# EC121 Mathematical Techniques A Revision Notes

Size: px
Start display at page:

Transcription

2 EC Mathematical Techniques A Revision Notes in order to calculate the difference between them, which is (+) dollars. In other words, Anne has two dollars more than John. Cases (c) and (d) are a little more difficult. In case (c), Anne's bank balance is (+) dollars as before, but John's account is in the red; his balance is (-) dollars. When they add their balances, therefore, they find that their combined balance is only (+) dollars. In case (d), Anne's bank balance is (+) dollars and John's is (-), as in case (c). This time, though, they are subtracting John's balance from Anne's in order to calculate the difference between them, which is (+) dollars. In other words, Anne has eight dollars more than John. John would need to pay eight dollars into his account in order to have the same balance as Anne. In summary, adding a positive number [case (a)] and subtracting a negative number [case (d)] both result in addition of the two numbers. Subtracting a positive number [case (b)] and adding a negative number [case(c)] both result in subtraction of the second number from the first. In practice of course we don't usually bother to place a + sign in front of positive numbers, nor bother with the brackets. So cases (a) - (c) above would actually be written as: (a) (b) (c) (d) +. The result is (Anne and John's combined bank balance). -. The result is (the difference between Anne' balance and John's). + (-). The result is (their combined balances, when John is in the red). - (-). The result is (the difference between Anne's balance and John's, when John is in the red). It makes no difference to the application of the rules if the first number (, in our examples) is negative. Of course, it changes the answers, because we are now assuming that Anne's bank balance is (-) instead of. [Check for yourself that the answers to (a), (b), (c) and (d) in this case are -, -, - and - respectively.]. Multiplication and division with negative numbers One approach to understanding the rules for multiplication of positive and negative numbers is as follows: If a number is multiplied by +, the number is left unchanged. If a number is multiplied by -, the number is left unchanged in absolute magnitude but its sign is reversed; that is, if it was previously positive it becomes negative and vice versa. Thus: (+) x (+) (+) (+) x (-) (-) (-) x (+) (-) (-) x (-) (+) However, consistency clearly requires that these rules must hold for multiplication by any number, not just -. This implies that: D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

3 EC Mathematical Techniques A Revision Notes () (+) x (+) + () (+) x (-) - () (-) x (+) - () (-) x (-) + The way to remember these rules is that if the two numbers have the same sign (cases and ), the result is positive, while if they have different signs (cases and ), the result is negative. Division Since division simply reverses multiplication, consistency requires that division should obey the same sign rules as apply to multiplication. Thus: () (+) (+ ) () (+) ( ) () (-) ( + ) () (-) ( ) - + Notice that the division operation can be written in two ways: either using the sign, or by writing a fraction in which the first number appears in the top (called the numerator) and the second in the bottom (the denominator). This second way is more convenient, hence is more common than the first. Again, the way to remember these rules is that if the two numbers have the same sign (cases and ), the result is positive, while if they have different signs (cases and ), the result is negative. From now on we shall follow the normal convention and omit the + sign and the brackets in front of positive numbers.. Brackets and when we need them When addition and subtraction are mixed with multiplication and division, you get different answers depending on which part of the calculation you do first. For example, 6 + D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

4 EC Mathematical Techniques A Revision Notes is ambiguous. If you do the addition first you get 7, but if you do the division first you get Mathematicians have set up certain conventions in an attempt to avoid ambiguity in such cases. Specifically, the convention is that you work from left to right in an expression, carrying out any division operations first, then any multiplication, then addition and finally subtraction. Applying these conventions to the expression above tells us to do the division first, so the correct answer is 0. But, these conventions are not followed universally, and so the only way to be sure of avoiding ambiguity is to use brackets to indicate which of any alternatives is correct in the particular case. The rule is that whatever is in the brackets must be done first. Thus but (6 + ) ( ) However, because as noted earlier the division sign ( ) is not often used, these two expressions would not be written in this way. Instead we would write 6 + to denote (6 + ) 6 + When we write be divided by. That is why In contrast, we write it is understood that the whole of the numerator (that is, 6 + ) must to denote 6 + ( ). has the same meaning as (6 + ). 6 + To summarise, has the same meaning as (6 + ). In both expressions this means that the 6 and the must both be divided by. Overlooking this is one of the most common mistakes made by students taking Maths A. This rule will be considered further below, when we look at fractions.. Fractions In arithmetic the idea of fractions such as,, is familiar. As noted earlier, we often write, for example, to denote Thus a fractions (better described as a ratio or quotient) results whenever we divide one number by another. So we need to be clear about the rules for manipulating these fractions. Note that in any expression such as (also often written as /), is called the numerator and the denominator. D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

