This document contains NCERT solutions for Class 8 Maths Chapter 1 on Rational Numbers. It includes solutions to 11 questions from Exercise 1.1 that involve finding additive and multiplicative inverses of rational numbers, identifying properties used in multiplication, determining if a number is its own reciprocal, and filling in blanks about rational numbers and their reciprocals. The questions cover key concepts about rational numbers such as their properties and operations involving addition, subtraction, multiplication and division.
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. 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.
The document discusses the process of long division for both numbers and polynomials. It demonstrates long division of numbers step-by-step using the example of 78 divided by 2. It then explains that long division of polynomials follows the same process, setting up the division with the numerator polynomial inside and denominator polynomial outside. An example problem divides the polynomial 2x^2 - 3x + 20 by the polynomial x - 4 using the long division process.
This document is a lecture on real numbers presented by Ms. Cherry Rose R. Estabillo. It introduces the different subsets of real numbers from natural numbers to irrational numbers. Diagrams are shown to illustrate the relationships between these number sets. Examples are provided to demonstrate classifying numbers within these sets. Properties of real numbers like closure, commutativity, associativity and distributivity are discussed. The order of operations and examples applying it are also covered.
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 operations with integers. It defines absolute value and covers addition, subtraction, multiplication, and division of integers through examples. Rules for integer operations are that the sign of the product is the product of the signs of the factors, and the sign of the quotient is the sign of the dividend. Division by zero is undefined.
1) Rational numbers are numbers that can be written as a quotient of two integers, such as a/b where b does not equal 0. They include integers as well as fractions and terminating or repeating decimals.
2) The document provides examples of rational numbers and asks students to determine if examples are rational numbers and to plot them on a number line.
3) Students are given practice locating rational numbers on a number line, such as -5/3, and asked to plot multiple rational numbers on a single number line.
This document contains NCERT solutions for Class 8 Maths Chapter 1 on Rational Numbers. It includes solutions to 11 questions from Exercise 1.1 that involve finding additive and multiplicative inverses of rational numbers, identifying properties used in multiplication, determining if a number is its own reciprocal, and filling in blanks about rational numbers and their reciprocals. The questions cover key concepts about rational numbers such as their properties and operations involving addition, subtraction, multiplication and division.
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. 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.
The document discusses the process of long division for both numbers and polynomials. It demonstrates long division of numbers step-by-step using the example of 78 divided by 2. It then explains that long division of polynomials follows the same process, setting up the division with the numerator polynomial inside and denominator polynomial outside. An example problem divides the polynomial 2x^2 - 3x + 20 by the polynomial x - 4 using the long division process.
This document is a lecture on real numbers presented by Ms. Cherry Rose R. Estabillo. It introduces the different subsets of real numbers from natural numbers to irrational numbers. Diagrams are shown to illustrate the relationships between these number sets. Examples are provided to demonstrate classifying numbers within these sets. Properties of real numbers like closure, commutativity, associativity and distributivity are discussed. The order of operations and examples applying it are also covered.
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 operations with integers. It defines absolute value and covers addition, subtraction, multiplication, and division of integers through examples. Rules for integer operations are that the sign of the product is the product of the signs of the factors, and the sign of the quotient is the sign of the dividend. Division by zero is undefined.
1) Rational numbers are numbers that can be written as a quotient of two integers, such as a/b where b does not equal 0. They include integers as well as fractions and terminating or repeating decimals.
2) The document provides examples of rational numbers and asks students to determine if examples are rational numbers and to plot them on a number line.
3) Students are given practice locating rational numbers on a number line, such as -5/3, and asked to plot multiple rational numbers on a single number line.
1. The document discusses real numbers and their properties, including subsets such as rational and irrational numbers.
2. Key topics covered include using a number line to graph and order real numbers, properties of number operations like closure and commutativity, and defining operations like addition, subtraction, multiplication and division.
3. Unit analysis is also introduced to check that units make sense when performing number operations for real-life applications.
This document defines and categorizes different types of numbers and explains their representation on a number line. It discusses natural numbers, negative numbers, rational numbers, irrational numbers, and real numbers. It also describes properties of the number line including that numbers increase in value as you move right and decrease as you move left, and how to determine the distance between points on the number line using absolute value.
