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Integers | Study
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English Version Integers | Speed Notes
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Whole Numbers The numbers 1,2, 3, ……which we use for counting are known as natural numbers. If you add 1 to a natural number, we get its successor. If you subtract 1 from a natural number, you get its predecessor. (Scroll down to continue …)
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Integers
Definition
Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. The set of integers can be represented as: Integers={…,−3,−2,−1,0,1,2,3,…}Integers={…,−3,−2,−1,0,1,2,3,…}
Key Properties of Integers
- Closure Properties:
- Addition: The sum of any two integers is an integer.
- Examples:
- 2+3=52+3=5
- −1+4=3−1+4=3
- −2+(−3)=−5−2+(−3)=−5
- Examples:
- Subtraction: The difference between any two integers is an integer.
- Examples:
- 5−3=25−3=2
- −2−1=−3−2−1=−3
- 0−(−4)=40−(−4)=4
- Examples:
- Multiplication: The product of any two integers is an integer.
- Examples:
- 3×2=63×2=6
- −4×5=−20−4×5=−20
- −3×−2=6−3×−2=6
- Examples:
- Addition: The sum of any two integers is an integer.
- Identity Elements:
- Additive Identity: The integer 0 is the identity element for addition.
- Examples:
- 7+0=77+0=7
- −5+0=−5−5+0=−5
- 0+0=00+0=0
- Examples:
- Multiplicative Identity: The integer 1 is the identity element for multiplication.
- Examples:
- 4×1=44×1=4
- −3×1=−3−3×1=−3
- 0×1=00×1=0
- Examples:
- Additive Identity: The integer 0 is the identity element for addition.
- Inverse Elements:
- Additive Inverse: For every integer a, there exists an integer −a such that a+(−a)=a+−a=0.
- Examples:
- The additive inverse of 5 is -5: 5+(−5)=5+−5=0
- The additive inverse of -3 is 3: −3+3=0
- The additive inverse of 0 is 0: 0+0=0
- Examples:
- Multiplicative Inverse: Integers do not have multiplicative inverses within the set of integers (except for 1 and -1).
- Additive Inverse: For every integer a, there exists an integer −a such that a+(−a)=a+−a=0.
- Commutative and Associative Properties:
- Commutative Property:
- Addition: a+b = b+a
- Examples:
- 2+3=3+2
- −1+4 = 4+(−1) = 4-1 = 3
- 0+5 = 5+0 = 5
- Examples:
- Multiplication: a×b=b×a
- Examples:
- 3×4 = 4×3 = 12
- −2×1 = 1×−2 = -2
- 0×5 = 5×0 = 0
- Examples:
- Addition: a+b = b+a
- Associative Property:
- Addition: (a+b)+c = a+(b+c) = (a+c)+b
- Examples:
- (1+2)+3 = 1+(2+3) = (1+3)+2
- [0+(−4)]+2 = 0+[−4+2] = [(0+2)+(-4)]
- [-2+(−3)]+(-1) = -2+[−3+(-1)] = [-2+(−1)]+(-3)
- Examples:
- Multiplication: (a×b)×c=a×(b×c)(a×c)×b
- Examples:
- (2×3)×4 = 2×(3×4) = (2×4)×3
- (0×−1)×5 = 0×(−1×5) = (0×5)×−1
- (−2×3)×−1 = −2×(3×−1) = (−2×-1)×3
- Examples:
- Addition: (a+b)+c = a+(b+c) = (a+c)+b
- Commutative Property:
- Distributive Property:
- Multiplication distributes over addition:
- Example: a×(b+c)=(a×b)+(a×c) Or a×(b+c)=a×b+a×c
- Examples:
- 2×(3+4) = (2×3)+(2×4) = 6+12 = 14 Or (2×7) = 14
- −3×(1+2) = (−3×1)+(−3×2) = -3-6 = -9 Or −3×3 = −9
- 0×(5+7) = (0×5)+(0×7) = 0×(5+7) = 0×5+0×7 = 0+0 =0
- Examples:
- Example: a×(b+c)=(a×b)+(a×c) Or a×(b+c)=a×b+a×c
- Multiplication distributes over addition:
Ordering of Integers
- Integers can be ordered on a number line, where:
- Negative integers are to the left of 0.