5 EC Mathematical Techniques A Revision Notes "Cancelling" and Fractions It is easy to see that. 6 What we have done here (almost without thinking) is to divide both the numerator and the denominator by ; that is: 6 6 We divided numerator and denominator by because we saw that both numerator and denominator were divisible by without remainder; in other words we saw that was a factor of both the numerator and the denominator. This process is called cancelling and can be done whenever you can find something which is a factor of both the numerator and the denominator. Before cancelling, though, you must make sure that the whole of the numerator and the whole of the denominator are divisible by the factor, without remainder. The process of cancelling can also be reversed; that is, we can multiply numerator and denominator by any number. For example, x x Cancelling and its reverse are permissible because multiplying or dividing both numerator and denominator by the same thing leave a fraction unchanged in its value.. Addition and Subtraction of fractions Addition Consider: + The way we tackle this is to see, intuitively, that a half is the same thing as two quarters, so the answer is + + Another example is: + Again we see, almost without thinking, that 0 and 0, so What we need to do now is to make this intuitive process systematic, as follows. In the previous example, 0 is called the common denominator. A common denominator is any number which is divisible (without remainder) by the denominators of the two fractions to be added. Thus 0 is a common denominator because both and divide into it without D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

6 EC Mathematical Techniques A Revision Notes 6 remainder. (This follows from the fact that 0 x ). However, 0, 0, 0 etc are also common denominators because they too can be divided by both five and two without remainder. But ten is the lowest common denominator. It is the smallest number which is divisible by both and without remainder. (Using the lowest common denominator when adding fractions is not essential but simplifies the calculation.) How do we find a common denominator? Consider the example: + 7 We have to find a number which is divisible by both 7 and without remainder. It occurs to us that if we take the product of 7 and, that is 7 X or 77, this is certain to be divisible by both 7 and without remainder. So we can then write: and 7 x 7 x x 7 x so The pattern here is that we took each fraction in turn and multiplied both its numerator and its denominator by the denominator of the other fraction. Thus in the example above where we took + becomes x x 0 and becomes x x 0 so Things are a little more tricky when there are three or more fractions to be added. The easiest method is to add them in pairs. For example, given + + we first add the / and the /, then add the result of this to /, as follows x x + + x x D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

7 EC Mathematical Techniques A Revision Notes 7 Alternatively we can multiply numerator and denominator of each fraction by the product of the other two (or more) denominators. This gives: + + xx xx + xx xx + xx xx This second method does not always result in the lowest common denominator being found; in our example the common denominator is 60, whereas the lowest common denominator is 0. Consequently may lead to some unnecessary calculation, but this method does have the merit of being reliable and reasonably easy to remember. Subtraction The method is exactly the same as the method for addition. For example Multiplication and Division of fractions Multiplication The rule for multiplying one fraction by another is very simple. You simply multiply the two numerators together to obtain the numerator of the answer, and multiply the two denominators together to obtain the denominator of the answer. Thus in general: X x x This generalises to any number of terms, e.g.: X X x x x x Reciprocals From the multiplication rule, we can get X X X X D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

8 EC Mathematical Techniques A Revision Notes Here, and are said to be reciprocals. By definition, their product is unity (in other words, when you multiply one by the other the result is ). "of" Sometimes we meet a question such as: what is half of? Obviously the answer is 9, but we need a procedure for answering this kind of question so that we don't have to rely on intuition (which can lead us astray) The key insight is to see that "half of " is found by multiplying by. That is, "half of " x 9. So "of" just means " multiplied by". For example, of x x x Division of Fractions From the multiplication rule above, we can get 0 X 0 X 0 X X But we also know that: 0 Combining these, we see that 0 0 X So dividing by is equivalent to multiplying by its reciprocal, This gives us the fundamental rule for dividing by a fraction: you simply invert the fraction and multiply instead of dividing. For example, Consider: The rule is that we first invert the fraction ( i.e. becomes ) it instead of dividing. Thus: and then multiply by X X D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

10 EC Mathematical Techniques A Revision Notes 0 x x Summarising, whenever you have something like + ; that is, something with addition or subtraction in the numerator or denominator, you must do the addition or subtraction first before doing any cancelling. But when you have something like x ; that is, something with only multiplication (or division) in the numerator and denominator, you can cancel individual elements in the numerator and denominator, if you wish. But you will also get the right answer if you choose instead to perform the multiplication or division in numerator or denominator first, and then do the cancelling.. Powers and roots Squares and square roots The product X is often written as. We read this as three squared and means threes multiplied together. Thus X 9. (It is called a square because is the area of a square with sides of length a.) Similarly X X, X X X X and so on. The process which is the reverse of squaring is called taking (or finding) the square root. Thus is the square root of 9 because 9. We write the square root of 9 as 9. Thus 9 because 9. Because squaring and taking the square root are mutually reversing processes, it follows that if you square a number and then take the square root, you arrive back where you started; that is: ( ). Similarly ( ). Negative square roots. Every positive number has square roots, one positive and the other negative. For example, 9 has two square roots. One, as we have just seen, is +. This is because (+) (+) X (+) 9. But (-) is also a square root of 9, because (-) (-) X (-) 9. (This is because when two negative numbers are multiplied together, the result is a positive number - see section. above). So the two square roots of 9 are + and - (sometimes written more compactly as ± ). In symbols, we write: 9 ±. We normally consider only the positive root unless the context of the problem requires us to consider both. D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