I am Jayson L. I am a Mathematical Statistics Homework Expert at excelhomeworkhelp.com. I hold a Master's in Statistics, from Liverpool, UK. I have been helping students with their homework for the past 5 years. I solve homework related to Mathematical Statistics.
Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call +1 678 648 4277 for any assistance with the Mathematical Statistics Homework.
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.
What is four times three? 12 you might say, but no longer! In a new type of math— intersection math—
we will see that four times three is 18, two times two is 1, and that two times five is 10 (Hang on! That’s
not new!) Let’s spend some time together exploring this new math and answering the question: What is
1001 times 492?
Rational number for class VIII(Eight) by G R AHMED , K V KHANAPARAMD. G R Ahmed
1. A rational number is any number that can be expressed as the ratio of two integers.
2. Examples of rational numbers given in the document include fractions like 3/5, 4/5, and terminating or repeating decimals that can be written as fractions.
3. To find 5 rational numbers between 3/5 and 4/5, we can write fractions that increment by 1/5: 3/5, 11/15, 13/15, 17/15, 19/15, 4/5.
In this slide we are going to study about Rational number, which is the first chapter of NCERT Class 8th Mathematics.
You can watch the complete description in video form on YouTube, in my channel
This document discusses rational numbers. It defines rational numbers as numbers that can be written as fractions p/q where p and q are integers and q is not equal to 0. Some key properties of rational numbers are discussed, including that they are closed under addition, subtraction, and multiplication. Rational numbers exhibit commutativity and associativity with addition and multiplication, as well as distributivity of multiplication over addition and subtraction. The document also shows the locations of different number types including rational numbers on the number line.
The document discusses the distributive property in algebra. It provides definitions of key terms like term, coefficient, and like terms. It gives examples of using the distributive property to simplify expressions and solve problems involving perimeter. The distributive property allows multiplying a number by the sum of two other numbers, distributing the number factor across the addition.
The document discusses solving equations. It defines key terms like open sentence and equation. It explains that an open sentence with variables is neither true nor false until the variables are replaced with numbers, with each valid replacement called a solution. It outlines properties of equality like reflexive, symmetric, and transitive properties that can be used to solve equations, such as adding or subtracting the same number to both sides.
The document introduces key concepts in algebra including variables, constants, types of numbers (counting, integers, rational, irrational, real), graphs, averages, and positive and negative numbers. It provides examples and guidelines for understanding these concepts. Variables represent quantities that can vary, while constants represent fixed values. Different number sets are explained and visualized on a number line. Averages are calculated by adding values and dividing by the total count. Positive numbers are greater than zero, while negative numbers are less than zero.
1 s2 addition and subtraction of signed numbersmath123a
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
This document defines and provides examples of different types of real numbers including rational and irrational numbers. It discusses how rational numbers can be represented as fractions or decimals, and how to convert between fraction and decimal representations. Irrational numbers are defined as having non-terminating, non-repeating decimal representations. The key types of real numbers - natural numbers, integers, rational numbers, and irrational numbers - are related in a Venn diagram with their union being the set of all real numbers.
Real numbers follow rules of equality and substitution. If two numbers are equal, then they are equal regardless of any addition, subtraction, multiplication, or division operations performed on them. Equality is also reflexive, symmetric, and transitive - a number equals itself; if a equals b then b equals a; and if a equals b and b equals c, then a equals c.
1 s3 multiplication and division of signed numbersmath123a
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules to determine the sign of the product: two numbers with the same sign yield a positive product, while two numbers with opposite signs yield a negative product. It also discusses that in algebra, operations are often implied rather than written out. The even-odd rule determines the sign of products with multiple factors: an even number of negative factors yields a positive product, while an odd number yields a negative product.