- Positive integers are to the right of 0.
- Examples of ordering:
- …−3<−2<−1<0<1<2<3−3<−2<−1<0<1<2<3…
- −5,−2,0,4,3−5,−2,0,4,3 arranged in order: −5<−2<0<3<4−5<−2<0<3<4
Absolute Value
- The absolute value of an integer is its distance from zero on the number line, regardless of direction.
- Notation: ∣a∣∣a∣
- Examples:
- ∣3∣=3∣3∣=3
- ∣−3∣=3∣−3∣=3
- ∣0∣=0∣0∣=0
Conclusion
Understanding integers and their properties is fundamental in mathematics. They play a critical role in various areas, including algebra, number theory, and real-world applications. Mastery of integer operations is essential for higher-level mathematics.
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- WHOLE NUMBERS | Study
- KNOWING OUR NUMBERS | Study
- PLAYING WITH NUMBERS | Study
- BASIC GEOMETRICAL IDEAS | Study
- UNDERSTANDING ELEMENTARY SHAPES | Study
- INTEGERS | Study
- FRACTIONS | Study
- DECIMALS | Study
- DATA HANDLING | Study
- MENSURATION | Study
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60501001)Who is credited with inventing the world wide web?
Bill GatesTim Bernes-LeeMark ZuckerbergElon MuskWhat technology is used to secure online communication and protect sensitive data?
FirewallEncryptionBiometricsCloud computingWhat is the term used to describe the practice of using algorithms to analyze large sets of data?
Machine LearningArtificial Intelligence (AI)Big DataData MiningWhich technology has enabled the development of decentralized digital currencies like Bitcoin?
BlockchainVirtual realityAIQuantum computingWhat is the term used to describe a digital currency that relies on cryptography for secure transactions and operates independently of a central bank?
EtherumLitecoinCryptocurrencyRipple1)Who is credited with inventing the world wide web?
Bill GatesTim Bernes-LeeMark ZuckerbergElon MuskWhat technology is used to secure online communication and protect sensitive data?
FirewallEncryptionBiometricsCloud computingWhat is the term used to describe the practice of using algorithms to analyze large sets of data?
Machine LearningArtificial Intelligence (AI)Big DataData MiningWhich technology has enabled the development of decentralized digital currencies like Bitcoin?
BlockchainVirtual realityAIQuantum computingWhat is the term used to describe a digital currency that relies on cryptography for secure transactions and operates independently of a central bank?
EtherumLitecoinCryptocurrencyRipple1)Who is credited with inventing the world wide web?
Bill GatesTim Bernes-LeeMark ZuckerbergElon MuskWhat technology is used to secure online communication and protect sensitive data?
FirewallEncryptionBiometricsCloud computingWhat is the term used to describe the practice of using algorithms to analyze large sets of data?
Machine LearningArtificial Intelligence (AI)Big DataData MiningWhich technology has enabled the development of decentralized digital currencies like Bitcoin?
BlockchainVirtual realityAIQuantum computingWhat is the term used to describe a digital currency that relies on cryptography for secure transactions and operates independently of a central bank?
EtherumLitecoinCryptocurrencyRipple1)Who is credited with inventing the world wide web?
Bill GatesTim Bernes-LeeMark ZuckerbergElon MuskWhat technology is used to secure online communication and protect sensitive data?
FirewallEncryptionBiometricsCloud computingWhat is the term used to describe the practice of using algorithms to analyze large sets of data?
Machine LearningArtificial Intelligence (AI)Big DataData MiningWhich technology has enabled the development of decentralized digital currencies like Bitcoin?
BlockchainVirtual realityAIQuantum computingWhat is the term used to describe a digital currency that relies on cryptography for secure transactions and operates independently of a central bank?
