Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0. Rational numbers are closed under addition, subtraction, and multiplication but not division. Addition and multiplication of rational numbers are commutative, but subtraction and division are not. Addition is associative for rational numbers, but subtraction is not.
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0.
Rational numbers are closed under addition, subtraction, and multiplication, but not division. Addition, subtraction and multiplication of rational numbers are commutative, but division is not. Addition of rational numbers is associative, but subtraction is not.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, but not division. They are commutative for addition and multiplication, but not for subtraction or division. Addition is associative for rational numbers, but subtraction is not.
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
This document discusses key concepts in the real number system including:
- Rational numbers that can be expressed as ratios of integers, and irrational numbers that cannot.
- Integers, including positive, negative and whole numbers.
- Properties of addition like commutativity, associativity and closure.
- Properties of multiplication like commutativity, associativity and distributivity.
- Absolute value and rules for performing operations on signed numbers like addition, subtraction, multiplication and division.
This document discusses several mathematical concepts including sets, real numbers, inequalities, and absolute value. It provides examples and properties for each concept. Sets can be defined by listing elements or with a common characteristic. Real numbers include natural numbers, integers, rationals, and irrationals. Properties of real numbers include closure under addition and multiplication. Inequalities can be solved using the same methods as equations while maintaining the inequality sign. Absolute value gives the distance of a number from zero and has properties related to products and sums.
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0.
Rational numbers are closed under addition, subtraction, and multiplication, but not division. Addition, subtraction and multiplication of rational numbers are commutative, but division is not. Addition of rational numbers is associative, but subtraction is not.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, but not division. They are commutative for addition and multiplication, but not for subtraction or division. Addition is associative for rational numbers, but subtraction is not.
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
This document discusses key concepts in the real number system including:
- Rational numbers that can be expressed as ratios of integers, and irrational numbers that cannot.
- Integers, including positive, negative and whole numbers.
- Properties of addition like commutativity, associativity and closure.
- Properties of multiplication like commutativity, associativity and distributivity.
- Absolute value and rules for performing operations on signed numbers like addition, subtraction, multiplication and division.
This document discusses several mathematical concepts including sets, real numbers, inequalities, and absolute value. It provides examples and properties for each concept. Sets can be defined by listing elements or with a common characteristic. Real numbers include natural numbers, integers, rationals, and irrationals. Properties of real numbers include closure under addition and multiplication. Inequalities can be solved using the same methods as equations while maintaining the inequality sign. Absolute value gives the distance of a number from zero and has properties related to products and sums.
CLASS VII -operations on rational numbers(1).pptxRajkumarknms
This document discusses properties of operations on rational numbers. It covers:
1) Addition of rational numbers, including having the same or different denominators. Properties include closure, commutativity, and additive identity.
2) Subtraction of rational numbers and its properties, noting the difference property and lack of an identity element.
3) Multiplication of rational numbers by multiplying numerators and denominators. Properties are closure, commutativity, associativity, identity of 1, and annihilation by 0.
4) Distributive property relating multiplication and addition/subtraction of rational numbers.
The document discusses number systems and provides examples of different types of numbers. It begins by explaining how early humans counted items without a formal system of numbers. The key developments were the creation of numbers and the number zero, which allowed people to answer questions about quantities.
The document then reviews natural numbers, whole numbers, and integers. It introduces rational numbers as numbers that can be expressed as fractions. Rational numbers can be positive or negative. Any number that cannot be expressed as a rational number, such as the square root of 2, is considered irrational. Real numbers include all rational and irrational numbers.
This document defines and explains key concepts related to real numbers and algebraic expressions. It introduces sets and subsets of real numbers like integers, rational numbers, and irrational numbers. It describes properties of real numbers including addition, multiplication, order, and absolute value. It also covers representing real numbers on a number line, algebraic expressions, and properties of negatives.
The document discusses various properties of real numbers including the commutative, associative, identity, inverse, zero, and distributive properties. It also covers topics such as combining like terms, translating word phrases to algebraic expressions, and simplifying algebraic expressions. Examples are provided to illustrate each concept along with explanations of key terms like coefficients, variables, and like terms.
1. Natural numbers include counting numbers like 1, 2, 3, and continue indefinitely. Whole numbers include natural numbers plus zero. Integers include whole numbers and their opposites.
2. Rational numbers can be written as a fraction, like 1.5 = 3/2. Irrational numbers cannot be written as a fraction, like π.
3. The four basic operations are addition, subtraction, multiplication, and division. Addition and subtraction follow rules about sign and order. Multiplication and division rules depend on the signs of the factors or dividend and divisor.
This document provides learning objectives and content about rational and irrational numbers for a Class 9 mathematics lesson. It begins by defining different types of numbers - natural, whole, integers, rational, and irrational - and provides examples. It then explains rational numbers as those that can be written as fractions p/q, and irrational numbers as those that cannot be expressed as fractions. Various methods are provided for representing and finding rational numbers between two given rational numbers, as well as representing irrational numbers on the number line. Finally, the document discusses operations involving rational and irrational numbers.
This document provides a summary of key terms in mathematical English for concepts in arithmetic, algebra, geometry, number theory, and more. Some key points covered include:
- Terms for integers, fractions, real and complex numbers, exponents, and basic arithmetic operations.
- Algebraic expressions, indices, matrices, inequalities, polynomial equations, and congruences.
- The use of definite and indefinite articles for theorems, conjectures, and mathematical concepts.
- Concepts in number theory like Fermat's Little Theorem and the Chinese Remainder Theorem.