11 EC Mathematical Techniques A Revision Notes Square roots of negative numbers. Suppose we are seeking the square root of (-9). Let us denote the number we are looking for by the symbol y (this is just a label). By definition, y is a number such that y (-9). Thus y is negative. But, from section XX above we know that if y is a positive number then y positive. So y cannot be positive if y is negative. y X y is Similarly we know from section XX above that if y is a negative number then y is positive. So y cannot be negative if y is negative. y X y Combining these two, we can say that y can be neither positive nor negative if y is negative. Therefore the square root of a negative number does not exist. (This is not quite true. In more advanced maths the concept of an imaginary number is introduced, and this number has the property that its square is negative. Thus to be completely precise we have to say that the square root of a negative number does exist, but is not a real number (but rather, is an imaginary number). This need not concern us greatly in this module.) 6. Decimals, fractions and percentages 6. Converting decimal numbers to fractions and vice versa To quickly refresh our memories as to the meaning of decimal numbers, recall that by definition 0. 0, 0.0, , (which after cancelling between top and bottom) 000 and so on. The pattern is that the number of zeros in the denominator of the fraction is always equal to the number of digits after the decimal point. (Except that a zero or zeros at the right hand end don't count. E.g ; and so on. 00 Some conversions from fractions to decimals and vice versa are fairly obvious, e.g. 0. ; 0. ; 0. 0 The underlying rule for converting a fraction into a decimal is that you divide the numerator by the denominator, using decimal long division (being careful to get the decimal point in the right place!). For example, given we first divide into. This won t go, so we add a D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

12 EC Mathematical Techniques A Revision Notes zero and simultaneously enter a decimal point (see figure ). We then divide into 0, which goes times, so the answer is 0.. Figure : 0 ) 0 Consider as another example. To convert this to a decimal number, we divide into. This won t go, so we add a zero and enter a decimal point (see figure ). We then divide into 0, which goes once with remainder. We add another zero and divide into 0, which goes twice with remainder. We add another zero and divide into 0, which goes times without remainder, giving us our answer: 0. Figure 0 ) In practice you will probably use your calculator for this, but it's worth seeing how it is done without a calculator. To convert a decimal number into a fraction, simply apply the definitions above. For example, 0., which after cancelling, 0 Similarly 0., which after cancelling, 000 A strange feature of decimal numbers is that some fractions ahve no exact decimal equivalent. For example, (approximately) with the 6's recurring indefinitely. 6. Percentages (i) To convert a decimal fraction into a percentage, simply multiply by 00. For example in the case of 0. we get: D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

13 EC Mathematical Techniques A Revision Notes x 00 0 (per cent) 0 0 (ii) To convert any fraction into a percentage, simply multiply by 00. For example,. We get: 00 X 00 0 (per cent). So, 0%. Similarly 9 as a percentage is: 00 X 00.% (approx.) 9 9 (iii) To convert any percentage into a fraction, simply reverse the above, i.e. divide by 00. Thus 0% becomes X (iii) To find, say % of 7, recall that % simply means five-hundredths. So we have to divide 7 into 00 parts and then take of them. To do this, we divide 7 by 00, then multiply the result by. This gives: 7 X (per cent). (With some practice, you ll find you can amaze your friends by calculating, say, % of 6 (.%) in your head). 7. Some Additional Symbols To conclude these notes, here are some symbols which we will be using later and which may be new to you: means "is NOT equal to" means "is approximately equal to" means "is identically equal to". This conveys a stronger meaning than the sign. The difference between and is important in Economics and we shall return to it later. means "infinity". (Note that infinity is NOT "a very large number". Any number, however large, is finite; while the essence of infinity is that it is not finite) - means "minus infinity" x means the absolute value of x, i.e. ignoring its sign. For example, > means "greater than" (for example, > ) < means "less than" (for example, < ) The symbols > and < are called inequalities and can be used in a variety of ways. For example if we want to say that some number called z is positive (i.e. greater than zero), we can simply write: z > 0. If on the other hand we want to say the some number k is negative, we can write: k < 0. A third possibility is that we might want to say that some D:\Temp\ECa.RevN.doc.doc G.T.Renshaw 00

### Topic 3: Fractions. Topic 1 Integers. Topic 2 Decimals. Topic 3 Fractions. Topic 4 Ratios. Topic 5 Percentages. Topic 6 Algebra