EtherumLitecoinCryptocurrencyRipple+1 to +Infinity is….
Natural NumbersWhole NumbersNegitive Integers(1/2) + (3/4) = 1 whole
TrueFalse+1 to +Infinity is….
Natural NumbersWhole NumbersNegitive Integers(1/2) + (3/4) = 1 whole
TrueFalseAll whole numbers are natural numbers except the zero.
TrueFalseAll natural numbers are not whole numbers
TrueFalseAre all natural numbers whole numbers but all whole numbers are not natural numbers?
NoYesAll whole numbers are natural numbers except the zero.
TrueFalseAll natural numbers are not whole numbers
TrueFalseAre all natural numbers whole numbers but all whole numbers are not natural numbers?
NoYesQuestion
When is Teachers’ Day celebrated in India?
Sep/05Oct/05Aug/15Nov/14Who is honored on Teachers’ Day in India?
Mahatma GandhiDr. Sarvepalli RadhakrishnanJawaharlal NehruB.R. AmbedkarWhat was Dr. Sarvepalli Radhakrishnan’s profession?
ScientistPoliticianTeacherEngineerIn which year was Teachers’ Day first celebrated in India?
1950196219471985What is the main purpose of celebrating Teachers’ Day?
To honor teachersTo promote educationTo celebrate studentsTo recognize school achievementsWhich of the following is a common activity on Teachers’ Day?
Sports competitionsCultural programsScience fairsField tripsDr. Radhakrishnan was the first Vice President of India. What position did he hold before that?
Prime MinisterPresidentGovernorMinister of EducationWhat is a popular way students show appreciation to their teachers on Teachers’ Day?
Giving giftsOrganizing a strikeIgnoring themWriting complaintsWhich Indian state celebrates Teachers’ Day with a state holiday?
MaharashtraKeralaTamil NaduAll of theseWhat is the theme for Teachers’ Day typically focused on?
Teacher welfareTechnology in educationStudent achievementsEnvironmental awarenessWhat is the significance of the date September 5?
It marks the start of the school yearIt is the birth anniversary of Dr. RadhakrishnanIt is the end of the academic yearIt is the day schools close for summerWhich organization is known for promoting Teachers’ Day celebrations?
UNESCONCERTCBSEAll of the aboveWhat is often given to teachers as a token of gratitude?
FlowersChocolatesCardsAll of the aboveWhich famous quote is often associated with Dr. Radhakrishnan?
“Education is the most powerful weapon.”“Teachers are the builders of the nation.”“The roots of education are bitter, but the fruits are sweet.”“Knowledge is power.”What do students often perform for teachers on this day?
DanceSkitsSongsAll of the aboveWhat is a common theme for speeches given on Teachers’ Day?
The importance of teachersHistorical eventsSports achievementsCultural diversityWhich subject did Dr. Radhakrishnan specialize in?
PhysicsPhilosophyMathematicsLiteratureWhat is the traditional greeting students use for teachers on this day?
NamasteHelloGood morningGreetingsIn which educational institution did Dr. Radhakrishnan serve as a professor?
University of DelhiBanaras Hindu UniversityAligarh Muslim UniversityJadavpur UniversityWhat is a common gift students give teachers on Teachers’ Day?
BooksPensPersonalized mugsAll of the aboveWhich of the following is NOT a reason to celebrate Teachers’ Day?
To appreciate teachers’ effortsTo promote student-teacher relationshipsTo criticize the education systemTo encourage students to value educationWhat role do schools play on Teachers’ Day?
They remain closedThey organize events to honor teachersThey focus on examsThey conduct field tripsWhich of the following is a common sentiment expressed on Teachers’ Day?
“Thank you for your guidance.”“We don’t need teachers.”“Education is overrated.”“Teachers are not important.”What is a popular song often played on Teachers’ Day?
“We Are the Champions”“Ain’t No Mountain High Enough”“Thank You, Teacher”“Imagine”What type of events do schools typically hold on Teachers’ Day?
