STRAND A: NUMBER. UNIT A6 Number Systems: Text. Contents. Section. A6.1 Roman Numerals. A6.2 Number Classification* A6.


 Magdalen Fox
 7 months ago
 Views:
Transcription
1 STRAND A: NUMBER A6 Number Systems Text Contents Section A6.1 Roman Numerals A6. Number Classification* A6.3 Binary Numbers* A6.4 Adding and Subtracting Binary Numbers*
2 A6 Number Systems A6.1 Roman Numerals The number system that we use in daily life is the base 10 (Arabic) system. Roman numerals, the numeric system used in ancient Rome, uses letters from the Latin alphabet in an equivalent system of representing numbers. In the Roman system we use I for 1 V for 5 X for 10 L for 50 C for 100 D for 500 M for 1000 The values of consecutive letters are ADDED unless a letter with a lower value appears in front of a letter with a higher value. When this is the case, the lower value letter is SUBTRACTED from the higher value, as shown in the following examples: placing I before V or before X to make the numbers 4 (IV) or 9 (IX) placing X before L or C to make the numbers 40 (XL) or 90 (XC) placing C before D or M to make 400 (CD) or 900 (CM). Examples of the effects of the order in which letters are placed: VII = = 7 XXV = = 5 IV = 5 1= 4 IC = = 99 VL = 50 5 = 45 1
3 A6.1 Example 1 Complete these number lines using Roman numerals. (a) I V X (b) X L C (a) I II III IV V VI VII VIII IX X XI XII (b) X XX XXX XL L LX LXX LXXX XC C CX CXX Example Convert the following Roman numerals to base 10. (a) II (b) IX (c) DI (d) LIX (a) 11 = 1+ 1= (b) IX = 10 1 = 9 (c) DI = = 501 (d) LIX = 50 + ( 10 1) = 59 Example 3 Convert the following base 10 numbers to Roman numerals. (a) 14 (b) 84 (c) 7 (d) 1010
4 A6.1 (a) 14 = = 10 + ( 5 1) = XIV 84 = = ( 5 1) = LXXXIV 7 = = XXVII 1010 = = MX Exercises 1. Convert the following Roman numerals to base 10. (a) VIII (b) CD (c) DVI (d) CDXC. Convert the following base 10 numbers to Roman numerals. (a) 6 (b) 109 (c) 56 (d) Complete the number line using Roman numerals. C D M Change the Roman numerals to base 10 numbers. (a) (i) DIX (ii) MCMXLV (iii) CMIV (iv) CDXVI (v) MCXI (vi) CMXCIX (b) (i) List the numbers in (a) above in decreasing order. (ii) Divide the second number in your list by 11 and write the answer in Roman numerals. 5. Above the entrance to a church there is a Roman number MDCCXCI When do you think the church was built? The crypt was built 153 years before the main church. What Roman number is carved in the wall of the crypt? 3
5 A Which Roman numeral can written instead of the shapes to make the statements true? (a) CDLXXIX < < CDLXXXIII (b) CMXCVIII < < MIV Give your answers in Roman numerals. 7. Continue the sequences, giving the next 3 terms, using Roman numerals. (a) XLVII, LXVII, LXXXVII,...,...,..., (b) CMI, DCCCI, DCCI,...,...,..., 8. The sum of any two adjacent numbers is the number directly above them. Fill in the missing numbers. (a) (b) XC L XL XX LXXV L CCCL C A6. Number Classification* The number system is classified into various categories. Natural numbers (or 'counting numbers') These are the positive whole numbers, namely 1,, 3,,,... Integers These are the set of positive and negative whole numbers, e.g. 1, 10, 364, 7, 10. The set of integers is denoted by Z. Rational Numbers A rational number is a number which can be written in the form m, where m and n are integers. n For example, 4 5 is a rational number. A rational number is in its simplest form if m and n have no common factor and n is positive. The set of rational numbers is denoted by Q. 4
6 A6. Irrational Numbers There are numbers which cannot be written in the form m, where m and n are integers. n Examples of irrational numbers are π,, 3, 5. Terminating Decimals These are decimal numbers which stop after a certain number of decimal places. For example, 7 = is a terminating decimal because it stops (terminates) after 3 decimal places. Recurring Decimals These are decimal numbers which keep repeating a digit or group of digits; for example = is a recurring decimal. The six digits repeat in this order. Recurring decimals are written with dots over the first and last digit of the repeating digits, e.g Note All terminating and recurring decimals can be written in the form m, so they are n rational numbers. Real Numbers These are made up of all possible rational and irrational numbers; the set of real numbers is denoted by R. We can use a Venn diagram to illustrate this classification of number, as shown below. R 1+ Q + Q π π N Z 3 3 5
7 A6. R : Real numbers N : Natural numbers Z : Integers Q : Rational numbers Q + : Positive rational numbers Some numbers have been inserted to illustrate each set. Worked Example 1 Classify the following numbers as integers, rational, irrational, recurring decimals, terminating decimals. 5, 7, 0.6, , 8, 11, 10, π 4, 49 Number Rational Irrational Integer Recurring Terminating Decimal Decimal π 4 49 Worked Example Show that the numbers and are rational. 6