Topic : Fractions Topic Integers Topic Decimals Topic Fractions Topic Ratios Topic Percentages Duration / weeks Content Outline PART (/ week) Introduction Converting Fractions to Decimals Converting Decimals

### Math 7 Notes Unit Three: Applying Rational Numbers

Math 7 Notes Unit Three: Applying Rational Numbers Strategy note to teachers: Typically students need more practice doing computations with fractions. You may want to consider teaching the sections on

### FRACTIONS AND DECIMALS

Mathematics Revision Guides Fractions and Decimals Page of MK HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier FRACTIONS AND DECIMALS Version: Date: -0-0 Mathematics Revision Guides

### NON-CALCULATOR ARITHMETIC

Mathematics Revision Guides Non-Calculator Arithmetic Page 1 of 30 M.K. HOME TUITION Mathematics Revision Guides: Level: GCSE Foundation Tier NON-CALCULATOR ARITHMETIC Version: 3.2 Date: 21-10-2016 Mathematics

### Divisibility Rules and Their Explanations

Divisibility Rules and Their Explanations Increase Your Number Sense These divisibility rules apply to determining the divisibility of a positive integer (1, 2, 3, ) by another positive integer or 0 (although

### Skill 1: Multiplying Polynomials

CS103 Spring 2018 Mathematical Prerequisites Although CS103 is primarily a math class, this course does not require any higher math as a prerequisite. The most advanced level of mathematics you'll need

### Chapter 3 Data Representation

Chapter 3 Data Representation The focus of this chapter is the representation of data in a digital computer. We begin with a review of several number systems (decimal, binary, octal, and hexadecimal) and

### EXAMPLE 1. Change each of the following fractions into decimals.

CHAPTER 1. THE ARITHMETIC OF NUMBERS 1.4 Decimal Notation Every rational number can be expressed using decimal notation. To change a fraction into its decimal equivalent, divide the numerator of the fraction

### Section 1.2 Fractions

Objectives Section 1.2 Fractions Factor and prime factor natural numbers Recognize special fraction forms Multiply and divide fractions Build equivalent fractions Simplify fractions Add and subtract fractions

### Learning Log Title: CHAPTER 3: ARITHMETIC PROPERTIES. Date: Lesson: Chapter 3: Arithmetic Properties

Chapter 3: Arithmetic Properties CHAPTER 3: ARITHMETIC PROPERTIES Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 3: Arithmetic Properties Date: Lesson: Learning Log Title:

### Dr. Relja Vulanovic Professor of Mathematics Kent State University at Stark c 2008

MATH-LITERACY MANUAL Dr. Relja Vulanovic Professor of Mathematics Kent State University at Stark c 2008 1 Real Numbers 1.1 Sets 1 1.2 Constants and Variables; Real Numbers 7 1.3 Operations with Numbers

### Switching Circuits and Logic Design Prof. Indranil Sengupta Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Switching Circuits and Logic Design Prof. Indranil Sengupta Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 02 Octal and Hexadecimal Number Systems Welcome

### MA 1128: Lecture 02 1/22/2018

MA 1128: Lecture 02 1/22/2018 Exponents Scientific Notation 1 Exponents Exponents are used to indicate how many copies of a number are to be multiplied together. For example, I like to deal with the signs

### Integers are whole numbers; they include negative whole numbers and zero. For example -7, 0, 18 are integers, 1.5 is not.

What is an INTEGER/NONINTEGER? Integers are whole numbers; they include negative whole numbers and zero. For example -7, 0, 18 are integers, 1.5 is not. What is a REAL/IMAGINARY number? A real number is

### Gateway Regional School District VERTICAL ALIGNMENT OF MATHEMATICS STANDARDS Grades 3-6

NUMBER SENSE & OPERATIONS 3.N.1 Exhibit an understanding of the values of the digits in the base ten number system by reading, modeling, writing, comparing, and ordering whole numbers through 9,999. Our

### GENERAL MATH FOR PASSING

GENERAL MATH FOR PASSING Your math and problem solving skills will be a key element in achieving a passing score on your exam. It will be necessary to brush up on your math and problem solving skills.

### 1. Let n be a positive number. a. When we divide a decimal number, n, by 10, how are the numeral and the quotient related?

Black Converting between Fractions and Decimals Unit Number Patterns and Fractions. Let n be a positive number. When we divide a decimal number, n, by 0, how are the numeral and the quotient related?.

