MAT Mathematics in Today's World Winter 2015
|
|
- Berenice Pierce
- 7 years ago
- Views:
Transcription
1 MAT 1000 Mathematics in Today's World Winter 2015
2 Last Time Identification numbers often include check digits: extra digits that allow us to catch errors. There are several different methods for finding check digits used in practice. We looked at sytems which are used for UPC codes, credit card numbers, and ISBNs as well as bar codes.
3 Today Binary codes are strings consisting of either 0s or 1s. We will look at a specific way to encode binary messages using Venn diagrams Then we consider a more general method, called parity check sums
4 Binary codes Binary codes are messages which are represented using only the digits 0 and 1. Some examples of binary codes of length three are 101, or 110, or 000 Computers use binary codes internally.
5 Binary codes We can append extra digits to binary codes to help catch errors. In fact, we can either identify where the error occurs, or we can fix the error. This requires appending more than a single digit.
6 Binary codes One advantage to binary codes: each position is either a 1 or a 0, so there are only two possible errors: 0 can be received as 1 1 can be received as 0
7 Venn diagram encoding If our binary codes have length 4, there is an encoding/decoding system which uses Venn diagrams. Use three circles in the following configuration:
8 Venn diagram encoding Note that we have four sections of overlap: Our four digit binary message will be placed in these 4 spaces, in this order. Then we append three digits, based on the other 3 spaces.
9 Example Venn diagram encoding Let s encode the message 1011 First, fill in the numbered spaces in the Venn diagram using the digits of the message, in order: 1 st digit in the 1 st space, 2 nd digit in the second space, and so on.
10 Example Venn diagram encoding We have three more spaces to fill: We will put either a 1 or a 0 in each of these.
11 Example Venn diagram encoding To decide whether to put in a 0 or a 1, we choose whichever makes the total number of 1s in each circle even: Adding these three extra digits (in order) gives
12 Example Venn diagram encoding Now we will see how to correct errors using this method. Suppose our message is mistakenly received as When we fill in the Venn diagram, we can see there is a mistake because some of the circles have an odd number of 1s in them.
13 Example Venn diagram encoding The upper left circle has an even number of 1s:
14 Example Venn diagram encoding But the other two circles both have an odd number of 1s:
15 Example Venn diagram encoding But the other two circles both have an odd number of 1s:
16 Example Venn diagram encoding Which digit is incorrect? The incorrect digit must appear in both of the circles with an odd number of 1s, but it is not in the circle with the correct number of 1s. This tells us exactly where the error must be:
17 Venn diagram encoding Example Moreover, once we know the location of the error, we can fix it. After all, this is a binary code: if 0 is not the correct digit, then 1 must be. Fixing the mistake recovers our original message:
18 Venn diagram encoding Some disadvantages of this method: Only works on messages of length four If there are two or more errors, they may go undetected, or they may be fixed incorrectly Let s see an example that shows how two errors can be fixed incorrectly.
19 Example Venn diagram encoding The original message is Encode this message So the encoded message is:
20 Venn diagram encoding Example Suppose the message is received with two errors as What happens when we decode this message?
21 Example Venn diagram encoding The two upper circles have an odd number of 1s in them, but the lower circle has an even number of 1s. So the method tells us there is an error in this place:
22 Venn diagram encoding Example This gives the corrected message Of course this is not actually the correct message. That was So this method may fail when there are two or more errors in the message.