- Terms for geometric concepts like points, lines, intersections, and rectangles.
english mathematics dictionary
kamus bahassa inggris untuk matematika
oleh neneng
Nurwaningsih
(06081281520066)
Nurwaningsih30@gmail.com
PROGRAM STUDI PENDIDIKAN MATEMATIKA
FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN
UNIVERSITAS SRIWIJAYA
INDRALAYA
2017
semoga bermanfaat
This document defines integers and the four basic integer operations - addition, subtraction, multiplication, and division. It provides rules for performing each operation on integers, such as the product of two integers with the same sign is positive and the product of two integers with different signs is negative. Examples are included to demonstrate applying the rules to solve integer operation problems.
The document discusses real numbers and their subsets. It defines natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. Examples are provided of different types of numbers. Classification of numbers using Venn diagrams is demonstrated. Rounding and truncating decimals is also covered.
This document provides an overview of the topics covered in an introductory mathematics analysis course for business, economics, and social sciences. It includes:
1) A review of key concepts like algebra, subsets of real numbers, properties of operations, and graphing numbers on a number line.
2) An outline of course structure with sections on algebra, algebraic expressions, fractions, and mathematical systems.
3) Examples of problems and their step-by-step solutions covering topics like simplifying expressions, factoring, addition/subtraction of fractions, and properties of real numbers.
This document provides an overview of irrational numbers and surds for a GCSE maths class. It begins by defining rational and irrational numbers, then discusses surds and how to manipulate them within expressions and fractions. Examples are provided of simplifying surds, expanding expressions involving surds, rationalizing denominators, and solving worksheet questions involving surds. The document aims to help students understand the difference between rational and irrational numbers and how to work with surds.
- Rational numbers include integers and can be written as a ratio of two integers. Rational numbers in decimal form either terminate or repeat.
- Irrational numbers cannot be written as a ratio of integers and their decimal representations do not terminate or repeat, such as sqrt(2).
- The document discusses the properties of operations like addition, subtraction, multiplication and division when performed on different number sets - integers, rational numbers, whole numbers. It is shown that some operations are commutative and associative while others are not, depending on the number set.
This document discusses integers and the four basic operations that can be performed on them - addition, subtraction, multiplication, and division. It defines an integer as a positive or negative whole number including 0. It provides rules for performing each operation, such as the product of two integers with the same sign is positive and with different signs is negative for multiplication. Examples are worked through for each operation to demonstrate how to apply the rules.
At the end of the lesson, the learner will be able to:
divide integers (with non-zero divisor)
interpret quotients of rational numbers by describing real-world contexts
1. The document discusses subsets of real numbers including natural numbers, whole numbers, integers, and rational numbers.
2. Natural numbers are used for counting and start at 1. Whole numbers are formed by adding 0 to the natural numbers. Integers are formed by adding the negatives of natural numbers to whole numbers.
3. Rational numbers can be expressed as fractions a/b where a and b are integers and b is not equal to 0. Their decimal representations either terminate or repeat.
This document provides an introduction to integers through five parts:
Part I defines key integer vocabulary like positive and negative numbers. It discusses integer properties like opposites and compares/orders integers on number lines. Real world applications like temperature, sea level, and money are explored.
Part II covers integer addition rules - signs the same means keep the sign, signs different means subtract the numbers and keep the larger absolute value sign. Number lines demonstrate adding integers visually.
Part III explains that subtracting a negative number is the same as adding a positive number through changing operation and number signs. More examples solidify this rule.
Part IV proves this subtraction rule is true by using the same checking method as regular subtraction equations
The document discusses various types of real numbers including rational and irrational numbers. It provides examples and classifications of numbers as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also discusses properties of real numbers such as closure, commutativity, associativity, identity, and inverses for addition and multiplication. Examples are provided to demonstrate how to classify numbers and identify properties of real numbers.
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers:
1) When adding integers with the same sign, add their absolute values and use the common sign. When adding integers with opposite signs, take the absolute difference and use the sign of the larger number.
2) To subtract an integer, add its opposite and then follow the addition rules.
3) When multiplying an even number of negatives, the result is positive. With an odd number of negatives, the result is negative.
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers.
It begins by explaining the rules for adding integers with the same sign and integers with different signs, providing examples such as -6 + -2 = -8. It then explains that subtracting integers uses the rule of "adding the opposite" and provides examples like 7 - (-6) = 13.
The document also covers multiplying and dividing integers, noting that an even number of negatives yields a positive result and an odd number yields a negative result. It provides examples such as -2(-2)(-2)= 16 and 2 (-5)= -10.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
CLASS VII -operations on rational numbers(1).pptxRajkumarknms
This document discusses properties of operations on rational numbers. It covers:
1) Addition of rational numbers, including having the same or different denominators. Properties include closure, commutativity, and additive identity.
2) Subtraction of rational numbers and its properties, noting the difference property and lack of an identity element.
3) Multiplication of rational numbers by multiplying numerators and denominators. Properties are closure, commutativity, associativity, identity of 1, and annihilation by 0.
4) Distributive property relating multiplication and addition/subtraction of rational numbers.
The document discusses number systems and provides examples of different types of numbers. It begins by explaining how early humans counted items without a formal system of numbers. The key developments were the creation of numbers and the number zero, which allowed people to answer questions about quantities.
The document then reviews natural numbers, whole numbers, and integers. It introduces rational numbers as numbers that can be expressed as fractions. Rational numbers can be positive or negative. Any number that cannot be expressed as a rational number, such as the square root of 2, is considered irrational. Real numbers include all rational and irrational numbers.