8 A6. For a terminating decimal the proof is straightforward = = which is rational because m = 69 and n = 00 are integers. For a recurring decimal, we multiply by a power of ten so that after the decimal point we have only the repeating digits = Also = Subtracting the second equation from the first gives = 917 which is rational because m = 917 and n = Worked Example ( ) = = Prove that 3 is irrational. Assume that 3 is rational so we can find integers m and n such that 3 = m n, and m and n have no common factors. Square both sides, 3 = m n Multiply both sides by n, m = 3n Since 3n is divisible by 3, then m is divisible by 3. Since m is an integer, m is an integer and 3 is prime, then m must be divisible by 3. Let m = 3 p, where p is an integer. 3n = m = ( 3p) = 9p n = 3p Since 3p is divisible by 3, n is divisible by 3 and hence n is divisible by 3. We have shown that m and n are both divisible by 3. This contradicts the original assumption that 3 = m, where m and n are integers with n no factor common. Our original assumption is wrong. 3 is not rational it is irrational. 7
9 A6. Exercises 1. Classify the following numbers as rational or irrational, terminating or recurring decimals. (a) 100 (b) 06. (c) π 0. (d) (e) 075. (f) 1 π (g) 11 (h) (i) π (j) 5 11 (k) sin 45 (l) cos60. Where possible, write each of the following numbers in the form and n are integers with no common factors (a) 0.49 (b) 03. (c) 49 4 (e) (f) 01. (g) 009. (i) 0.15 (j) 0.96 (d) 7 (h) m n, where m Write each of the following recurring decimals in the form integers with no common factors. m, where m and n are n (a) 041 (b) (c) (d) 08. (e) (f) 05. (g) Given the rational numbers show that a a = 3, b = 7 9, c = 11 15, d = b, c+ d, a+ c+ d, ab, cd, bc and abc are rational numbers. 5. Prove that the following numbers are irrational,, 5, What happens if you try to prove that 4 is irrational using the proof by a contradiction method. 8
10 A6. 7. Write down two rational numbers and two irrational numbers which lie in each of the following intervals. (a) 0< x < 1 (b) 1< x < 9 (c) 4< x < (d) 98 < x < Which of these expressions can be written as recurring decimals and which can be written as nonrecurring decimals? 7, π, 1 17, 7, x = a + b State whether x is rational or irrational in each of the following cases, and show sufficient working to justify each answer. (a) a = 5 and b = 1 (b) a = 5 and b = 6 (c) a = and b = 7 (d) a = 3 7 and b = (a) Write down two irrational numbers that multiply together to give a rational number. (b) Which of the following numbers are rational? 1 1,, 4, 4 1, π, π 11. (a) A Mathematics student attempted to define an irrational number as follows: "An irrational number is a number which, in its decimal form, goes on and on." (i) Give an example to show that this definition is not correct. (ii) What must be added to the definition to make it correct? (b) Which of the following are rational and which are irrational? 4 1, 6 1, , 1 3 Express each of the rational numbers in the form integers. 3 p, where p and q are q 1. (a) Write down an irrational number that lies between 4 and 5. (b) N is a rational number which is not equal to zero. Show clearly why 1 N must also be rational. 13. n is a positive integer such that n = correct to 1 decimal place. (a) (i) Find a value of n. (b) (ii) Explain why n is irrational. Write down a number between 10 and 11 that has a rational square root. 9
11 A6.3 Binary Numbers* We normally work with numbers in base 10. In this section we consider numbers in base, often called binary numbers. In base 10 we use the digits 0, 1,, 3, 4, 5, 6, 7, 8, and 9. In base we use only the digits 0 and 1. Binary numbers are at the heart of all computing systems since, in an electrical circuit, 0 represents no current flowing whereas 1 represents a current flowing. In base 10 we use a system of place values as shown below: Note that, to obtain the place value for the next digit to the left, we multiply by 10. If we were to add another digit to the front (left) of the numbers above, that number would represent s. In base we use a system of place values as shown below: = = 73 Note that the place values begin with 1 and are multiplied by as you move to the left. Once you know how the place value system works, you can convert binary numbers to base 10, and vice versa. Worked Example 1 Convert the following binary numbers to base 10: (a) 111 (b) 101 (c) For each number, consider the place value of every digit. (a) = 7 The binary number 111 is 7 in base 10. (b) = 5 The binary number 101 is 5 in base