23 Parity Check Sums Here are some possible improvements on the Venn diagram method: 1. Longer messages 2. Correct more errors To describe these improved methods, we need to look at the Venn diagram method in a different way, using parity check sums
24 Parity Check Sums The parity of a number is whether it is even or odd. Suppose we have a four-digit binary message a 1 a 2 a 3 a 4 We decide to append three digits, c 1, c 2, c 3 We choose c 1 to have the same parity as the sum a 1 + a 2 + a 3 This means that If a 1 + a 2 + a 3 is even, c 1 = 0 If a 1 + a 2 + a 3 is odd, c 1 = 1
25 Parity Check Sums We choose c 2 to have the same parity as the sum a 1 + a 3 + a 4 And c 3 will have the same parity as the sum a 2 + a 3 + a 4
26 Example Parity Check Sums Use these parity check sums to encode the message The first check sum is a 1 +a 2 + a 3 which is = 2. So the first appended digit is 0 The next check sum is a 1 +a 3 + a 4 which is = 3. So the second appended digit is 1 The last check sum is a 2 + a 3 + a 4 which is = 2. So the third appended digit is 0 This makes the message:
27 Parity Check Sums If you compare this method with the Venn diagram method we used earlier, you will see that they are identical (for any four digit message they give the same code) The advantage of parity check sums over Venn diagrams is that we have more flexibility: we can now work with longer messages we can add more digits (which can catch more errors)
28 Example Parity Check Sums In addition to the digits c 1, c 2, c 3 defined above, we could add another digit c 4 using the sum a 1 + a 2 + a 4 (which was not used in the Venn diagram method)
29 Example Parity Check Sums For a message of length five a 1 a 2 a 3 a 4 a 5 we could use check digits c 1, c 2, c 3, c 4 which have the same parity as a 1 + a 2 + a 3 + a 4 a 2 + a 3 + a 5 a 1 + a 2 + a 4 a 1 + a 3 There are lots of other possibilities.
30 Parity Check Sums If we encode a message with parity check sums, how should we decode it? The method used is called nearest neighbor decoding. To use this, we have to discuss the distance between binary strings.
31 Parity Check Sums The distance between two binary strings is the number of places in which they differ. Example: and have a distance of 1 Example: and have a distance of 3 Example: and have a distance of 0 Note if two strings have different lengths, it doesn t make sense to talk about their distance. The distance between and doesn t make sense
32 Nearest neighbor decoding: Parity Check Sums Receive an encoded message, which may have some errors. Find the nearest correct message (meaning the one which is the smallest distance from the received message). Do not decode if there is a tie.
33 Parity Check Sums In order to use nearest neighbor decoding, we need to make a list of every possible correct message. In the next example, we will take a parity check sum method, list every possible correct message, and then use that list to decode a message.
34 Parity Check Sums Example Messages will be of length three a 1 a 2 a 3. We will add three parity sum digits using the sums a 1 + a 2 + a 3 and a 1 + a 3 and a 2 + a 3 Messages a 1 + a 2 + a 3 a 1 + a 3 a 2 + a 3 Coded Messages
35 Parity Check Sums Example Decode the message We will find the distance from this message to each valid message: Coded Messages Distance The message is decoded to be
36 Parity Check Sums We have lots of choice for different parity check sums. How many should we use? Which ones should we use? We will see next time how to analyze different choices of parity check sums.
ELFRING FONTS UPC BAR CODES
ELFRING FONTS UPC BAR CODES This package includes five UPC-A and five UPC-E bar code fonts in both TrueType and PostScript formats, a Windows utility, BarUPC, which helps you make bar codes, and Visual
More informationColored Hats and Logic Puzzles
Colored Hats and Logic Puzzles Alex Zorn January 21, 2013 1 Introduction In this talk we ll discuss a collection of logic puzzles/games in which a number of people are given colored hats, and they try
More informationCheck Digit Schemes and Error Detecting Codes. By Breanne Oldham Union University
Check Digit Schemes and Error Detecting Codes By Breanne Oldham Union University What are Check Digit Schemes? Check digit schemes are numbers appended to an identification number that allow the accuracy
More informationFACTORS, PRIME NUMBERS, H.C.F. AND L.C.M.