This document defines and explains key concepts related to real numbers and algebraic expressions. It introduces sets and subsets of real numbers like integers, rational numbers, and irrational numbers. It describes properties of real numbers including addition, multiplication, order, and absolute value. It also covers representing real numbers on a number line, algebraic expressions, and properties of negatives.
The document discusses various properties of real numbers including the commutative, associative, identity, inverse, zero, and distributive properties. It also covers topics such as combining like terms, translating word phrases to algebraic expressions, and simplifying algebraic expressions. Examples are provided to illustrate each concept along with explanations of key terms like coefficients, variables, and like terms.
1. Natural numbers include counting numbers like 1, 2, 3, and continue indefinitely. Whole numbers include natural numbers plus zero. Integers include whole numbers and their opposites.
2. Rational numbers can be written as a fraction, like 1.5 = 3/2. Irrational numbers cannot be written as a fraction, like π.
3. The four basic operations are addition, subtraction, multiplication, and division. Addition and subtraction follow rules about sign and order. Multiplication and division rules depend on the signs of the factors or dividend and divisor.
This document provides learning objectives and content about rational and irrational numbers for a Class 9 mathematics lesson. It begins by defining different types of numbers - natural, whole, integers, rational, and irrational - and provides examples. It then explains rational numbers as those that can be written as fractions p/q, and irrational numbers as those that cannot be expressed as fractions. Various methods are provided for representing and finding rational numbers between two given rational numbers, as well as representing irrational numbers on the number line. Finally, the document discusses operations involving rational and irrational numbers.
This document provides a summary of key terms in mathematical English for concepts in arithmetic, algebra, geometry, number theory, and more. Some key points covered include:
- Terms for integers, fractions, real and complex numbers, exponents, and basic arithmetic operations.
- Algebraic expressions, indices, matrices, inequalities, polynomial equations, and congruences.
- The use of definite and indefinite articles for theorems, conjectures, and mathematical concepts.
- Concepts in number theory like Fermat's Little Theorem and the Chinese Remainder Theorem.
- Terms for geometric concepts like points, lines, intersections, and rectangles.
english mathematics dictionary
kamus bahassa inggris untuk matematika
oleh neneng
Nurwaningsih
(06081281520066)
Nurwaningsih30@gmail.com
PROGRAM STUDI PENDIDIKAN MATEMATIKA
FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN
UNIVERSITAS SRIWIJAYA
INDRALAYA
2017
semoga bermanfaat
This document defines integers and the four basic integer operations - addition, subtraction, multiplication, and division. It provides rules for performing each operation on integers, such as the product of two integers with the same sign is positive and the product of two integers with different signs is negative. Examples are included to demonstrate applying the rules to solve integer operation problems.
The document discusses real numbers and their subsets. It defines natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. Examples are provided of different types of numbers. Classification of numbers using Venn diagrams is demonstrated. Rounding and truncating decimals is also covered.
This document provides an overview of the topics covered in an introductory mathematics analysis course for business, economics, and social sciences. It includes:
1) A review of key concepts like algebra, subsets of real numbers, properties of operations, and graphing numbers on a number line.
2) An outline of course structure with sections on algebra, algebraic expressions, fractions, and mathematical systems.
3) Examples of problems and their step-by-step solutions covering topics like simplifying expressions, factoring, addition/subtraction of fractions, and properties of real numbers.
This document provides an overview of irrational numbers and surds for a GCSE maths class. It begins by defining rational and irrational numbers, then discusses surds and how to manipulate them within expressions and fractions. Examples are provided of simplifying surds, expanding expressions involving surds, rationalizing denominators, and solving worksheet questions involving surds. The document aims to help students understand the difference between rational and irrational numbers and how to work with surds.
- Rational numbers include integers and can be written as a ratio of two integers. Rational numbers in decimal form either terminate or repeat.
- Irrational numbers cannot be written as a ratio of integers and their decimal representations do not terminate or repeat, such as sqrt(2).
- The document discusses the properties of operations like addition, subtraction, multiplication and division when performed on different number sets - integers, rational numbers, whole numbers. It is shown that some operations are commutative and associative while others are not, depending on the number set.
This document discusses integers and the four basic operations that can be performed on them - addition, subtraction, multiplication, and division. It defines an integer as a positive or negative whole number including 0. It provides rules for performing each operation, such as the product of two integers with the same sign is positive and with different signs is negative for multiplication. Examples are worked through for each operation to demonstrate how to apply the rules.
At the end of the lesson, the learner will be able to:
divide integers (with non-zero divisor)
interpret quotients of rational numbers by describing real-world contexts
1. The document discusses subsets of real numbers including natural numbers, whole numbers, integers, and rational numbers.
2. Natural numbers are used for counting and start at 1. Whole numbers are formed by adding 0 to the natural numbers. Integers are formed by adding the negatives of natural numbers to whole numbers.
3. Rational numbers can be expressed as fractions a/b where a and b are integers and b is not equal to 0. Their decimal representations either terminate or repeat.
This document provides an introduction to integers through five parts:
Part I defines key integer vocabulary like positive and negative numbers. It discusses integer properties like opposites and compares/orders integers on number lines. Real world applications like temperature, sea level, and money are explored.
Part II covers integer addition rules - signs the same means keep the sign, signs different means subtract the numbers and keep the larger absolute value sign. Number lines demonstrate adding integers visually.
Part III explains that subtracting a negative number is the same as adding a positive number through changing operation and number signs. More examples solidify this rule.