12 A6.3 (c) = 10 The binary number is 10 in base 10. Worked Example Convert the following base 10 numbers into binary numbers: (a) 3 (b) 11 (c) 140 We need to write these numbers in terms of the binary place value headings 1,, 4, 8, 16, 3, 64, 18,..., etc. (a) 1 3 = The base 10 number 3 is written as 11 in base. (b) = The base 10 number 11 is written as 1011 in base. (c) = The base 10 number 140 is written as in base. Exercises 1. Convert the following binary numbers to base 10: (a) 110 (b) 1111 (c) 1001 (d) 1101 (e) (f) (g) (h) (i) (j) (k) (l) Convert the following base 10 numbers to binary numbers: (a) 9 (b) 8 (c) 14 (d) 17 (e) 18 (f) 30 (g) 47 (h) 5 (i) 67 (j) 84 (k) 00 (l)
13 A Convert the following base 10 numbers to binary numbers: (a) 5 (b) 9 (c) 17 (d) 33 Describe any pattern that you notice in these binary numbers. What will be the next base 10 number that will fit this pattern? 4. Convert the following base 10 numbers to binary numbers: (a) 3 (b) 7 (c) 15 (d) 31 What is the next base 10 number that will continue your binary pattern? 5. A particular binary number has 3 digits. (a) What are the largest and smallest possible binary numbers? (b) Convert these numbers to base When a particular base 10 number is converted it gives a 4digit binary number. What could the original base 10 number be? 7. A 4digit binary number has zeros and ones. (a) List all the possible binary numbers with these digits. (b) Convert these numbers to base A binary number has 8 digits and is to be converted to base 10. (a) (b) What is the largest possible base 10 answer? What is the smallest possible base 10 answer? 9. The base 10 number 999 is to be converted to binary. How many more digits does the binary number have than the number in base 10? 10. Calculate the difference between the base 10 number and the binary number 11111, giving your answer in base 10. 1
14 A6.4 Adding and Subtracting Binary Numbers It is possible to add and subtract binary numbers in a similar way to base 10 numbers. For example, = 3 in base 10 becomes = 11 in binary. In the same way, 3 1= in base 10 becomes 11 1 = 10 in binary. When you add and subtract binary numbers you will need to be careful when 'carrying' or 'borrowing' as these will take place more often. * Key Addition Results for Binary Numbers = = = 11 Key Subtraction Results for Binary Numbers 1 0 = = = 10 Worked Example 1 Calculate, using binary numbers: (a) (b) (c) (a) 111 (b) 101 (c) Note how important it is to 'carry' correctly. Worked Example Calculate the binary numbers: (a) (b) (c) (a) 111 (b) 110 (c)
15 A6.4 Exercises 1. Calculate the binary numbers: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Calculate the binary numbers: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Calculate the binary numbers: (a) (b) (c) (d) Describe any patterns that you observe in your answers. 4. Calculate the binary numbers: (a) (b) (c) (d) Describe any patterns that you observe in your answers. 5. Solve the following equations, where all numbers, including x, are binary: (a) x + 11 = 1101 (b) x 10 = 101 (c) x 1101 = (d) x = (e) x = (f) x 1001 = Calculate the binary numbers: (a) 10 1 (b) (c) (d) Describe any patterns that you observe in your answers. 14
16 A (a) Convert the binary numbers and 1110 to base 10. (b) (c) (d) Add together the two base 10 numbers. Add together the two binary numbers. Convert your answer to base 10 and compare with your answer to (b). 8. (a) Convert the binary numbers and to base 10. (b) Calculate the difference between the two base 10 numbers. (c) Convert your answer to (b) into a binary number. (d) Calculate the difference between the two binary numbers and compare with your answer to (c). 9. Here are 3 binary numbers: Working in binary, (a) add together the two smaller numbers, (b) add together the two larger numbers, (c) take the smallest number away from the largest number, (d) add together all three numbers. 10. Calculate the binary numbers: (a) (b)
MEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationRoman Numerals. Symbol Decimal Value I One (1) V Five (5) X Ten (10) L Fifty (50) C Hundred (100) D Five Hundred (500) M Thousand (1000)
Roman Numerals Prerequisite: Number System, Addition and Subtraction. Target Audience: Standard V to Standard VII. In Syllabus Standard V. Revision Standard VI & VII. Symbol Decimal Value I One (1) V
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On wellordering and induction: (a) Prove the induction principle from the wellordering principle. (b) Prove the wellordering
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationCubes and Cube Roots
CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
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 informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More informationAccuplacer Arithmetic Study Guide
Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationOct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: Our standard number system is base, also
More informationWarm up. Connect these nine dots with only four straight lines without lifting your pencil from the paper.