### Rational Numbers CHAPTER Introduction

RATIONAL NUMBERS Rational Numbers CHAPTER. Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + () is solved when x, because this value of

### Real Numbers finite subset real numbers floating point numbers Scientific Notation fixed point numbers

Real Numbers We have been studying integer arithmetic up to this point. We have discovered that a standard computer can represent a finite subset of the infinite set of integers. The range is determined

### 2.Simplification & Approximation

2.Simplification & Approximation As we all know that simplification is most widely asked topic in almost every banking exam. So let us try to understand what is actually meant by word Simplification. Simplification

### MAT 003 Brian Killough s Instructor Notes Saint Leo University

MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample

### Topic C. Communicating the Precision of Measured Numbers

Topic C. Communicating the Precision of Measured Numbers C. page 1 of 14 Topic C. Communicating the Precision of Measured Numbers This topic includes Section 1. Reporting measurements Section 2. Rounding

### Pre-Algebra Notes Unit Five: Rational Numbers and Equations

Pre-Algebra Notes Unit Five: Rational Numbers and Equations Rational Numbers Rational numbers are numbers that can be written as a quotient of two integers. Since decimals are special fractions, all the

### Mathematics. Name: Class: Transforming Life chances

Mathematics Name: Class: Transforming Life chances Children first- Aspire- Challenge- Achieve Aspire: To be the best I can be in everything that I try to do. To use the adults and resources available both

### Math 340 Fall 2014, Victor Matveev. Binary system, round-off errors, loss of significance, and double precision accuracy.

Math 340 Fall 2014, Victor Matveev Binary system, round-off errors, loss of significance, and double precision accuracy. 1. Bits and the binary number system A bit is one digit in a binary representation

### Chapter 1: Number and Operations

Chapter 1: Number and Operations 1.1 Order of operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply

### Basic Arithmetic Operations

Basic Arithmetic Operations Learning Outcome When you complete this module you will be able to: Perform basic arithmetic operations without the use of a calculator. Learning Objectives Here is what you

### Pre-Algebra Notes Unit Five: Rational Numbers and Equations

Pre-Algebra Notes Unit Five: Rational Numbers and Equations Rational Numbers Rational numbers are numbers that can be written as a quotient of two integers. Since decimals are special fractions, all the

### Pre-Algebra Notes Unit Five: Rational Numbers and Equations

Pre-Algebra Notes Unit Five: Rational Numbers and Equations Rational Numbers Rational numbers are numbers that can be written as a quotient of two integers. Since decimals are special fractions, all the

### Chapter 1 Operations With Numbers

Chapter 1 Operations With Numbers Part I Negative Numbers You may already know what negative numbers are, but even if you don t, then you have probably seen them several times over the past few days. If

### Course Learning Outcomes for Unit I. Reading Assignment. Unit Lesson. UNIT I STUDY GUIDE Number Theory and the Real Number System

UNIT I STUDY GUIDE Number Theory and the Real Number System Course Learning Outcomes for Unit I Upon completion of this unit, students should be able to: 2. Relate number theory, integer computation, and

### Rev Name Date. . Round-off error is the answer to the question How wrong is the rounded answer?

Name Date TI-84+ GC 7 Avoiding Round-off Error in Multiple Calculations Objectives: Recall the meaning of exact and approximate Observe round-off error and learn to avoid it Perform calculations using

### !"!!!"!!"!! = 10!!!!!(!!) = 10! = 1,000,000

Math Review for AP Chemistry The following is a brief review of some of the math you should remember from your past. This is meant to jog your memory and not to teach you something new. If you find you

### Section 1.1 Definitions and Properties

Section 1.1 Definitions and Properties Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Abbreviate repeated addition using Exponents and Square

### Gateway Regional School District VERTICAL ARTICULATION OF MATHEMATICS STANDARDS Grades K-4

NUMBER SENSE & OPERATIONS K.N.1 Count by ones to at least 20. When you count, the last number word you say tells the number of items in the set. Counting a set of objects in a different order does not

### 3.4 Equivalent Forms of Rational Numbers: Fractions, Decimals, Percents, and Scientific Notation

3.4 Equivalent Forms of Rational Numbers: Fractions, Decimals, Percents, and Scientific Notation We know that every rational number has an infinite number of equivalent fraction forms. For instance, 1/

### 0001 Understand the structure of numeration systems and multiple representations of numbers. Example: Factor 30 into prime factors.

NUMBER SENSE AND OPERATIONS 0001 Understand the structure of numeration systems and multiple representations of numbers. Prime numbers are numbers that can only be factored into 1 and the number itself.

### BASIC MATH CONTENTS. Section 1... Whole Number Review. Section 2... Decimal Review. Section 3... Fraction Review. Section 4...

BASIC MATH The purpose of this booklet is to refresh the reader s skills in basic mathematics. There are basic mathematical processes, which must be followed throughout all areas of math applications.