Part IV proves this subtraction rule is true by using the same checking method as regular subtraction equations
The document discusses various types of real numbers including rational and irrational numbers. It provides examples and classifications of numbers as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also discusses properties of real numbers such as closure, commutativity, associativity, identity, and inverses for addition and multiplication. Examples are provided to demonstrate how to classify numbers and identify properties of real numbers.
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers:
1) When adding integers with the same sign, add their absolute values and use the common sign. When adding integers with opposite signs, take the absolute difference and use the sign of the larger number.
2) To subtract an integer, add its opposite and then follow the addition rules.
3) When multiplying an even number of negatives, the result is positive. With an odd number of negatives, the result is negative.
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers.
It begins by explaining the rules for adding integers with the same sign and integers with different signs, providing examples such as -6 + -2 = -8. It then explains that subtracting integers uses the rule of "adding the opposite" and provides examples like 7 - (-6) = 13.
The document also covers multiplying and dividing integers, noting that an even number of negatives yields a positive result and an odd number yields a negative result. It provides examples such as -2(-2)(-2)= 16 and 2 (-5)= -10.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
1. RATIONAL NUMBERS 1
1.1 Introduction
InMathematics,wefrequentlycomeacrosssimpleequationstobesolved.Forexample,
theequation x + 2 = 13 (1)
is solved when x =11, because this value of x satisfies the given equation.The solution
11 is a natural number. On the other hand, for the equation
x + 5 = 5 (2)
the solution gives the whole number 0 (zero). If we consider only natural numbers,
equation (2) cannot be solved.To solve equations like (2), we added the number zero to
thecollectionofnaturalnumbersandobtainedthewholenumbers.Evenwholenumbers
willnotbesufficienttosolveequationsoftype
x + 18 = 5 (3)
Do you see ‘why’? We require the number –13 which is not a whole number.This
led us to think of integers, (positive and negative). Note that the positive integers
correspondtonaturalnumbers.Onemaythinkthatwehaveenoughnumberstosolveall
simpleequationswiththeavailablelistofintegers.Nowconsidertheequations
2x = 3 (4)
5x + 7 = 0 (5)
forwhichwecannotfindasolutionfromtheintegers.(Checkthis)
Weneedthenumbers
3
2
tosolveequation(4)and
7
5
−
tosolve
equation(5).Thisleadsustothecollectionofrationalnumbers.
We have already seen basic operations on rational
numbers.Wenowtrytoexploresomepropertiesofoperations
on the different types of numbers seen so far.
Rational Numbers
CHAPTER
1
2022-23
2. 2 MATHEMATICS
1.2 Properties of Rational Numbers
1.2.1 Closure
(i) Wholenumbers
Let us revisit the closure property for all the operations on whole numbers in brief.
Operation Numbers Remarks
Addition 0 + 5 = 5, a whole number Whole numbers are closed
4 + 7 = ... . Is it a whole number? underaddition.
In general, a + b is a whole
number for any two whole
numbers a and b.
Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed
wholenumber. undersubtraction.
Multiplication 0 × 3 = 0, a whole number Whole numbers are closed
3×7=....Isitawholenumber? undermultiplication.
In general, if a and b are any two
whole numbers, their product ab
isawholenumber.
Division 5 ÷ 8 =
5
8
, which is not a
wholenumber.
Check for closure property under all the four operations for natural numbers.
(ii) Integers
Let us now recall the operations under which integers are closed.
Operation Numbers Remarks
Addition – 6 + 5 = – 1, an integer Integers are closed under
Is – 7 + (–5) an integer? addition.
Is 8 + 5 an integer?
In general, a + b is an integer
for any two integers a and b.
Subtraction 7 – 5 = 2, an integer Integers are closed under
Is 5 – 7 an integer? subtraction.
– 6 – 8 = – 14, an integer
Whole numbers are not closed
underdivision.
2022-23
3. RATIONAL NUMBERS 3
– 6 – (– 8) = 2, an integer
Is 8 – (– 6) an integer?
Ingeneral,foranytwointegers
a and b, a – b is again an integer.
Check if b – a is also an integer.
Multiplication 5 × 8 = 40, an integer Integers are closed under
Is – 5 × 8 an integer? multiplication.
– 5 × (– 8) = 40, an integer
Ingeneral,foranytwointegers
a and b, a × b is also an integer.
Division 5 ÷ 8 =
5
8
, which is not Integers are not closed
aninteger.
underdivision.
You have seen that whole numbers are closed under addition and multiplication but
notundersubtractionanddivision.However,integersareclosedunderaddition,subtraction
andmultiplicationbutnotunderdivision.
(iii) Rational numbers
Recallthatanumberwhichcanbewrittenintheform
p
q ,wherepandqareintegers
and q ≠ 0 is called a rational number. For example,
2
3
− ,
6
7
,
9
5
−
are all rational
numbers. Since the numbers 0, –2, 4 can be written in the form
p
q , they are also
rationalnumbers.(Checkit!)
(a) You know how to add two rational numbers. Let us add a few pairs.
3 ( 5)
8 7
−
+ =
21 ( 40) 19
56 56
+ − −
= (arationalnumber)
3 ( 4)
8 5
− −
+ =
15 ( 32)
...
40
− + −
= Isitarationalnumber?
4 6
7 11
+ = ... Isitarationalnumber?
We find that sum of two rational numbers is again a rational number. Check it
for a few more pairs of rational numbers.
We say that rational numbers are closed under addition. That is, for any
two rational numbers a and b, a + b is also a rational number.
(b) Willthedifferenceoftworationalnumbersbeagainarationalnumber?