Mathematics Revision Guides Factors, Prime Numbers, H.C.F. and L.C.M. Page 1 of 16 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M. Version:
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationLesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationIE1204 Digital Design F12: Asynchronous Sequential Circuits (Part 1)
IE1204 Digital Design F12: Asynchronous Sequential Circuits (Part 1) Elena Dubrova KTH / ICT / ES dubrova@kth.se BV pp. 584-640 This lecture IE1204 Digital Design, HT14 2 Asynchronous Sequential Machines
More informationMATHEMATICAL EXCURSIONS Math and Bar Codes
MATHEMATICAL EXCURSIONS Math and Bar Codes If you examine most consumer products, you will find a small rectangular area with a pattern of black bars and white spaces of various widths on the packaging
More informationOA4-13 Rounding on a Number Line Pages 80 81
OA4-13 Rounding on a Number Line Pages 80 81 STANDARDS 3.NBT.A.1, 4.NBT.A.3 Goals Students will round to the closest ten, except when the number is exactly halfway between a multiple of ten. PRIOR KNOWLEDGE
More informationLocating and Decoding EAN-13 Barcodes from Images Captured by Digital Cameras
Locating and Decoding EAN-13 Barcodes from Images Captured by Digital Cameras W3A.5 Douglas Chai and Florian Hock Visual Information Processing Research Group School of Engineering and Mathematics Edith
More informationN O T E S. A Reed Solomon Code Magic Trick. The magic trick T O D D D. M A T E E R
N O T E S A Reed Solomon Code Magic Trick T O D D D. M A T E E R Howard Community College Columbia, Maryland 21044 tmateer@howardcc.edu Richard Ehrenborg [1] has provided a nice magic trick that can be
More information. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
More informationWe can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b
In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should
More informationELFRING FONTS BAR CODES EAN 8, EAN 13, & ISBN / BOOKLAND
ELFRING FONTS BAR CODES EAN 8, EAN 13, & ISBN / BOOKLAND This package includes ten EAN bar code fonts in scalable TrueType and PostScript formats, a Windows utility (BarEAN) to help you make bar codes,
More informationBasics of Digital Recording
Basics of Digital Recording CONVERTING SOUND INTO NUMBERS In a digital recording system, sound is stored and manipulated as a stream of discrete numbers, each number representing the air pressure at a
More informationFactorizations: Searching for Factor Strings
" 1 Factorizations: Searching for Factor Strings Some numbers can be written as the product of several different pairs of factors. For example, can be written as 1, 0,, 0, and. It is also possible to write
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More information1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:
Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of
More informationReading 13 : Finite State Automata and Regular Expressions
CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model
More informationIntroduction to Algebraic Coding Theory
Introduction to Algebraic Coding Theory Supplementary material for Math 336 Cornell University Sarah A. Spence Contents 1 Introduction 1 2 Basics 2 2.1 Important code parameters..................... 4
More informationA Survey of the Theory of Error-Correcting Codes
Posted by permission A Survey of the Theory of Error-Correcting Codes by Francis Yein Chei Fung The theory of error-correcting codes arises from the following problem: What is a good way to send a message
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationEncoding Text with a Small Alphabet
Chapter 2 Encoding Text with a Small Alphabet Given the nature of the Internet, we can break the process of understanding how information is transmitted into two components. First, we have to figure out
More informationPRINT AWARENESS ASSESSMENT. Instructions and Teacher Recording Form
PRINT AWARENESS ASSESSMENT Instructions and Teacher Recording Form Instructions: This form includes instructions for each section and allows room for the teacher or assessor to write comments while the
More informationJust the Factors, Ma am
1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
More informationError Detection and Correction
Error Detection and Correction Outline for Today 1. Parity Check Code 2. Bounds based on Hamming distance 3. Hamming Code Can You Raed Tihs? I cnduo t bvleiee taht I culod aulaclty uesdtannrd waht I was
More informationMII stands for major industry identifier; 4 and 5 indicate Banking and Financial. VISA cards begin with 4 and MasterCard cards with 5.