### Excerpt from "Art of Problem Solving Volume 1: the Basics" 2014 AoPS Inc.

Chapter 5 Using the Integers In spite of their being a rather restricted class of numbers, the integers have a lot of interesting properties and uses. Math which involves the properties of integers is

### Number System. Introduction. Natural Numbers (N) Whole Numbers (W) Integers (Z) Prime Numbers (P) Face Value. Place Value

1 Number System Introduction In this chapter, we will study about the number system and number line. We will also learn about the four fundamental operations on whole numbers and their properties. Natural

### ( 3) ( 4 ) 1. Exponents and Radicals ( ) ( xy) 1. MATH 102 College Algebra. still holds when m = n, we are led to the result

Exponents and Radicals ZERO & NEGATIVE EXPONENTS If we assume that the relation still holds when m = n, we are led to the result m m a m n 0 a = a = a. Consequently, = 1, a 0 n n a a a 0 = 1, a 0. Then

### Radar. Electronics. Television UNITED ELECTRONICS LABORATORIES ARITHMETIC FOR ELECTRONICS ASSIGNMENT 4 REVISED 1967

Electronics Television Radar UNITED ELECTRONICS LABORATORIES REVISED 967 COPYRIGHT 956 UNITED ELECTRONICS LABORATORIES ARITHMETIC FOR ELECTRONICS ASSIGNMENT 4 ASSIGNMENT 4 ARITHMETIC FOR ELECTRONICS In

### Lesson 1: Arithmetic Review

In this lesson we step back and review several key arithmetic topics that are extremely relevant to this course. Before we work with algebraic expressions and equations, it is important to have a good

### Real Numbers finite subset real numbers floating point numbers Scientific Notation fixed point numbers

Real Numbers We have been studying integer arithmetic up to this point. We have discovered that a standard computer can represent a finite subset of the infinite set of integers. The range is determined

### CHAPTER 1: INTEGERS. Image from CHAPTER 1 CONTENTS

CHAPTER 1: INTEGERS Image from www.misterteacher.com CHAPTER 1 CONTENTS 1.1 Introduction to Integers 1. Absolute Value 1. Addition of Integers 1.4 Subtraction of Integers 1.5 Multiplication and Division

### Watkins Mill High School. Algebra 2. Math Challenge

Watkins Mill High School Algebra 2 Math Challenge "This packet will help you prepare for Algebra 2 next fall. It will be collected the first week of school. It will count as a grade in the first marking

### NUMBERS AND NUMBER RELATIONSHIPS

MODULE MODULE CHAPTERS Numbers and number patterns 2 Money matters KEY SKILLS writing rational numbers as terminating or recurring decimals identifying between which two integers any irrational number

### DLD VIDYA SAGAR P. potharajuvidyasagar.wordpress.com. Vignana Bharathi Institute of Technology UNIT 1 DLD P VIDYA SAGAR

UNIT I Digital Systems: Binary Numbers, Octal, Hexa Decimal and other base numbers, Number base conversions, complements, signed binary numbers, Floating point number representation, binary codes, error

### Chapter 1. Math review. 1.1 Some sets

Chapter 1 Math review This book assumes that you understood precalculus when you took it. So you used to know how to do things like factoring polynomials, solving high school geometry problems, using trigonometric

### Topic 2: Decimals. Topic 1 Integers. Topic 2 Decimals. Topic 3 Fractions. Topic 4 Ratios. Topic 5 Percentages. Topic 6 Algebra

41 Topic 2: Decimals Topic 1 Integers Topic 2 Decimals Topic 3 Fractions Topic 4 Ratios Duration 1/2 week Content Outline Introduction Addition and Subtraction Multiplying and Dividing by Multiples of

### Manipulate expressions containing surds and rationalise denominators (A*) Able to simplify surds (A)

Moving from A to A* Manipulate expressions containing surds and rationalise denominators (A*) Solve using surds (A*) A* Solve direct and inverse variation three variables (A*) A* Find formulae describing

### Rational Number is a number that can be written as a quotient of two integers. DECIMALS are special fractions whose denominators are powers of 10.

PA Ch 5 Rational Expressions Rational Number is a number that can be written as a quotient of two integers. DECIMALS are special fractions whose denominators are powers of 0. Since decimals are special

### 1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM

1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM 1.1 Introduction Given that digital logic and memory devices are based on two electrical states (on and off), it is natural to use a number

### GCSE-AS Mathematics Bridging Course. Chellaston School. Dr P. Leary (KS5 Coordinator) Monday Objectives. The Equation of a Line.