We have,
5 2
7 3
−
− =
5 3 – 2 7 29
21 21
− × × −
= (arationalnumber)
2022-23
4. 4 MATHEMATICS
TRY THESE
5 4
8 5
− =
25 32
40
−
= ... Isitarationalnumber?
3
7
8
5
−
−
= ... Isitarationalnumber?
Trythisforsomemorepairsof rationalnumbers.Wefindthatrationalnumbers
are closed under subtraction. That is, for any two rational numbers a and
b, a – b is also a rational number.
(c) Let us now see the product of two rational numbers.
2 4
3 5
−
× =
8 3 2 6
;
15 7 5 35
−
× = (both the products are rational numbers)
4 6
5 11
−
− × = ... Isitarationalnumber?
Takesomemorepairsofrationalnumbersandcheckthattheirproductisagain
arationalnumber.
We say that rational numbers are closed under multiplication. That
is, for any two rational numbers a and b, a × b is also a rational
number.
(d) We note that
5 2 25
3 5 6
− −
÷ = (arationalnumber)
2 5
...
7 3
÷ = . Is it a rational number?
3 2
...
8 9
− −
÷ = . Is it a rational number?
Canyousaythatrationalnumbersareclosedunderdivision?
We find that for any rational number a, a ÷ 0 is not defined.
So rational numbers are not closed under division.
However,ifweexcludezerothenthecollectionof,allotherrationalnumbersis
closedunderdivision.
Fillintheblanksinthefollowingtable.
Numbers Closed under
addition subtraction multiplication division
Rationalnumbers Yes Yes ... No
Integers ... Yes ... No
Wholenumbers ... ... Yes ...
Naturalnumbers ... No ... ...
2022-23
5. RATIONAL NUMBERS 5
1.2.2 Commutativity
(i) Wholenumbers
Recall the commutativity of different operations for whole numbers by filling the
followingtable.
Operation Numbers Remarks
Addition 0 + 7 = 7 + 0 = 7 Additioniscommutative.
2 + 3 = ... + ... = ....
For any two whole
numbers a and b,
a + b = b + a
Subtraction ......... Subtractionisnotcommutative.
Multiplication ......... Multiplicationiscommutative.
Division ......... Divisionisnotcommutative.
Checkwhetherthecommutativityoftheoperationsholdfornaturalnumbersalso.
(ii) Integers
Fill in the following table and check the commutativity of different operations for
integers:
Operation Numbers Remarks
Addition ......... Additioniscommutative.
Subtraction Is 5 – (–3) = – 3 – 5? Subtractionisnotcommutative.
Multiplication ......... Multiplicationiscommutative.
Division ......... Divisionisnotcommutative.
(iii) Rational numbers
(a) Addition
You know how to add two rational numbers. Let us add a few pairs here.
2 5 1 5 2 1
and
3 7 21 7 3 21
− −
+ = + =
So,
2 5 5 2
3 7 7 3
− −
+ = +
Also,
−
+
−
6
5
8
3
= ... and
Is
−
+
−
=
−
+
−
6
5
8
3
6
5
8
3
?
2022-23
6. 6 MATHEMATICS
TRY THESE
Is
3 1 1 3
8 7 7 8
− −
+ = +
?
You find that two rational numbers can be added in any order. We say that
addition is commutative for rational numbers. That is, for any two rational
numbers a and b, a + b = b + a.
(b) Subtraction
Is
2 5 5 2
3 4 4 3
− = − ?
Is
1 3 3 1
2 5 5 2
− = − ?
Youwillfindthatsubtractionisnotcommutativeforrationalnumbers.
Note thatsubtractionisnotcommutativeforintegersandintegersarealsorational
numbers.So,subtractionwillnotbecommutativeforrationalnumberstoo.
(c) Multiplication
We have,
−
× =
−
= ×
−
7
3
6
5
42
15
6
5
7
3
Is
−
×
−
=
−
×
−
8
9
4
7
4
7
8
9
?
Check for some more such products.
You will find that multiplication is commutative for rational numbers.
In general, a × b = b × a for any two rational numbers a and b.
(d) Division
Is
5 3 3 5
?
4 7 7 4
− −
÷ = ÷
You will find that expressions on both sides are not equal.
Sodivisionisnotcommutativeforrationalnumbers.
Completethefollowingtable:
Numbers Commutative for
addition subtraction multiplication division
Rationalnumbers Yes ... ... ...
Integers ... No ... ...
Wholenumbers ... ... Yes ...
Naturalnumbers ... ... ... No
2022-23
7. RATIONAL NUMBERS 7
1.2.3 Associativity
(i) Wholenumbers
Recalltheassociativityofthefouroperationsforwholenumbersthroughthistable:
Operation Numbers Remarks
Addition ......... Additionisassociative
Subtraction ......... Subtractionisnotassociative
Multiplication Is 7 × (2 × 5) = (7 × 2) × 5? Multiplicationisassociative
Is 4 × (6 × 0) = (4 × 6) × 0?
For any three whole
numbers a, b and c
a × (b × c) = (a × b) × c
Division ......... Divisionisnotassociative
Fillinthistableandverifytheremarksgiveninthelastcolumn.
Checkforyourselftheassociativityofdifferentoperationsfornaturalnumbers.
(ii) Integers
Associativityofthefouroperationsforintegerscanbeseenfromthistable
Operation Numbers Remarks
Addition Is (–2) + [3 + (– 4)] Additionisassociative
= [(–2) + 3)] + (– 4)?
Is (– 6) + [(– 4) + (–5)]
= [(– 6) +(– 4)] + (–5)?
