b) 12 4m c) 60 cm Definition : A rate is a quotient, that compares two quantities that cannot be converted to the same unit of measure. 60 miles 12 teaspoons Example 4: 3gallons 2quarts Definition : An equation that states that two ratios are equal is called a proportion . a c = b d The first and last terms ( a and d ) of the proportion are the extremes . The second and third terms are the means . Property 1: (Means - Extremes Property) In a proportion, the product of the means equals the product of the extremes. a c If = , b ≠ 0 and d ≠ 0 , then a × d = b × c b d Example 5: Use the means-extremes property to solve each proportion for x. x 5 a) = 8 12 x 5 b) = 20 x x + 2 4 c) = 5 x +1 Property 2: In a proportion, the means or the extremes (or both) may be interchanged. -
Division of Whole Numbers
The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 10 DIVISION OF WHOLE NUMBERS A guide for teachers - Years 4–7 June 2011 4YEARS 7 Polynomials (Number and Algebra: Module 10) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑ NonCommercial‑NoDerivs 3.0 Unported License. 2011. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 10 DIVISION OF WHOLE NUMBERS A guide for teachers - Years 4–7 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge 4YEARS 7 {4} A guide for teachers DIVISION OF WHOLE NUMBERS ASSUMED KNOWLEDGE • An understanding of the Hindu‑Arabic notation and place value as applied to whole numbers (see the module Using place value to write numbers). • An understanding of, and fluency with, forwards and backwards skip‑counting. • An understanding of, and fluency with, addition, subtraction and multiplication, including the use of algorithms. -
MATH TODAY Grade 5, Module 4, Topic G
MATH TODAY Grade 5, Module 4, Topic G 5th Grade Math Focus Area– Topic G Module 4: Multiplication and Division of Fractions and Decimal Fractions Module 4: Multiplication and Division of Fractions and Decimal Fractions Divide a whole number by a unit fraction Math Parent Letter Garret is running a 5-K race. There are water stops every This document is created to give parents and students a better kilometer, including at the finish line. How many water stops will understanding of the math concepts found in the Engage New there be? Number Sentence: 5 ÷ York material which is taught in the classroom. Module 4 of Step 1: Draw a tape diagram to model the problem. Engage New York covers multiplication and division of fractions 5 and decimal fractions. This newsletter will discuss Module 4, Topic G. In this topic students will explore the meaning of division with fractions and decimal fractions. The tape diagram is partitioned into 5 equal units. Each unit Topic G: Division of Fractions and Decimal Fractions represents 1 kilometer of the race. Words to know: Step 2: Since water stops are every kilometer, each unit of the divide/division quotient tape diagram is divided into 2 equal parts. divisor dividend 5 unit fraction decimal fraction decimal divisor tenths hundredths Things to Remember! 1st Last Quotient – the answer of dividing one quantity by another stop stop Unit Fraction – a fraction with a numerator of 1 When you count the number of halves in the tape diagram, you Decimal Fraction – a fraction whose denominator is a power of will determine that there are a total of 10.