GCSE-AS Mathematics Bridging Course Chellaston School Dr (KS5 Coordinator) Monday Objectives The Equation of a Line Surds Linear Simultaneous Equations Tuesday Objectives Factorising Quadratics & Equations

### 3.1 DATA REPRESENTATION (PART C)

3.1 DATA REPRESENTATION (PART C) 3.1.3 REAL NUMBERS AND NORMALISED FLOATING-POINT REPRESENTATION In decimal notation, the number 23.456 can be written as 0.23456 x 10 2. This means that in decimal notation,

### For more information, see the Math Notes box in Lesson of the Core Connections, Course 1 text.

Number TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have eactly two factors, namely, one and itself, are called

### Section A Arithmetic ( 5) Exercise A

Section A Arithmetic In the non-calculator section of the examination there might be times when you need to work with quite awkward numbers quickly and accurately. In particular you must be very familiar

### Summer Assignment Glossary

Algebra 1.1 Summer Assignment Name: Date: Hour: Directions: Show all work for full credit using a pencil. Circle your final answer. This assignment is due the first day of school. Use the summer assignment

### SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Numbers & Number Systems

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Numbers & Number Systems Introduction Numbers and Their Properties Multiples and Factors The Division Algorithm Prime and Composite Numbers Prime Factors

### Properties and Definitions

Section 0.1 Contents: Operations Defined Multiplication as an Abbreviation Visualizing Multiplication Commutative Properties Parentheses Associative Properties Identities Zero Product Answers to Exercises

### Fraction to Percents Change the fraction to a decimal (see above) and then change the decimal to a percent (see above).

PEMDAS This is an acronym for the order of operations. Order of operations is the order in which you complete problems with more than one operation. o P parenthesis o E exponents o M multiplication OR

### Mini-Lesson 1. Section 1.1: Order of Operations PEMDAS

Name: Date: 1 Section 1.1: Order of Operations PEMDAS If we are working with a mathematical expression that contains more than one operation, then we need to understand how to simplify. The acronym PEMDAS

### Example: Which of the following expressions must be an even integer if x is an integer? a. x + 5

8th Grade Honors Basic Operations Part 1 1 NUMBER DEFINITIONS UNDEFINED On the ACT, when something is divided by zero, it is considered undefined. For example, the expression a bc is undefined if either

### Helping Students Understand Pre-Algebra

Helping Students Understand Pre-Algebra By Barbara Sandall, Ed.D., & Mary Swarthout, Ph.D. COPYRIGHT 2005 Mark Twain Media, Inc. ISBN 10-digit: 1-58037-294-5 13-digit: 978-1-58037-294-7 Printing No. CD-404021

### KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS

DOMAIN I. COMPETENCY 1.0 MATHEMATICS KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS Skill 1.1 Compare the relative value of real numbers (e.g., integers, fractions, decimals, percents, irrational

### Slide Set 1. for ENEL 339 Fall 2014 Lecture Section 02. Steve Norman, PhD, PEng

Slide Set 1 for ENEL 339 Fall 2014 Lecture Section 02 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2014 ENEL 353 F14 Section

### Table of Laplace Transforms

Table of Laplace Transforms 1 1 2 3 4, p > -1 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Heaviside Function 27 28. Dirac Delta Function 29 30. 31 32. 1 33 34. 35 36. 37 Laplace Transforms

### Fractions. There are several terms that are commonly used when working with fractions.

Chapter 0 Review of Arithmetic Fractions There are several terms that are commonly used when working with fractions. Fraction: The ratio of two numbers. We use a division bar to show this ratio. The number

### Lesson 1: Arithmetic Review

Lesson 1: Arithmetic Review Topics and Objectives: Order of Operations Fractions o Improper fractions and mixed numbers o Equivalent fractions o Fractions in simplest form o One and zero Operations on

### Algebra 2 Semester 1 (#2221)

Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the 2016-2017 Course Guides for the following course: Algebra 2 Semester

### Mathematical Data Operators

Mathematical Data Operators Programming often requires numbers to be manipulated. This usually involves standard mathematical operators: addition, subtraction, multiplication and division. It can also

### SECTION 3. ROUNDING, ESTIMATING, AND USING A CALCULATOR

SECTION 3. ROUNDING, ESTIMATING, AND USING A CALCULATOR Exact numbers are not always necessary or desirable. Sometimes it may be necessary to express the number which is a result of a calculation to a

### 9 abcd = dcba b + 90c = c + 10b b = 10c.

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

### Math 7 Notes Unit 2B: Rational Numbers

Math 7 Notes Unit B: Rational Numbers Teachers Before we move to performing operations involving rational numbers, we must be certain our students have some basic understandings and skills. The following

### Chapter 2: Number Systems

Chapter 2: Number Systems Logic circuits are used to generate and transmit 1s and 0s to compute and convey information. This two-valued number system is called binary. As presented earlier, there are many

### NUMBER SENSE AND OPERATIONS. Competency 0001 Understand the structure of numeration systems and multiple representations of numbers.

