The document discusses various types of real numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It provides definitions and key properties of each type of number. Examples are also given to illustrate concepts like finding the highest common factor and lowest common multiple using the Euclid division algorithm and prime factorization method. Theorems are presented to prove that certain numbers like the square root of 2 are irrational.
In this slide you get to know the all the detail and in depth knowledge of the chapter Real Number, 1st chapter of CBSE class 10th. Here you get all the variety of questions.
You can watch the video lecture on YouTube-
https://youtu.be/T2N-NObDf8w
In this slide you get to know the all the detail and in depth knowledge of the chapter Real Number, 1st chapter of CBSE class 10th. Here you get all the variety of questions.
You can watch the video lecture on YouTube-
https://youtu.be/T2N-NObDf8w
Elementary Algebra's Course in the 1th semester.
Roots and Radicals
1. Understanding Roots and Radicals
2. Undestanding Rational and Irrational Number
3. Finding The Square Root of a Number by Using a Graph
4. Finding The Square Root of a Number by Using a Table
5. Simplifying The Square Root of a Product
6. Simplifying The Square Root of a Fraction
7. Adding and Substraction Square Roots of a Numbers
8. Multiplying and Dividing Square Roots of a Numbers
(. Rationalizing The Dominator of Fraction
10. Solving Radical Equation
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Elementary Algebra's Course in the 1th semester.
Roots and Radicals
1. Understanding Roots and Radicals
2. Undestanding Rational and Irrational Number
3. Finding The Square Root of a Number by Using a Graph
4. Finding The Square Root of a Number by Using a Table
5. Simplifying The Square Root of a Product
6. Simplifying The Square Root of a Fraction
7. Adding and Substraction Square Roots of a Numbers
8. Multiplying and Dividing Square Roots of a Numbers
(. Rationalizing The Dominator of Fraction
10. Solving Radical Equation
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Acetabularia Information For Class 9 .docxvaibhavrinwa19
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Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
3. NATURAL NUMBER
Natural numbers were first studied seriously by such Greek philosopher and
mathematics as Pythagoras (582-500BC) and Archimedes(287-212BC).
The natural numbers are those numbers use for counting and ordering.
Natural numbers are a part of the number system which includes all the positive
integers from 1till infinity and also used for counting purpose.
Natural number do not negative or zero.
1 is smallest Natural number.
It is denoted by N.
4. WHOLE NUMBER
Whole number is firstly discovered by Bob Sinclair in 1968.
The whole numbers start from 0,1,2,3,4 and so on.
All the natural numbers are considered as whole number,but all the whe numbers
are not natural numbers.
The whole number does not contain any decimal or fractional part.
It is denoted by W.
5. INTEGERS
Integers is discovered by Leopold kronecker
An integers is the number zero,a positive or a negative integers with a minus sign.
It can be represented in a number line excluding the fractional part.
It is denoted by Z.
6. RATIONAL NUMBERS
Rational numbers is discovered by Pythagoras.
A rational number that can be expressed as the quotient or fraction p/q.
0 is a rational number because it is an integer that can be written in 0/1,0/2 etc.
A rational number is a number whose decimal form is finite or recurring in nature
for e.g.2.67 and 5.66…
It is denoted by Q.
7. IRRATIONAL NUMBERS
Irrational numbers is discovered by Greek mathematician Hippasus in 5th century
Irrational number are those number that cannot be represented in the form of a
ratio.
The irrational number are all the real number that are not rational number.
8. EUCLID DIVISON ALGORITHM
It is the process of dividing one integer by another,In a way that produces an
integer quotient and an integer remainder smaller than the divisor.A
fundamental property is that the quotient and the remainder exist and are
unique,under some condition.
a = bq +r, where 0_<r<b.
9. EXAMPLES
EXAMPLE1. Use EUCLUD’S ALGORITHM TO FIND THE HCF OF 4052 AND 12576.
SOL. Step1. Since 12576>4052,we apply th division algorithm to 12576 and 4052 to get
12576 = 4052×3+ 420
Step 2. Since the remainder 420 is not equal to 0,we apply the division algorithm to 4052 And 420 to get.
4052=420×9+272
STEP 3. We consider the new divisor 420 and the new remainder 272 and apply the division algorithm to get
420 = 272×1+148
We consider the new division 272 and the new remainder 148 and apply the division algorithm to get
272 = 148×1+124
We consider the new divisor 148 and the new divisor 148 and the new remainder 124 and apply the divisor algorithm to get
148 = 124×1+124
We consider the new divisor 124 and the new remainder 24 and apply division algorithm to get
124 = 24×5+4
We consider the new divisor 24 and the new remainder 4 and apply division algorithm to get
24 = 4×6+0
NOTICE THAT 4 = HCF( 24,4) = HCF ( 124, 24) = HCF ( 148, 124) = HCF(272,148) = HCF ( 420,272) = HCF (4052,420) R HCF (12576,4052).
10. PRIME FACTORISATION METHOD
WE HAVE ALREADY LEARNT THAT HOW TO FIND THE HCF AND
LCM OF TWO POSITIVE INTEGER USING THE FUNDAMENTAL
THEORM OF ARTHMETIC IN EARLIER CLASSES WITHOUT
REALISING IT. THIS METHOD IS ALSO CALLED THR PRIME
FACTORISATION METHOD.
11. EXAMPLE
EXAMPLE:- Find the LCM and HCF of 6 and 20 by the prime
factorization method.
Sol. We have 6 = 2¹×3¹ and 20 = 2×2×5 = 2²×5¹.
We can find HCF(6,20) = 2 and LCM(6,20) = 2×2×3×5 = 60, as done
in your earlier classes.
NOTE THAT HCF(6,20)= 2¹ = product of the smallest power of each
common prime factor in the numbers.
LCM(6,20) = product of the greatest power of each prime factor
involved in the number.
12. EXAMPLE 2. Find the HCF of 96 and 404 by the prime
factorization method. Hence find the LCM.
Sol. The prime factorization of 96 and 404 gives:
96= 2⁵×3, 404 = 2²× 101
Therefore,the HCF of these two integer is 2² = 4.
LCM(96,404) = 96×404/HCF(96,404)= 96×404/4 = 9696.
13. REVISITING IRRATIONAL NUMBER
THEORM 1.3: Let p be a prime number.if p divides a²,the p divides a, where a is a positive
integers.
Proof:- let the prime factorization of a be as follows:
a = p¹p²….p,where p¹p² and p are primes , not necessary distinct.
Therefore,a² = (p¹p²…..p)(p¹p²…..p).
NOW, We are given that p divides a².Therefore from the fundamental theorem of arithmetic,
it follows that p is one of the prime factor of a². However using the uniqueness part of
fundamental theoerm of arithmetic we realize that only prime factor of a² are p²p²….p.
NOW, since a = p¹p²…p,p divides a.
14. Theorem 1.4 : √2 is irrational.
Proof : Let us assume, to the contrary, that √2 is rational
So, we can find integers r and s (≠ 0) such that √2 = r/s
Suppose r and s have a common factor other than 1. Then, we
divide by the common factor to get √2=a/b, where a and b are
coprime.
So, b√2=a
Squaring on both sides and rearranging, we get 2b²=a². Therefore,
2 divides a².
Now, by Theorem 1.3, it follows that 2 divides a.
So, we can write a=2c for some integer c.