For any three integers a, b and c
a + (b + c) = (a + b) + c
Subtraction Is 5 – (7 – 3) = (5 – 7) – 3? Subtractionisnotassociative
Multiplication Is 5 × [(–7) × (– 8) Multiplicationisassociative
= [5 × (–7)] × (– 8)?
Is (– 4) × [(– 8) × (–5)]
= [(– 4) × (– 8)] × (–5)?
For any three integers a, b and c
a × (b × c) = (a × b) × c
Division Is [(–10) ÷ 2] ÷ (–5) Divisionisnotassociative
= (–10) ÷ [2 ÷ (– 5)]?
2022-23
8. 8 MATHEMATICS
(iii) Rational numbers
(a) Addition
We have
2 3 5 2 7 27 9
3 5 6 3 30 30 10
− − − − − −
+ + = + = =
2 3 5 1 5 27 9
3 5 6 15 6 30 10
− − − − − −
+ + = + = =
So,
2 3 5 2 3 5
3 5 6 3 5 6
− − − −
+ + = + +
Find
−
+ +
−
−
+
+
−
1
2
3
7
4
3
1
2
3
7
4
3
and .Arethetwosumsequal?
Take some more rational numbers, add them as above and see if the two sums
are equal. We find that addition is associative for rational numbers. That
is, for any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
(b) Subtraction
You already know that subtraction is not associative for integers, then what
aboutrationalnumbers.
Is
−
−
−
−
= −
−
−
2
3
4
5
1
2
2
3
4
5
1
2
?
Checkforyourself.
Subtraction is not associative for rational numbers.
(c) Multiplication
Letuschecktheassociativityformultiplication.
7 5 2 7 10 70 35
3 4 9 3 36 108 54
− − − −
× × = × = =
−
×
× =
7
3
5
4
2
9
...
Wefindthat
7 5 2 7 5 2
3 4 9 3 4 9
− −
× × = × ×
Is
2 6 4 2 6 4
?
3 7 5 3 7 5
− −
× × = × ×
Take some more rational numbers and check for yourself.
We observe that multiplication is associative for rational numbers. That is
for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.
2022-23
11. RATIONAL NUMBERS 11
THINK, DISCUSS AND WRITE
You have done such additions earlier also. Do a few more such additions.
Whatdoyouobserve?Youwillfindthatwhenyouadd0toawholenumber,thesum
isagainthatwholenumber.Thishappensforintegersandrationalnumbersalso.
Ingeneral, a + 0 = 0 + a = a, where a is a whole number
b + 0 = 0 + b = b, where b is an integer
c + 0 = 0 + c = c, wherec is a rational number
Zero is called the identity for the addition of rational numbers. It is the additive
identity for integers and whole numbers as well.
1.2.5 The role of 1
We have,
5 × 1 = 5 = 1 × 5 (Multiplicationof1withawholenumber)
2
7
−
× 1 = ... × ... =
2
7
−
3
8
× ... = 1 ×
3
8
=
3
8
What do you find?
Youwillfindthatwhenyoumultiplyanyrationalnumberwith1,yougetbackthesame
rationalnumberastheproduct.Checkthisforafewmorerationalnumbers.Youwillfind
that, a × 1 = 1 × a = a for any rational number a.
We say that 1 is the multiplicative identity for rational numbers.
Is1themultiplicativeidentityforintegers?Forwholenumbers?
If a property holds for rational numbers, will it also hold for integers? For whole
numbers?Whichwill?Whichwillnot?
1.2.6 Negative of a number
Whilestudyingintegersyouhavecomeacrossnegativesofintegers.Whatisthenegative
of 1? It is – 1 because 1 + (– 1) = (–1) + 1 = 0
So, what will be the negative of (–1)? It will be 1.
Also, 2 + (–2) = (–2) + 2 = 0, so we say 2 is the negative or additive inverse of
–2 and vice-versa. In general, for an integer a, we have, a + (– a) = (– a) + a = 0; so, a
is the negative of – a and – a is the negative of a.
For the rational number
2
3
, we have,
2
3
2
3
+ −
=
2 ( 2)
0
3
+ −
=
2022-23
12. 12 MATHEMATICS
Also, −
+
2
3
2
3
= 0 (How?)
Similarly,
8
...
9
−
+ = ... +
−
=
8
9
0
... +
−
11
7
=
−
+ =
11
7
0
...
Ingeneral,forarationalnumber
a
b
,wehave,
a
b
a
b
a
b
a
b
+ −
= −
+ = 0 .Wesay
that
a
b
− is the additive inverse of
a
b
and
a
b
is the additive inverse of −
a
b
.
1.2.7 Reciprocal
By which rational number would you multiply
8
21
, to get the product 1? Obviously by
21 8 21
, since 1
8 21 8
× = .
Similarly,
5
7
−
must be multiplied by
7
5
−
so as to get the product 1.
We say that
21
8
is the reciprocal of
8
21
and
7
5
−
is the reciprocal of
5
7
−
.
Can you say what is the reciprocal of 0 (zero)?
Istherearationalnumberwhichwhenmultipliedby0gives1? Thus,zerohasnoreciprocal.
We say that a rational number
c
d
is called the reciprocal or multiplicative inverse of
another non-zero rational number
a
b
if 1
a c
b d
× = .
1.2.8 Distributivity of multiplication over addition for rational
numbers
To understand this, consider the rational numbers
3 2
,
4 3
−
and
5
6
−
.