SUBAREA I. NUMBER SENSE AND OPERATIONS Competency 0001 Understand the structure of numeration systems and multiple representations of numbers. Prime numbers are numbers that can only be factored into 1

### Decimals should be spoken digit by digit eg 0.34 is Zero (or nought) point three four (NOT thirty four).

Numeracy Essentials Section 1 Number Skills Reading and writing numbers All numbers should be written correctly. Most pupils are able to read, write and say numbers up to a thousand, but often have difficulty

### Basics of Computational Geometry

Basics of Computational Geometry Nadeem Mohsin October 12, 2013 1 Contents This handout covers the basic concepts of computational geometry. Rather than exhaustively covering all the algorithms, it deals

### Multiply Decimals Multiply # s, Ignore Decimals, Count # of Decimals, Place in Product from right counting in to left

Multiply Decimals Multiply # s, Ignore Decimals, Count # of Decimals, Place in Product from right counting in to left Dividing Decimals Quotient (answer to prob), Dividend (the # being subdivided) & Divisor

### Section 2.3 Rational Numbers. A rational number is a number that may be written in the form a b. for any integer a and any nonzero integer b.

Section 2.3 Rational Numbers A rational number is a number that may be written in the form a b for any integer a and any nonzero integer b. Why is division by zero undefined? For example, we know that

### Increasing/Decreasing Behavior

Derivatives and the Shapes of Graphs In this section, we will specifically discuss the information that f (x) and f (x) give us about the graph of f(x); it turns out understanding the first and second

### Know how to use fractions to describe part of something Write an improper fraction as a mixed number Write a mixed number as an improper fraction

. Fractions Know how to use fractions to describe part of something Write an improper fraction as a mixed number Write a mixed number as an improper fraction Key words fraction denominator numerator proper

### 9 abcd = dcba b + 90c = c + 10b b = 10c.

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

### NFC ACADEMY MATH 600 COURSE OVERVIEW

NFC ACADEMY MATH 600 COURSE OVERVIEW Math 600 is a full-year elementary math course focusing on number skills and numerical literacy, with an introduction to rational numbers and the skills needed for

### Engineering Mechanics Prof. Siva Kumar Department of Civil Engineering Indian Institute of Technology, Madras Statics - 4.3

Engineering Mechanics Prof. Siva Kumar Department of Civil Engineering Indian Institute of Technology, Madras Statics - 4.3 In this case let s say delta B and delta C are the kinematically consistent displacements.

### Pre-Algebra Notes Unit Five: Rational Numbers; Solving Equations & Inequalities

Pre-Algebra Notes Unit Five: Rational Numbers; Solving Equations & Inequalities Rational Numbers Rational numbers are numbers that can be written as a quotient of two integers. Since decimals are special

### Binary, Hexadecimal and Octal number system

Binary, Hexadecimal and Octal number system Binary, hexadecimal, and octal refer to different number systems. The one that we typically use is called decimal. These number systems refer to the number of

### Counting shapes 1.4.6

GRADE R_TERM 1 WEEK TOPIC CONTENT CAMI KEYSTROKE CAMI Program Count in ones 1.1.1.1; 1.1.1.2; 1.1.1.3 1.1.1.4 Cami Math Count pictures 1.1.3.1; 1.1.3.2; 1 & 2 Counting 1.1.3.3; 1.1.3.4; Counting in units

### Mathematics; Gateshead Assessment Profile (MGAP) Year 6 Understanding and investigating within number

Year 6 Understanding and investigating within number Place value, ordering and rounding Counting reading, writing, comparing, ordering and rounding whole numbers using place value Properties of numbers

### Raising achievement Foundation/Higher Tier Grades 1 9

Year 8 Maths Revision List Summer 018 The list below should give an indication of the material that Year 8 pupils will be tested on in Summer 018. Pupils should rate their confidence in each of the skills

### Accuplacer Arithmetic Study Guide

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 (the number on the bottom) Example: parts

### ROUNDING ERRORS LAB 1. OBJECTIVE 2. INTRODUCTION

ROUNDING ERRORS LAB Imagine you are traveling in Italy, and you are trying to convert \$27.00 into Euros. You go to the bank teller, who gives you 20.19. Your friend is with you, and she is converting \$2,700.00.

### Unit 7 Number System and Bases. 7.1 Number System. 7.2 Binary Numbers. 7.3 Adding and Subtracting Binary Numbers. 7.4 Multiplying Binary Numbers

Contents STRAND B: Number Theory Unit 7 Number System and Bases Student Text Contents Section 7. Number System 7.2 Binary Numbers 7.3 Adding and Subtracting Binary Numbers 7.4 Multiplying Binary Numbers

### Project 2: How Parentheses and the Order of Operations Impose Structure on Expressions

MAT 51 Wladis Project 2: How Parentheses and the Order of Operations Impose Structure on Expressions Parentheses show us how things should be grouped together. The sole purpose of parentheses in algebraic

### Accuplacer Arithmetic Review

Accuplacer Arithmetic Review Hennepin Technical College Placement Testing for Success Page Overview The Arithmetic section of ACCUPLACER contains 7 multiple choice questions that measure your ability to

### Rational and Irrational Numbers

LESSON. Rational and Irrational Numbers.NS. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion;... lso.ns.2,.ee.2? ESSENTIL QUESTION