Sports eventsCultural programsAcademic competitionsNone of the aboveWhich Indian Prime Minister emphasized the importance of education and teachers by adding a slogan as Jai Vignan In addition to jai javan, Jai Kisan?
Indira GandhiRajiv GandhiAtal Bihari VajpayeeManmohan SinghWhat is the primary goal of Teachers’ Day celebrations?
To entertain studentsTo promote teacher-student relationshipsTo increase school fundingTo prepare for examsWhat is one way teachers are recognized on this day?
Award ceremoniesPublic speechesCommunity serviceAll of the aboveWhat is a common theme in Teachers’ Day decorations?
Books and learningSports and gamesNature and environmentTechnology and innovationWhich of the following is a famous quote by Dr. Radhakrishnan?
“The best way to find yourself is to lose yourself.”“You cannot shake hands with a clenched fist.”“Teachers should be the best minds in the country.”“Success is not final, failure is not fatal.”What is a common activity for students to do in honor of their teachers?
Write essaysOrganize a surprise partyHave a picnicGo on a field tripWhat is a significant contribution of Dr. Radhakrishnan to education?
Establishing new universitiesPromoting educational reformsWriting textbooksAll of the aboveWhat is commonly included in Teachers’ Day speeches?
Personal anecdotesHistorical factsFuture goalsAll of the aboveWhat is a popular form of expression used by students on Teachers’ Day?
PoetryArtMusicAll of the aboveHow do teachers typically feel on Teachers’ Day?
OverwhelmedAppreciatedIndifferentFrustratedWhich of the following is a traditional gift for teachers in India?
SweetsClothingJewelryElectronicsWhat is a common theme in Teachers’ Day cards?
HumorGratitudeCriticismIndifferenceWhat is the role of student councils on Teachers’ Day?
To ignore teachersTo organize eventsTo criticize school policiesTo plan field tripsWhich famous Indian author wrote about the importance of education?
R.K. NarayanRabindranath TagoreChetan BhagatArundhati RoyWhat is a popular way for schools to celebrate Teachers’ Day?
By closing earlyBy organizing a talent showBy conducting examsBy having a picnicWhat is the significance of Dr. Radhakrishnan in the context of education?
He was a famous politicianHe was a renowned philosopher and educatorHe was a scientistHe was an artistWhat is a common theme in Teachers’ Day decorations?
Learning and knowledgeSports and gamesNature and environmentTechnology and innovationWhat do students often do to express their gratitude?
Write lettersOrganize eventsCreate artAll of the aboveWhat is one of the key messages of Teachers’ Day?
Education is essential for growthTeachers are unimportantStudents should not respect teachersExams are the only measure of successWhich of the following is a common sentiment expressed on Teachers’ Day?
“Thank you for your guidance.”“We don’t need teachers.”“Education is overrated.”“Teachers are not important.”What is a common activity for students to do in honor of their teachers?
Write essaysOrganize a surprise partyHave a picnicGo on a field tripWhat is often included in Teachers’ Day speeches?
Personal anecdotesHistorical factsFuture goalsAll of the aboveWhat is a traditional greeting students use for teachers on this day?
NamasteHelloGood morningGreetingsWhat is a popular way students show appreciation to their teachers on Teachers’ Day?
Giving giftsOrganizing a strikeIgnoring themWriting complaintsWhat is the main purpose of celebrating Teachers’ Day?
To honor teachersTo promote educationTo celebrate studentsTo recognize school achievementsQuestion
When is Teachers’ Day celebrated in India?
Sep/05Oct/05Aug/15Nov/14Who is honored on Teachers’ Day in India?
Mahatma GandhiDr. Sarvepalli RadhakrishnanJawaharlal NehruB.R. AmbedkarWhat was Dr. Sarvepalli Radhakrishnan’s profession?
ScientistPoliticianTeacherEngineerIn which year was Teachers’ Day first celebrated in India?