−
× +
−
3
4
2
3
5
6
=
−
×
+ −
3
4
4 5
6
( ) ( )
=
−
×
−
3
4
1
6
=
3 1
24 8
=
Also
3 2
4 3
−
× =
3 2 6 1
4 3 12 2
− × − −
= =
×
2022-23
13. RATIONAL NUMBERS 13
TRY THESE
And
3 5
4 6
− −
× =
5
8
Therefore
−
×
+
−
×
−
3
4
2
3
3
4
5
6
=
1 5 1
2 8 8
−
+ =
Thus,
−
× +
−
3
4
2
3
5
6
=
−
×
+
−
×
−
3
4
2
3
3
4
5
6
Find using distributivity. (i)
7
5
3
12
7
5
5
12
×
−
+ ×
(ii)
9
16
4
12
9
16
3
9
×
+ ×
−
Example 3: Writetheadditiveinverseofthefollowing:
(i)
7
19
−
(ii)
21
112
Solution:
(i)
7
19
is the additive inverse of
7
19
−
because
7
19
−
+
7
19
=
7 7 0
19 19
− +
= = 0
(ii) The additive inverse of
21
112
is
21
112
−
(Check!)
Example 4: Verify that – (– x) is the same as x for
(i) x =
13
17
(ii)
21
31
x
−
=
Solution: (i) We have, x =
13
17
The additive inverse of x =
13
17
is – x =
13
17
−
since
13
17
13
17
0
+
−
= .
The same equality
13
17
13
17
0
+
−
= , shows that the additive inverse of
13
17
−
is
13
17
or −
−
13
17
=
13
17
, i.e., – (– x) = x.
(ii) Additive inverse of
21
31
x
−
= is – x =
21
31
since
21 21
0
31 31
−
+ = .
The same equality
21 21
0
31 31
−
+ = , shows that the additive inverse of
21
31
is
21
31
−
,
i.e., – (– x) = x.
Distributivity of Multi-
plication over Addition
and Subtraction.
For all rational numbers a, b
and c,
a (b + c) = ab + ac
a (b – c) = ab – ac
When you use distributivity, you
split a product as a sum or
difference of two products.
2022-23
15. RATIONAL NUMBERS 15
10. Write.
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) Therationalnumberthatisequaltoitsnegative.
11. Fillintheblanks.
(i) Zero has ________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals
(iii) The reciprocal of – 5 is ________.
(iv) Reciprocal of
1
x
, where x ≠ 0 is ________.
(v) The product of two rational numbers is always a _______.
(vi) The reciprocal of a positive rational number is ________.
1.3 Representation of Rational Numbers on the
Number Line
Youhavelearnttorepresentnaturalnumbers,wholenumbers,integers
andrationalnumbersonanumberline.Letusrevisethem.
Naturalnumbers
(i)
Wholenumbers
(ii)
Integers
(iii)
Rational numbers
(iv)
(v)
The point on the number line (iv) which is half way between 0 and 1 has been
labelled
1
2
.Also, the first of the equally spaced points that divides the distance between
0 and 1 into three equal parts can be labelled
1
3
, as on number line (v). How would you
labelthesecondofthesedivisionpointsonnumberline(v)?
The line extends
indefinitely only to the
right side of 1.
The line extends indefinitely
to the right, but from 0.
There are no numbers to the
left of 0.
The line extends
indefinitely on both sides.
Do you see any numbers
between –1, 0; 0, 1 etc.?
The line extends indefinitely
on both sides. But you can
now see numbers between
–1, 0; 0, 1 etc.
2022-23
16. 16 MATHEMATICS
The point to be labelled is twice as far from and to the right of 0 as the point
labelled
1
3
.Soitistwotimes
1
3
,i.e.,
2
3
.Youcancontinuetolabelequally-spacedpointson
the number line in the same way. In this continuation, the next marking is 1. You can
seethat1isthesameas
3
3
.
Then comes
4 5 6
, ,
3 3 3
(or 2),
7
3
and so on as shown on the number line (vi)
(vi)
Similarly, to represent
1
8
, the number line may be divided into eight equal parts as
shown:
We use the number
1
8
to name the first point of this division. The second point of
division will be labelled
2
8
, the third point
3
8
, and so on as shown on number
line(vii)
(vii)
Anyrationalnumbercanberepresentedonthenumberlineinthisway.Inarational
number,thenumeralbelowthebar,i.e.,thedenominator,tellsthenumberofequal
parts into which the first unit has been divided. The numeral above the bar i.e., the
numerator, tells ‘how many’of these parts are considered. So, a rational number
such as
4
9
means four of nine equal parts on the right of 0 (number line viii) and
for
7
4
−
, we make 7 markings of distance
1
4
each on the left of zero and starting
from0.Theseventhmarkingis
7
4
−
[numberline(ix)].
(viii)
(ix)
2022-23
17. RATIONAL NUMBERS 17
TRY THESE
Writetherationalnumberforeachpointlabelledwithaletter.
(i)
(ii)
1.4 Rational Numbers between Two Rational Numbers
Can you tell the natural numbers between 1 and 5? They are 2, 3 and 4.
How many natural numbers are there between 7 and 9? There is one and it is 8.
How many natural numbers are there between 10 and 11? Obviously none.
List the integers that lie between –5 and 4. They are – 4, – 3, –2, –1, 0, 1, 2, 3.
How many integers are there between –1 and 1?
How many integers are there between –9 and –10?
You will find a definite number of natural numbers (integers) between two natural
numbers(integers).
How many rational numbers are there between
3
10
and
7
10
?
You may have thought that they are only
4 5
,
10 10
and
6
10
.
But you can also write
3
10
as
30
100
and
7
10
as
70
100
. Now the numbers,
31 32 33
, ,
100 100 100
68 69
, ... ,
100 100
, are all between
3
10
and
7
10
. The number of these rational numbers is 39.
