The document provides information about divisibility and algorithms related to division. It begins by defining divisibility and providing properties of divisibility. It then discusses the division algorithm theorem, which states that for integers a and b, where b is not zero, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. It also notes that a number is divisible by b if and only if the remainder is 0. The document then provides examples of applying the division algorithm to find the quotient and remainder.
Discrete Mathematics is a collection of branches of mathematics that involves discrete elements using algebra and arithmetic. This is a tool being used to improve reasoning and problem-solving capabilities. It involves distinct values; i.e. between any two points, there are several number of points.
There are so many mathematical symbols that are important for students. To make it easier for you we’ve given here the mathematical symbols table with definitions and examples
This is an interactive presentation which contains the information about Algebra for student-teacher , who are going to teach maths. Further, it contains information about the curriculum alignment and objectives of algebraic teaching which are mentioned in Curriculum of Pakistan.
Obj. 18 Isosceles and Equilateral Trianglessmiller5
* Identify isosceles and equilateral triangles by side length and angle measure
* Use the Isosceles Triangle Theorem to solve problems
* Use the Equilateral Triangle Corollary to solve problems
Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. These entities are what are typically called sets. The technique of using the concept of a set to answer questions is hardly new. It has been in use since ancient times. However, the rigorous treatment of sets happened only in the 19-th century due to the German mathematician Georg Cantor. He was solely responsible in ensuring that sets had a home in mathematics. Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator. He developed two types of transfinite numbers, namely, transfinite ordinals and transfinite cardinals. His new and path-breaking ideas were not well received by his contemporaries. Further, from his definition of a set, a number of contradictions and paradoxes arose. One of the most famous paradoxes is the Russell’s Paradox, due to Bertrand Russell in 1918. This paradox amongst others, opened the stage for the development of axiomatic set theory. The interested reader may refer to Katz [
Discrete Mathematics is a collection of branches of mathematics that involves discrete elements using algebra and arithmetic. This is a tool being used to improve reasoning and problem-solving capabilities. It involves distinct values; i.e. between any two points, there are several number of points.
There are so many mathematical symbols that are important for students. To make it easier for you we’ve given here the mathematical symbols table with definitions and examples
This is an interactive presentation which contains the information about Algebra for student-teacher , who are going to teach maths. Further, it contains information about the curriculum alignment and objectives of algebraic teaching which are mentioned in Curriculum of Pakistan.
Obj. 18 Isosceles and Equilateral Trianglessmiller5
* Identify isosceles and equilateral triangles by side length and angle measure
* Use the Isosceles Triangle Theorem to solve problems
* Use the Equilateral Triangle Corollary to solve problems
Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. These entities are what are typically called sets. The technique of using the concept of a set to answer questions is hardly new. It has been in use since ancient times. However, the rigorous treatment of sets happened only in the 19-th century due to the German mathematician Georg Cantor. He was solely responsible in ensuring that sets had a home in mathematics. Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator. He developed two types of transfinite numbers, namely, transfinite ordinals and transfinite cardinals. His new and path-breaking ideas were not well received by his contemporaries. Further, from his definition of a set, a number of contradictions and paradoxes arose. One of the most famous paradoxes is the Russell’s Paradox, due to Bertrand Russell in 1918. This paradox amongst others, opened the stage for the development of axiomatic set theory. The interested reader may refer to Katz [
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
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
2. “The laws of nature are
but the mathematical
thoughts of God ”
-Euclid
2
3. DIVISIBILITY
Definition: if a and b are integers with a≠ 0,
we say that a divides b if there is an integer c
such that b=ac. If a divides b, we also say
that a is a divisor or factor of b and that b is a
multiple of a.
If a divides b, we write ȁ
𝑎 𝑏 , while if a does
not divide b, we write a ∤ 𝑏
4. DIVISIBILITY
The notation a | b to mean that a divides b. Be
careful not to confuse “a | b” with “a/b” or
“a÷b”. The notation “a | b” is read “a divides
b”, which is a statement — a complete
sentence which could be either true or false.
On the other hand, “a ÷ b” is read “a divided
by b”. This is an expression, not a complete
sentence. Compare “6 divides 18” with “18
divided by 6” and be sure you understand the
difference
5. DIVISIBILITY
The properties in the next proposition are easy
consequences of the definition of divisibility;
Proposition.
Every nonzero number divides 0.
1 divides everything. So does −1.
Every nonzero number is divisible by itself.
6. Basic theorem of divisibility
1. If a|b and a|c, then a|(b + c).
2. If a|b, then a|(bc).
3. If a|b and b|c, then a|c.
6
7. Basic theorem of divisibility
THEOREM #1:If a|b and a|c, then a|(b + c).
3 | 6 and 3 | 9, so 3 | 15.
5| 35 and 5 | 25, so 5 | 60.
9 | 27 and 8 | 81, so 9 | 99.