Credit Cards Credit cards have 16-digit numbers, of which the first 15 digits identify the credit card and the sixteenth digit is the check digit. The following figure shows the significance of the digits:
More informationSession 6 Number Theory
Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationGrade 7 & 8 Math Circles October 19, 2011 Prime Numbers
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers Factors Definition: A factor of a number is a whole
More informationAIMS Education Foundation
Developed and Published by AIMS Education Foundation This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating Mathematics and Science) began in 1981 with a
More informationGrade 8 Mathematics Data Analysis, Probability, and Discrete Mathematics: Lesson 3
Grade 8 Mathematics Data Analysis, Probability, and Discrete Mathematics: Lesson 3 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type
More informationPrimes. Name Period Number Theory
Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in
More information1 ENGAGE. 2 TEACH and TALK GO. Round to the Nearest Ten or Hundred
Lesson 1.2 c Round to the Nearest Ten or Hundred Common Core Standard CC.3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. Lesson Objective Round 2- and 3-digit numbers
More information6. Block and Tackle* Block and tackle
6. Block and Tackle* A block and tackle is a combination of pulleys and ropes often used for lifting. Pulleys grouped together in a single frame make up what is called a pulley block. The tackle refers
More informationContemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More informationCOMBINATIONAL CIRCUITS
COMBINATIONAL CIRCUITS http://www.tutorialspoint.com/computer_logical_organization/combinational_circuits.htm Copyright tutorialspoint.com Combinational circuit is a circuit in which we combine the different
More informationThe finite field with 2 elements The simplest finite field is
The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0
More informationINTRODUCTION TO CREDIT
Grades 4-5 Lesson 3 INTRODUCTION TO CREDIT Key concepts: card companies. Borrowing money through credit, evaluating credit and credit Summary: This lesson introduces students to credit cards, credit card
More informationUnited States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1
United States Naval Academy Electrical and Computer Engineering Department EC262 Exam 29 September 2. Do a page check now. You should have pages (cover & questions). 2. Read all problems in their entirety.
More informationGrade 7/8 Math Circles Sequences and Series
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered
More informationCodes: How to Protect Your Data
Codes: How to Protect Your Data Michael Mitzenmacher 1 Introduction This chapter is going to be about error-correcting codes. Just to set the stage, we ll start with a quick little puzzle for you to think
More information2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are
More informationTHE LANGUAGE OF SETS AND SET NOTATION
THE LNGGE OF SETS ND SET NOTTION Mathematics is often referred to as a language with its own vocabulary and rules of grammar; one of the basic building blocks of the language of mathematics is the language
More informationMATH STUDENT BOOK. 8th Grade Unit 6
MATH STUDENT BOOK 8th Grade Unit 6 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
More informationSouth East of Process Main Building / 1F. North East of Process Main Building / 1F. At 14:05 April 16, 2011. Sample not collected
At 14:05 April 16, 2011 At 13:55 April 16, 2011 At 14:20 April 16, 2011 ND ND 3.6E-01 ND ND 3.6E-01 1.3E-01 9.1E-02 5.0E-01 ND 3.7E-02 4.5E-01 ND ND 2.2E-02 ND 3.3E-02 4.5E-01 At 11:37 April 17, 2011 At
More informationQuestions: Does it always take the same amount of force to lift a load? Where should you press to lift a load with the least amount of force?
Lifting A Load 1 NAME LIFTING A LOAD Questions: Does it always take the same amount of force to lift a load? Where should you press to lift a load with the least amount of force? Background Information:
More informationInternational Securities Identification Number (ISIN)
International Securities Identification Number (ISIN) An International Securities Identification Number (ISIN) uniquely identifies a security. An ISIN consists of three parts: Generally, a two letter country
More informationBAR CODE 2 OF 5 INTERLEAVED