Warm up Connect these nine dots with only four straight lines without lifting your pencil from the paper. Sometimes we need to think outside the box! Warm up Solution Warm up Insert the Numbers 1 8 into
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationLesson Plan. N.RN.3: Use properties of rational and irrational numbers.
N.RN.3: Use properties of rational irrational numbers. N.RN.3: Use Properties of Rational Irrational Numbers Use properties of rational irrational numbers. 3. Explain why the sum or product of two rational
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationSchneps, Leila; Colmez, Coralie. Math on Trial : How Numbers Get Used and Abused in the Courtroom. New York, NY, USA: Basic Books, 2013. p i.
New York, NY, USA: Basic Books, 2013. p i. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=2 New York, NY, USA: Basic Books, 2013. p ii. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=3 New
More informationFractions to decimals
Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of
More information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationParamedic Program PreAdmission Mathematics Test Study Guide
Paramedic Program PreAdmission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More informationAddition Methods. Methods Jottings Expanded Compact Examples 8 + 7 = 15
Addition Methods Methods Jottings Expanded Compact Examples 8 + 7 = 15 48 + 36 = 84 or: Write the numbers in columns. Adding the tens first: 47 + 76 110 13 123 Adding the units first: 47 + 76 13 110 123
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warmup problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationLINEAR EQUATIONS IN TWO VARIABLES
66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that
More informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
More informationSupplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Section 1 Real Numbers
Supplemental Worksheet Problems To Accompany: The PreAlgebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm
More informationACCUPLACER Arithmetic & Elementary Algebra Study Guide
ACCUPLACER Arithmetic & Elementary Algebra Study Guide Acknowledgments We would like to thank Aims Community College for allowing us to use their ACCUPLACER Study Guides as well as Aims Community College
More informationRational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have
8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationSection 33 Approximating Real Zeros of Polynomials
 Approimating Real Zeros of Polynomials 9 Section  Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationNUMBER SYSTEMS. 1.1 Introduction
NUMBER SYSTEMS 1.1 Introduction There are several number systems which we normally use, such as decimal, binary, octal, hexadecimal, etc. Amongst them we are most familiar with the decimal number system.
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More information47 Numerator Denominator
JH WEEKLIES ISSUE #22 20122013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational
More informationC A S I O f x  8 2 M S
C A S I O f x  8 2 M S MASTERING THE CALCULATOR USING THE CASIO fx82ms Learning and Teaching Support Unit (LTSU) The Learning Centre Guide book Written by Linda Galligan Published by University of Southern
More informationC A S I O f x  8 2 A U UNIVERSITY OF SOUTHERN QUEENSLAND. The Learning Centre Learning and Teaching Support Unit
C A S I O f x  8 2 A U UNIVERSITY OF SOUTHERN QUEENSLAND The Learning Centre Learning and Teaching Support Unit Mastering the calculator using the Casio fx82au Learning and Teaching Support Unit (LTSU)
More informationStudents will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationSquare Roots. Learning Objectives. PreActivity
Section 1. PreActivity Preparation Square Roots Our number system has two important sets of numbers: rational and irrational. The most common irrational numbers result from taking the square root of nonperfect
More informationGuidance paper  The use of calculators in the teaching and learning of mathematics
Guidance paper  The use of calculators in the teaching and learning of mathematics Background and context In mathematics, the calculator can be an effective teaching and learning resource in the primary
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationGrade 6 Math Circles. Binary and Beyond
Faculty of Mathematics Waterloo, Ontario N2L 3G1 The Decimal System Grade 6 Math Circles October 15/16, 2013 Binary and Beyond The cool reality is that we learn to count in only one of many possible number
More informationChapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
More informationM13/5/MATSD/SP1/ENG/TZ2/XX/M MARKSCHEME. May 2013 MATHEMATICAL STUDIES. Standard Level. Paper 1. 25 pages
M13/5/MATSD/SP1/ENG/TZ/XX/M MARKSCHEME May 013 MATHEMATICAL STUDIES Standard Level Paper 1 5 pages M13/5/MATSD/SP1/ENG/TZ/XX/M This markscheme is confidential and for the exclusive use of examiners in
More information2 When is a 2Digit Number the Sum of the Squares of its Digits?