7
8. Basic theorem of divisibility
THEOREM #2: If a|b, then a|(bc).
5 | 10, so 5 | 20, 5 | 30.
15 | 45, so 15 | 60, 5 | 2700.
11 | 66, so 11 | 121, 11 | 7986.
8
9. Basic theorem of divisibility
THEOREM #3 :If a|b and b|c, then a|c.
4 | 8 and 8 | 24, so 4 | 24
12 | 36 and 36 | 72, so 12 | 72
24 | 120 and 120 | 720, so 24 | 720
9
10. Complete the following statement using the theorem #1
15 | 45 and 15 | 30, so 3 | ?.
75
21| 63 and 21 | 441, so 21 | ?.
504
212 | 1060 and 212 | 2332, so 212 | ?.
3,392
10
11. Complete the following statement using the theorem #2
41 | 246, so 41 | 451, 41 | ?.
110,946
231 | 2772, so 231 | 4851, 231 | ?.
13,446,972
406 | 3248, so 406 | 5239, 406 | ?.
17,016,272
11
12. Complete the following statement using theorem #3
51 | 561 and 561 | 4488, so ?
51 | 4488
721 | 2884 and 2884 | 31724, so ?
721 | 31,724
1605 | 17,655 and 17655 | 88275, so ?
1,605 | 88,275
12
13. The Language in Which Mathematics Is Done
Even when talking about day to day life, it sometimes happens that one has a hard time
finding the words to express one’s thoughts. Furthermore, even when one believes that
one has been totally clear, it sometimes happens that there is a vagueness in one’s words
that results in the listener understanding something completely different from what the
speaker intended. Even the simplest mathematical reasoning is usually more complicated
than the most complicated things one attempts to discuss in everyday life. For this
reason, mathematics has developed an extremely specialized (and often stylized)
language. To see the value of this, one only has to think of how difficult it would be to
solve even the simple quadratic equation x2 − 5x = 14 if one’s calculation had to be done
without symbols, using only ordinary language.
“Suppose that a quantity has the property that when five times that quantity has the
property that when five times that quantity is subtracted from the product of the
quantity times itself then the result is 14. Determine the possible values of this quantity.”
13
14. The Language in Which Mathematics Is Done
During the middle ages, algebraic calculations were in fact done in this fashion — except
that the language was Latin, not English. But progress beyond the level of simple algebra
only became possible with the introduction of modern algebraic notation — which the West
learned from the Arabs. By combining symbolic notation with very precisely defined
concepts and a very formalized language, any mathematical proof can be expressed in a
way that can be understood by any “mathematically mature” reader. One is not dependent
on the reader’s ability to “see what is meant.”
14
15. DIVISION ALGORITHM
THEOREM
If a and b are integers such that b>0, then there
are unique integers q and r such that
a =bq + r with 0 ≤ 𝑟 < 𝑏.
In this theorem, q is the quotient and r is the
remainder, a the dividend and b the divisor.
We note that a is divisible by b if and only if
the remainder in the division algorithm is 0.
16. DIVISION ALGORITHM
THEOREM
• The Division Algorithm says that an integer can be divided by
another (nonzero) integer, with a unique quotient and
remainder.
18. DIVISION ALGORITHM
THEOREM
1. If a = 82 and b = 7, then 82 = (7)(11) + 5:
2. If a = 1 and b = 20, then 1 = (20)(0) + 1:
3. If a = 0 and b = 15, then 0 = (15)(0) + 0:
19. DIVISION ALGORITHM
THEOREM
a= -36, b=5
−36
5
= −7.2
round down, q=-8
-36=5(-8)+r
Find r:
Rewrite the equation
a=bq+r
r=-bq+a
r=-(5(-8))+(-36)
r=40-36
r=4
By the division
algorithm,
-36=5(-8)+4
20. DIVISION ALGORITHM
THEOREM
a= -413, b=3
−413
3
= −137.6
round down, q=-138
-413=3(-138)+r
Find r:
Rewrite the equation
a=bq+r
r=-bq+a
r=-(3(-138))+(-413)
r=414-413
r=1
By the division
algorithm,
-413=3(-138)+1
21. DIVISION ALGORITHM
THEOREM
a= -389, b=16
−389
16
= −24.31
round down, q=-25
-389=16(-25)+r
Find r:
Rewrite the equation
a=bq+r
r=-bq+a
r=-(16(-25))+(-389)
r=400-389
r=11
By the division
algorithm,
-389=16(-25)+11
22. DIVISION ALGORITHM
THEOREM
1. If a = -17 and b = 3,
then q = -6 and r = 1. −17 = 3 −6 + 1
2. If a = 18 and b = 6,
then q = 3 and r = 0. 18 = 6 3 + 0
3. If a = −79 and b = 9,
then q=-9 and r=2 −79 = 9 −9 + 2
23. Prime numbers
23
Prime numbers are the building blocks of arithmetic. At the moment there
are no efficient methods (algorithms) known that will determine whether a
given integer is prime or find its prime factors. This fact is the basis behind
many of the cryptosystems currently in use. One problem is that there is no
known procedure that will generate prime numbers, even recursively. In fact,
there are many things about prime numbers that we don’t know. For example,
there is a conjecture, known as Goldbach’s Conjecture, that there are
infinitely many prime pairs, that is, consecutive odd prime numbers, such as
5 and 7, or 41 and 43, which no one so far has been able to prove or disprove.