1950196219471985What is the main purpose of celebrating Teachers’ Day?
To honor teachersTo promote educationTo celebrate studentsTo recognize school achievementsWhich of the following is a common activity on Teachers’ Day?
Sports competitionsCultural programsScience fairsField tripsDr. Radhakrishnan was the first Vice President of India. What position did he hold before that?
Prime MinisterPresidentGovernorMinister of EducationWhat is a popular way students show appreciation to their teachers on Teachers’ Day?
Giving giftsOrganizing a strikeIgnoring themWriting complaintsWhich Indian state celebrates Teachers’ Day with a state holiday?
MaharashtraKeralaTamil NaduAll of theseWhat is the theme for Teachers’ Day typically focused on?
Teacher welfareTechnology in educationStudent achievementsEnvironmental awarenessWhat is the significance of the date September 5?
It marks the start of the school yearIt is the birth anniversary of Dr. RadhakrishnanIt is the end of the academic yearIt is the day schools close for summerWhich organization is known for promoting Teachers’ Day celebrations?
UNESCONCERTCBSEAll of the aboveWhat is often given to teachers as a token of gratitude?
FlowersChocolatesCardsAll of the aboveWhich famous quote is often associated with Dr. Radhakrishnan?
“Education is the most powerful weapon.”“Teachers are the builders of the nation.”“The roots of education are bitter, but the fruits are sweet.”“Knowledge is power.”What do students often perform for teachers on this day?
DanceSkitsSongsAll of the aboveWhat is a common theme for speeches given on Teachers’ Day?
The importance of teachersHistorical eventsSports achievementsCultural diversityWhich subject did Dr. Radhakrishnan specialize in?
PhysicsPhilosophyMathematicsLiteratureWhat is the traditional greeting students use for teachers on this day?
NamasteHelloGood morningGreetingsIn which educational institution did Dr. Radhakrishnan serve as a professor?
University of DelhiBanaras Hindu UniversityAligarh Muslim UniversityJadavpur UniversityWhat is a common gift students give teachers on Teachers’ Day?
BooksPensPersonalized mugsAll of the aboveWhich of the following is NOT a reason to celebrate Teachers’ Day?
To appreciate teachers’ effortsTo promote student-teacher relationshipsTo criticize the education systemTo encourage students to value educationWhat role do schools play on Teachers’ Day?
They remain closedThey organize events to honor teachersThey focus on examsThey conduct field tripsWhich of the following is a common sentiment expressed on Teachers’ Day?
“Thank you for your guidance.”“We don’t need teachers.”“Education is overrated.”“Teachers are not important.”What is a popular song often played on Teachers’ Day?
“We Are the Champions”“Ain’t No Mountain High Enough”“Thank You, Teacher”“Imagine”What type of events do schools typically hold on Teachers’ Day?
Sports eventsCultural programsAcademic competitionsNone of the aboveWhich Indian Prime Minister emphasized the importance of education and teachers by adding a slogan as Jai Vignan In addition to jai javan, Jai Kisan?
Indira GandhiRajiv GandhiAtal Bihari VajpayeeManmohan SinghWhat is the primary goal of Teachers’ Day celebrations?
To entertain studentsTo promote teacher-student relationshipsTo increase school fundingTo prepare for examsWhat is one way teachers are recognized on this day?
Award ceremoniesPublic speechesCommunity serviceAll of the aboveWhat is a common theme in Teachers’ Day decorations?
Books and learningSports and gamesNature and environmentTechnology and innovationWhich of the following is a famous quote by Dr. Radhakrishnan?
“The best way to find yourself is to lose yourself.”“You cannot shake hands with a clenched fist.”“Teachers should be the best minds in the country.”“Success is not final, failure is not fatal.”What is a common activity for students to do in honor of their teachers?
Write essaysOrganize a surprise partyHave a picnicGo on a field tripWhat is a significant contribution of Dr. Radhakrishnan to education?
Establishing new universitiesPromoting educational reformsWriting textbooksAll of the aboveWhat is commonly included in Teachers’ Day speeches?