This document is a summer math review packet for students entering 8th grade. It provides objectives and practice problems for key 8th grade math topics including order of operations, ratios and proportions, solving equations and inequalities, integers, fractions decimals and percents, geometry, statistics, mean median and mode, coordinate system and transformations, and GCF/LCM. The packet contains 50 total practice problems across these 10 math domains to help review and reinforce skills over the summer.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
The document discusses the real number system and different types of real numbers. It defines rational numbers as numbers that can be written as a fraction a/b, and irrational numbers as numbers that cannot be written as a fraction, such as the square root of 2. It provides examples of evaluating square roots and comparing real numbers using inequality symbols. It also covers ordering numbers from least to greatest and expressing decimals as rational numbers or repeating decimals.
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.
1. The document discusses real numbers and their properties, including subsets such as rational and irrational numbers.
2. Key topics covered include using a number line to graph and order real numbers, properties of number operations like closure and commutativity, and defining operations like addition, subtraction, multiplication and division.
3. Unit analysis is also introduced to check that units make sense when performing number operations for real-life applications.
This document defines and categorizes different types of numbers and explains their representation on a number line. It discusses natural numbers, negative numbers, rational numbers, irrational numbers, and real numbers. It also describes properties of the number line including that numbers increase in value as you move right and decrease as you move left, and how to determine the distance between points on the number line using absolute value.
I am Jayson L. I am a Mathematical Statistics Homework Expert at excelhomeworkhelp.com. I hold a Master's in Statistics, from Liverpool, UK. I have been helping students with their homework for the past 5 years. I solve homework related to Mathematical Statistics.
Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call +1 678 648 4277 for any assistance with the Mathematical Statistics Homework.
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.
What is four times three? 12 you might say, but no longer! In a new type of math— intersection math—
we will see that four times three is 18, two times two is 1, and that two times five is 10 (Hang on! That’s
not new!) Let’s spend some time together exploring this new math and answering the question: What is
1001 times 492?
Rational number for class VIII(Eight) by G R AHMED , K V KHANAPARAMD. G R Ahmed
1. A rational number is any number that can be expressed as the ratio of two integers.
2. Examples of rational numbers given in the document include fractions like 3/5, 4/5, and terminating or repeating decimals that can be written as fractions.
3. To find 5 rational numbers between 3/5 and 4/5, we can write fractions that increment by 1/5: 3/5, 11/15, 13/15, 17/15, 19/15, 4/5.
In this slide we are going to study about Rational number, which is the first chapter of NCERT Class 8th Mathematics.
You can watch the complete description in video form on YouTube, in my channel
This document discusses rational numbers. It defines rational numbers as numbers that can be written as fractions p/q where p and q are integers and q is not equal to 0. Some key properties of rational numbers are discussed, including that they are closed under addition, subtraction, and multiplication. Rational numbers exhibit commutativity and associativity with addition and multiplication, as well as distributivity of multiplication over addition and subtraction. The document also shows the locations of different number types including rational numbers on the number line.
The document discusses the distributive property in algebra. It provides definitions of key terms like term, coefficient, and like terms. It gives examples of using the distributive property to simplify expressions and solve problems involving perimeter. The distributive property allows multiplying a number by the sum of two other numbers, distributing the number factor across the addition.
The document discusses solving equations. It defines key terms like open sentence and equation. It explains that an open sentence with variables is neither true nor false until the variables are replaced with numbers, with each valid replacement called a solution. It outlines properties of equality like reflexive, symmetric, and transitive properties that can be used to solve equations, such as adding or subtracting the same number to both sides.
The document introduces key concepts in algebra including variables, constants, types of numbers (counting, integers, rational, irrational, real), graphs, averages, and positive and negative numbers. It provides examples and guidelines for understanding these concepts. Variables represent quantities that can vary, while constants represent fixed values. Different number sets are explained and visualized on a number line. Averages are calculated by adding values and dividing by the total count. Positive numbers are greater than zero, while negative numbers are less than zero.
1 s2 addition and subtraction of signed numbersmath123a
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
This document defines and provides examples of different types of real numbers including rational and irrational numbers. It discusses how rational numbers can be represented as fractions or decimals, and how to convert between fraction and decimal representations. Irrational numbers are defined as having non-terminating, non-repeating decimal representations. The key types of real numbers - natural numbers, integers, rational numbers, and irrational numbers - are related in a Venn diagram with their union being the set of all real numbers.