ELFRING FONTS INC BAR CODE 2 OF 5 INTERLEAVED This package includes 25 bar code 2 of 5 interleaved fonts in TrueType and PostScript formats, a Windows utility, Bar25i.exe, to help make your bar codes,
More informationSystems I: Computer Organization and Architecture
Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems - Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x
More informationMATHEMATICS: REPEATING AND GROWING PATTERNS First Grade. Kelsey McMahan. Winter 2012 Creative Learning Experiences
MATHEMATICS: REPEATING AND GROWING PATTERNS Kelsey McMahan Winter 2012 Creative Learning Experiences Without the arts, education is ineffective. Students learn more and remember it longer when they are
More informationLESSON 5 -- DEBIT CARDS VS CREDIT CARDS
LESSON 5 -- DEBIT CARDS VS CREDIT CARDS LESSON DESCRIPTION AND BACKGROUND Using the Better Money Habits video Credit and Debit: Two Very Different s (www.bettermoneyhabits.com), this lesson is designed
More informationACTIVITY: Identifying Common Multiples
1.6 Least Common Multiple of two numbers? How can you find the least common multiple 1 ACTIVITY: Identifying Common Work with a partner. Using the first several multiples of each number, copy and complete
More informationMedian, Mode, and Range Active Lesson
Median, Mode, and Range Active Lesson Teacher Candidate: Cory D Wilson_Dates: October 2007 Cooperating Teacher: Dr. Lori Engstrom Coop. Initials: Group Size: 25 students Allotted Time: 15 minutes _ Grade
More informationNon-Redundant (RAID Level 0)
There are many types of RAID and some of the important ones are introduced below: Non-Redundant (RAID Level 0) A non-redundant disk array, or RAID level 0, has the lowest cost of any RAID organization
More informationInternational Indian School, Riyadh SA1 Worksheet 2015-2016 Class: VI Mathematics
International Indian School, Riyadh SA1 Worksheet 2015-2016 Class: VI Mathematics CH KNOWING OUR NUMBERS I Fill In the blanks 1. 1km = mm 2. 1 gram = milligrams 3. The roman numeral M stands for the number
More informationIntegers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern.
INTEGERS Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe
More informationSilent data corruption in SATA arrays: A solution
Silent data corruption in SATA arrays: A solution Josh Eddy August 2008 Abstract Recent large academic studies have identified the surprising frequency of silent read failures that are not identified or
More informationNot for resale. 4.1 Divisibility of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples
CHAPTER 4 Number Theory 4.1 Divisibility of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples 4.4 Codes and Credit Card Numbers: Connections to Number Theory
More informationPrime Time: Homework Examples from ACE
Prime Time: Homework Examples from ACE Investigation 1: Building on Factors and Multiples, ACE #8, 28 Investigation 2: Common Multiples and Common Factors, ACE #11, 16, 17, 28 Investigation 3: Factorizations:
More informationBattleships Searching Algorithms
Activity 6 Battleships Searching Algorithms Summary Computers are often required to find information in large collections of data. They need to develop quick and efficient ways of doing this. This activity
More informationDC mesh current analysis
DC mesh current analysis This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationThe GMAT Guru. Prime Factorization: Theory and Practice
. Prime Factorization: Theory and Practice The following is an ecerpt from The GMAT Guru Guide, available eclusively to clients of The GMAT Guru. If you would like more information about GMAT Guru services,
More informationCoping with Bit Errors using Error Correction Codes
MIT 6.02 DRAFT Lecture Notes Last update: September 23, 2012 CHAPTER 5 Coping with Bit Errors using Error Correction Codes Recall our main goal in designing digital communication networks: to send information
More informationMeans, standard deviations and. and standard errors
CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard
More information10 th POLISH SUDOKU CHAMPIONSHIP INSTRUCTION BOOKLET. February 22, 2015 IMPORTANT INFORMATION:
10 th POLISH SUDOKU CHAMPIONSHIP February 22, 2015 INSTRUCTION BOOKLET IMPORTANT INFORMATION: 1. Answer form can be sent many times, but only the last version will be considered. 2. In case of a tie, the
More informationPrime Factorization 0.1. Overcoming Math Anxiety
0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF
More informationChapter 6: Graph Theory
Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.
More informationSQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationDecimals and other fractions
Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very
More informationAn in-building multi-server cloud system based on shortest Path algorithm depending on the distance and measured Signal strength
IOSR Journal of Computer Engineering (IOSR-JCE) e-issn: 2278-0661,p-ISSN: 2278-8727, Volume 17, Issue 1, Ver. I (Jan Feb. 2015), PP 38-42 www.iosrjournals.org An in-building multi-server cloud system based
More informationThe Assignment Problem and the Hungarian Method
The Assignment Problem and the Hungarian Method 1 Example 1: You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationChapter 2 Data Storage
Chapter 2 22 CHAPTER 2. DATA STORAGE 2.1. THE MEMORY HIERARCHY 23 26 CHAPTER 2. DATA STORAGE main memory, yet is essentially random-access, with relatively small differences Figure 2.4: A typical
More informationplc numbers - 13.1 Encoded values; BCD and ASCII Error detection; parity, gray code and checksums
plc numbers - 3. Topics: Number bases; binary, octal, decimal, hexadecimal Binary calculations; s compliments, addition, subtraction and Boolean operations Encoded values; BCD and ASCII Error detection;
More informationQuestion of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay
QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know
More informationThe Mathematics 11 Competency Test Percent Increase or Decrease
The Mathematics 11 Competency Test Percent Increase or Decrease The language of percent is frequently used to indicate the relative degree to which some quantity changes. So, we often speak of percent
More informationEE 261 Introduction to Logic Circuits. Module #2 Number Systems
EE 261 Introduction to Logic Circuits Module #2 Number Systems Topics A. Number System Formation B. Base Conversions C. Binary Arithmetic D. Signed Numbers E. Signed Arithmetic F. Binary Codes Textbook
More informationValor Christian High School Mrs. Bogar Biology Graphing Fun with a Paper Towel Lab
1 Valor Christian High School Mrs. Bogar Biology Graphing Fun with a Paper Towel Lab I m sure you ve wondered about the absorbency of paper towel brands as you ve quickly tried to mop up spilled soda from
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationEx. 2.1 (Davide Basilio Bartolini)
ECE 54: Elements of Information Theory, Fall 00 Homework Solutions Ex.. (Davide Basilio Bartolini) Text Coin Flips. A fair coin is flipped until the first head occurs. Let X denote the number of flips
More informationSection IV.1: Recursive Algorithms and Recursion Trees
Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller
More informationThe Wilcoxon Rank-Sum Test
1 The Wilcoxon Rank-Sum Test The Wilcoxon rank-sum test is a nonparametric alternative to the twosample t-test which is based solely on the order in which the observations from the two samples fall. We
More informationProblem of the Month: Once Upon a Time
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More informationSet operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE
Set operations and Venn Diagrams Set operations and Venn diagrams! = { x x " and x " } This is the intersection of and. # = { x x " or x " } This is the union of and. n element of! belongs to both and,
More informationOA4-13 Rounding on a Number Line
OA4-13 Rounding on a Number Line 1. Draw an arrow to show whether the circled number is closer to 0 or 10. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 c) 0 1 2 3 4 5 6 7 8 9 10 d) 0 1 2 3 4 5 6 7 8 9
More informationMATHEMATICS. Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples. Equipment. MathSphere
MATHEMATICS Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples Paper, pencil, ruler. Equipment MathSphere 5330 Square numbers, prime numbers, factors and multiples
More informationDescriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics
Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),
More informationnorth seattle community college
INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The
More informationCustomer Barcoding Technical Specifications
Customer Barcoding Technical Specifications June 1998 Contents Revised 3 Aug 2012 Introduction 2 Key features of the barcoding system 2 About this document 2 Why we are introducing Customer Barcoding 3
More informationGreatest Common Factors and Least Common Multiples with Venn Diagrams
Greatest Common Factors and Least Common Multiples with Venn Diagrams Stephanie Kolitsch and Louis Kolitsch The University of Tennessee at Martin Martin, TN 38238 Abstract: In this article the authors
More information1051-232 Imaging Systems Laboratory II. Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002
05-232 Imaging Systems Laboratory II Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002 Abstract: For designing the optics of an imaging system, one of the main types of tools used today is optical
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationUnit 6 Number and Operations in Base Ten: Decimals
Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,
More informationClass Notes CS 3137. 1 Creating and Using a Huffman Code. Ref: Weiss, page 433
Class Notes CS 3137 1 Creating and Using a Huffman Code. Ref: Weiss, page 433 1. FIXED LENGTH CODES: Codes are used to transmit characters over data links. You are probably aware of the ASCII code, a fixed-length
More information