Personal anecdotesHistorical factsFuture goalsAll of the aboveWhat is a popular form of expression used by students on Teachers’ Day?
PoetryArtMusicAll of the aboveHow do teachers typically feel on Teachers’ Day?
OverwhelmedAppreciatedIndifferentFrustratedWhich of the following is a traditional gift for teachers in India?
SweetsClothingJewelryElectronicsWhat is a common theme in Teachers’ Day cards?
HumorGratitudeCriticismIndifferenceWhat is the role of student councils on Teachers’ Day?
To ignore teachersTo organize eventsTo criticize school policiesTo plan field tripsWhich famous Indian author wrote about the importance of education?
R.K. NarayanRabindranath TagoreChetan BhagatArundhati RoyWhat is a popular way for schools to celebrate Teachers’ Day?
By closing earlyBy organizing a talent showBy conducting examsBy having a picnicWhat is the significance of Dr. Radhakrishnan in the context of education?
He was a famous politicianHe was a renowned philosopher and educatorHe was a scientistHe was an artistWhat is a common theme in Teachers’ Day decorations?
Learning and knowledgeSports and gamesNature and environmentTechnology and innovationWhat do students often do to express their gratitude?
Write lettersOrganize eventsCreate artAll of the aboveWhat is one of the key messages of Teachers’ Day?
Education is essential for growthTeachers are unimportantStudents should not respect teachersExams are the only measure of successWhich of the following is a common sentiment expressed on Teachers’ Day?
“Thank you for your guidance.”“We don’t need teachers.”“Education is overrated.”“Teachers are not important.”What is a common activity for students to do in honor of their teachers?
Write essaysOrganize a surprise partyHave a picnicGo on a field tripWhat is often included in Teachers’ Day speeches?
Personal anecdotesHistorical factsFuture goalsAll of the aboveWhat is a traditional greeting students use for teachers on this day?
NamasteHelloGood morningGreetingsWhat is a popular way students show appreciation to their teachers on Teachers’ Day?
Giving giftsOrganizing a strikeIgnoring themWriting complaintsWhat is the main purpose of celebrating Teachers’ Day?
To honor teachersTo promote educationTo celebrate studentsTo recognize school achievementsWhich of the following is not a type of contact force
Muscular ForceFrictional ForceMagnetic ForceLiquid Pressure depends upon
Height of its columnDepthAll of these1 kgf =_______ N
10520How much atmospheric pressure is exerted on 15×15 cm head?
1 kgf2250 N100 NWho invented Magdeburg Hemispheres
Otto von GuerickeSir Isaac NewtonGalileo GalileePressure = ___________
Force / TimeForce / Area of contactForce / DistanceForce = _______
Distance / TimeWeight / TimeWeight / AreaWhich of the following is / are not effects or force?
Change state of motionChange in shapeNone of theseIn magnets,same poles _______
RepelAttractAll of theseIn magnets,opposite poles
RepelAttractNone of theseCopy – In magnets,opposite poles
RepelAttractNone of theseCopy – 1 kgf =_______ N
105201231231
dfgdfgdfdfgfgdggddgdffdWhich of the following is not a type of contact force
Muscular ForceFrictional ForceMagnetic ForceLiquid Pressure depends upon
Height of its columnWidth of the coloumnShape of the container1 kgf =_______ N
10520How much atmospheric pressure is exerted on 15×15 cm head?
1 kgf2250 N100 NWho invented Magdeburg Hemispheres
Otto von GuerickeSir Isaac NewtonGalileo GalileePressure = ___________
Force / TimeForce / Area of contactForce / DistanceThrust Force = _______
Pressure x Area of contactrac{Force}{Area of contact}rac{Pressure}{Area of contact}Which of the following is / are not effects or force?
Change state of motionChange in shapeNone of theseIn magnets, same poles _______
RepelAttractAll of theseIn magnets, opposite poles
RepelAttractNone of theseForce and Pressure
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