When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch
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 informationCalculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationUnit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12
Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationMath Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
More informationPowerTeaching i3: Algebra I Mathematics
PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and
More informationKnowing and Using Number Facts
Knowing and Using Number Facts Use knowledge of place value and Use knowledge of place value and addition and subtraction of twodigit multiplication facts to 10 10 to numbers to derive sums and derive
More informationMath 10  Unit 3 Final Review  Numbers
Class: Date: Math 10  Unit Final Review  Numbers Multiple Choice Identify the choice that best answers the question. 1. Write the prime factorization of 60. a. 2 7 9 b. 2 6 c. 2 2 7 d. 2 7 2. Write the
More informationAnchorage School District/Alaska Sr. High Math Performance Standards Algebra
Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,
More informationPowers and roots 5.1. Previous learning. Objectives based on NC levels and (mainly level ) Lessons 1 Working with integer powers of numbers
N 5.1 Powers and roots Previous learning Before they start, pupils should be able to: recognise and use multiples, factors (divisors), common factor, highest common factor, lowest common multiple and primes
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationMathematics (Project Maths)
2010. M130 S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination Sample Paper Mathematics (Project Maths) Paper 2 Higher Level Time: 2 hours, 30 minutes 300 marks
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE  1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationSection 1.1 Systems of Numeration and Additive Systems of Numeration
Chapter 1 Counting and Measuring Section 1.1 Systems of Numeration and Additive Systems of Numeration Counting The language of mathematics began when human civilization began to develop words and later
More informationa cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
More informationC&A MM Inventory Relieve & Post to Job Cost Installation and Setup Instructions with Process Flow
4820 8 TH STREET SALEM, OREGON 97301 C&A MM Inventory Relieve & Post to Job Cost Installation and Setup Instructions with Process Flow The general purpose of this program is to relieve parts from inventory
More information6.1. The Exponential Function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The Exponential Function 6.1 Introduction In this block we revisit the use of exponents. We consider how the expression a x is defined when a is a positive number and x is irrational. Previously we have
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationPlaying with Numbers
PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationCommon Core Standards for Fantasy Sports Worksheets. Page 1
Scoring Systems Concept(s) Integers adding and subtracting integers; multiplying integers Fractions adding and subtracting fractions; multiplying fractions with whole numbers Decimals adding and subtracting
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More informationInfinite Algebra 1 supports the teaching of the Common Core State Standards listed below.
Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School
More information4.2 Euclid s Classification of Pythagorean Triples
178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationSECTION 16 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More information6 Proportion: Fractions, Direct and Inverse Variation, and Percent
6 Proportion: Fractions, Direct and Inverse Variation, and Percent 6.1 Fractions Every rational number can be written as a fraction, that is a quotient of two integers, where the divisor of course cannot
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationMATHCOUNTS TOOLBOX Facts, Formulas and Tricks
MATHCOUNTS TOOLBOX Facts, Formulas and Tricks MATHCOUNTS Coaching Kit 40 I. PRIME NUMBERS from 1 through 100 (1 is not prime!) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 II.
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationIrrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.
Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that
More informationUNCORRECTED PAGE PROOFS
number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.
More informationFind the Square Root
verview Math Concepts Materials Students who understand the basic concept of square roots learn how to evaluate expressions and equations that have expressions and equations TI30XS MultiView rational
More information