Also
3
10
can be expressed as
3000
10000
and
7
10
as
7000
10000
. Now, we see that the
rational numbers
3001 3002 6998 6999
, ,..., ,
10000 10000 10000 10000
are between
3
10
and
7
10
. These
are 3999 numbers in all.
In this way, we can go on inserting more and more rational numbers between
3
10
and
7
10
.Sounlikenaturalnumbersandintegers,thenumberofrationalnumbersbetween
tworationalnumbersisnotdefinite.Hereisonemoreexample.
How many rational numbers are there between
1
10
−
and
3
10
?
Obviously
0 1 2
, ,
10 10 10
arerationalnumbersbetweenthegivennumbers.
2022-23
18. 18 MATHEMATICS
If we write
1
10
−
as
10000
100000
−
and
3
10
as
30000
100000
, we get the rational numbers
9999 9998
, ,...,
100000 100000
− − 29998
100000
−
,
29999
100000
, between
1
10
−
and
3
10
.
You will find that you get countless rational numbers between any two given
rational numbers.
Example 6: Write any 3 rational numbers between –2 and 0.
Solution: –2 can be written as
20
10
−
and 0 as
0
10
.
Thus we have
19 18 17 16 15 1
, , , , , ...,
10 10 10 10 10 10
− − − − − −
between –2 and 0.
You can take any three of these.
Example 7: Find any ten rational numbers between
5
6
−
and
5
8
.
Solution:Wefirstconvert
5
6
−
and
5
8
torationalnumberswiththesamedenominators.
5 4 20
6 4 24
− × −
=
×
and
5 3 15
8 3 24
×
=
×
Thus we have
19 18 17 14
, , ,...,
24 24 24 24
− − −
as the rational numbers between
20
24
−
and
15
24
.
You can take any ten of these.
Another Method
Letusfindrationalnumbersbetween1and2.Oneofthemis 1.5or
1
1
2
or
3
2
.Thisisthe
mean of 1 and 2.You have studied mean in ClassVII.
We find that between any two given numbers, we need not necessarily get an
integer but there will always lie a rational number.
We can use the idea of mean also to find rational numbers between any two given
rationalnumbers.
Example 8: Find a rational number between
1
4
and
1
2
.
Solution:We find the mean of the given rational numbers.
1
4
1
2
2
+
÷ =
1 2
4
2
3
4
1
2
3
8
+
÷ = × =
3
8
lies between
1
4
and
1
2
.
This can be seen on the number line also.
2022-23
19. RATIONAL NUMBERS 19
We find the mid point ofAB which is C, represented by
1
4
1
2
2
+
÷ =
3
8
.
We find that
1 3 1
4 8 2
< < .
If a and b are two rational numbers, then
2
a b
+
is a rational number between a and
b such that a <
2
a b
+
< b.
This again shows that there are countless number of rational numbers between any
twogivenrationalnumbers.
Example 9: Find three rational numbers between
1
4
and
1
2
.
Solution: We find the mean of the given rational numbers.
As given in the above example, the mean is
3
8
and
1 3 1
4 8 2
< < .
Wenowfindanotherrationalnumberbetween
1 3
and
4 8
.Forthis,weagainfindthemean
of
1 3
and
4 8
. That is,
1
4
3
8
2
+
÷ =
5 1 5
8 2 16
× =
1 5 3 1
4 16 8 2
< < <
Now find the mean of
3 1
and
8 2
. We have,
3
8
1
2
2
+
÷ =
7 1
8 2
× =
7
16
Thus we get
1 5 3 7 1
4 16 8 16 2
< < < < .
Thus,
5 3 7
, ,
16 8 16
are the three rational numbers between
1 1
and
4 2
.
Thiscanclearlybeshownonthenumberlineasfollows:
In the same way we can obtain as many rational numbers as we want between two
givenrationalnumbers.Youhavenoticedthattherearecountlessrationalnumbersbetween
anytwogivenrationalnumbers.
2022-23
20. 20 MATHEMATICS
EXERCISE 1.2
1. Represent these numbers on the number line. (i)
7
4
(ii)
5
6
−
2. Represent
2 5 9
, ,
11 11 11
− − −
on the number line.
3. Writefiverationalnumberswhicharesmallerthan2.
4. Find ten rational numbers between
2 1
and
5 2
−
.
5. Findfiverationalnumbersbetween.
(i)
2
3
and
4
5
(ii)
3
2
−
and
5
3
(iii)
1
4
and
1
2
6. Writefiverationalnumbersgreaterthan–2.
7. Find ten rational numbers between
3
5
and
3
4
.
WHAT HAVE WE DISCUSSED?
1. Rationalnumbersareclosed undertheoperationsofaddition,subtractionandmultiplication.
2. Theoperationsadditionandmultiplicationare
(i) commutativeforrationalnumbers.
(ii) associative for rational numbers.
3. The rational number 0 is theadditive identity for rational numbers.
4. Therationalnumber1isthe multiplicativeidentityforrationalnumbers.
5. The additive inverse of the rational number
a
b
is
a
b
− and vice-versa.
6. The reciprocal or multiplicative inverse of the rational number
a
b
is
c
d
if 1
a c
b d
× = .
7. Distributivity of rational numbers: For all rational numbers a,b and c,
a(b + c) = ab + ac and a(b – c) = ab – ac
8. Rational numbers can be represented on a number line.
9. Betweenanytwogivenrationalnumberstherearecountlessrationalnumbers.Theideaofmean
helpsustofindrationalnumbersbetweentworationalnumbers.
2022-23