Real numbers follow rules of equality and substitution. If two numbers are equal, then they are equal regardless of any addition, subtraction, multiplication, or division operations performed on them. Equality is also reflexive, symmetric, and transitive - a number equals itself; if a equals b then b equals a; and if a equals b and b equals c, then a equals c.
1 s3 multiplication and division of signed numbersmath123a
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules to determine the sign of the product: two numbers with the same sign yield a positive product, while two numbers with opposite signs yield a negative product. It also discusses that in algebra, operations are often implied rather than written out. The even-odd rule determines the sign of products with multiple factors: an even number of negative factors yields a positive product, while an odd number yields a negative product.
This document is a summer math review packet for students entering 8th grade. It provides objectives and practice problems for key 8th grade math topics including order of operations, ratios and proportions, solving equations and inequalities, integers, fractions decimals and percents, geometry, statistics, mean median and mode, coordinate system and transformations, and GCF/LCM. The packet contains 50 total practice problems across these 10 math domains to help review and reinforce skills over the summer.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
The document discusses the real number system and different types of real numbers. It defines rational numbers as numbers that can be written as a fraction a/b, and irrational numbers as numbers that cannot be written as a fraction, such as the square root of 2. It provides examples of evaluating square roots and comparing real numbers using inequality symbols. It also covers ordering numbers from least to greatest and expressing decimals as rational numbers or repeating decimals.
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 and multiplication of rational numbers are commutative, but subtraction and division are not. Addition is associative for rational numbers, but subtraction is not.
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.
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.
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.
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 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.
- 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.
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.
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
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
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 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.
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 the Indian Constitution and the need for laws. It provides definitions of key terms like constitution, sovereignty, secularism, and democratic republic as they relate to India. The constitution establishes India as a sovereign, socialist, secular, democratic republic that assures justice, equality and liberty. Laws are important to ensure a safe and fair society where the rich and powerful don't dominate. The constitution gives legality to laws and the Supreme Court can rule on their constitutionality. Examples of dissent include Gandhi's Salt Satyagraha protest against the British salt tax law.
Fungi play an important role in making curd and bread through fermentation. Microorganisms have many commercial uses including in food production processes like making curd and bread. Chapter 2 of Class 8 NCERT book discusses microorganisms like fungi and their applications in commercial processes.
cheeck this class 8 maths ppt in class 8 students or below can refer this ppt and make their mind map for maths. thank you
and understant the table given in power point presentation
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This document summarizes the first chapter of Mary Shelley's novel Frankenstein. It describes how the young English captain Robert Walton's small sailing ship is in danger of being crushed by the frozen Arctic ice around it. As Walton ponders whether to risk the lives of his courageous crew for his personal goal of discovery, the crew spots through the lifting fog an unusual spectacle - a sledge with dogs and a gigantic driver in the middle of the frozen sea. The chapter leaves Walton doubtful of whether they will survive the harsh conditions of their perilous voyage attempting to find a sea route near the North Pole from Europe to Asia.
The 18th century saw major political developments in India with the arrival of European trading powers like the British, French, Dutch, and Portuguese who were attracted by the riches of Indian spice trade. Over time the British transformed from traders to political rulers, establishing control over large areas. British rule over India introduced sweeping changes administratively, politically, economically, and culturally and unified India under their control.
How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
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2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
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3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
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.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
A Free 200-Page eBook ~ Brain and Mind Exercise.pptxOH TEIK BIN
(A Free eBook comprising 3 Sets of Presentation of a selection of Puzzles, Brain Teasers and Thinking Problems to exercise both the mind and the Right and Left Brain. To help keep the mind and brain fit and healthy. Good for both the young and old alike.
Answers are given for all the puzzles and problems.)
With Metta,
Bro. Oh Teik Bin 🙏🤓🤔🥰
Accounting for Restricted Grants When and How To Record Properly
Hemh101
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
2019-20
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.
2019-20
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)
2019-20
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 ... ...
2019-20
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
?
2019-20
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
2019-20
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)]?
2019-20
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.
2019-20
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
+ −
=
2019-20
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
− × − −
= =
×
2019-20
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.
2019-20
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.
2019-20
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)
2019-20
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.
2019-20
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.
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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.
2019-20
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.
2019-20