As the next theorem illustrates, it is possible, however, to prove that there
are infinitely many prime numbers. Its proof, attributed to Euclid, is one of
the most elegant in all of mathematics.
24. Prime numbers
24
A prime is an integer with precisely two positive integer divisors. In
the past three centuries, the mathematicians have devoted countless
hours in exploring the world of primes. They have discovered many
fascinating properties, formulated diverse conjectures, and proved
interesting and surprising results.
25. Prime numbers
25
A prime is a positive integer greater than 1 that is divisible by no
positive integers other than 1 and itself.
e.g. 1,2,3,5,7,11, 179
27. GREATEST COMMON DIVISOR
27
The greatest common divisor (gcd) of two or more non- zero integers
is the largest positive integer that divides the numbers without a
remainder.
28. GREATEST COMMON DIVISOR
28
The common divisors of 36 and 60 are 1, 2, 3, 4, 6, 12.
The greatest common divisor gcd(36,60) = 12.
34. 34
Euclid
Euclid is often referred to as the “Father of
Geometry”, and he wrote perhaps the
most important and successful
mathematical textbook of all time, the
“Stoicheion” or “Elements”, which
represents the culmination of the
mathematical revolution which had taken
place in Greece up to that time. He also
wrote works on the division of geometrical
figures into into parts in given ratios, on
catoptrics (the mathematical theory of
mirrors and reflection), and on spherical
astronomy (the determination of the
location of objects on the “celestial
sphere”), as well as important texts on
optics and music.
35. 35
> The well known Euclidean algorithm finds the greatest
common divisor of two numbers using only
elementary mathematical operations - division and
subtraction
Euclidean Algorithm
36. 36
Euclidean Algorithm
> A divisor of a number a is an integer that divides it without
remainder
> For example the divisors of 12 are 1, 2, 3, 4, 6 and 12
> The divisors of 18 are 1, 2, 3, 6, 9 and 18.
37. 37
Euclidean Algorithm
> The greatest common divisor, or GCD, of two numbers is
the largest divisor that is common to both of them.
> For example GCD(12, 18) is the largest of the divisors
common to both 12 and 18.
38. 38
Euclidean Algorithm
> The greatest common divisor, or GCD, of two numbers is
the largest divisor that is common to both of them.
> For example GCD(12, 18) is the largest of the divisors
common to both 12 and 18.
40. 40
Euclidean Algorithm
> The Euclidean Algorithm to find GCD(a, b) relies upon
replacing one of a or b with the remainder after division.
> Thus the numbers we seek the GCD of are steadily
becoming smaller and smaller. We stop when one of them
becomes 0.
41. 41
Euclidean Algorithm
> Specifically, we assume that a is larger than b. If b is
larger than a, then we swap them around so that a becomes
the old b and b becomes the old a.
> We then look for numbers q and r so that a=bq+r. They
must have the properties that q0 and 0r<b.
> In other words, we seek the largest such q.
42. 42
Euclidean Algorithm
> As examples, consider the following.
> a=12, b=5; 12=5*2+2 so q=2, r=2
> a=24, b=18; 24=18*1+6 so q=1, r=6
> a=30, b=15; 30=15*2+0 so q=2, r=0
> a=27, b=14; 27=14*1+13 so q=1, r=13
> Try the ones on the next slide.
43. 43
Euclidean Algorithm
> Find q and r for the following sets of a and b. The answers
are on the next slide.
> a=28, b=12
> a=50, b=30
> a=35, b=14
> a=100, b=20
45. 45
Euclidean Algorithm
> The algorithm works in the following way.
> Given a and b, we find numbers q and r so that a=bq+r.
> We make sure that q is as large as possible (≥0), and 0≤r<b.
> For example, if a=18, b=12, then we write 18=12*1+6.
46. 46
Euclidean Algorithm
> Once the remainder r has been found we replace a by b and
b by r.
> This relies on the fact that GCD(a,b)=GCD(b,r).
> Hence we repeatedly find r, the remainder after a is divided
by b.
> Then replace a by b and b by r, and keep on in this way until
r=0.
47. 47
Euclidean Algorithm
> Let us look at a graphical interpretation of the Euclidean
algorithm.
> Obviously if p=GCD(a,b) then p|a and p|b, that is to say p
divides both a and b evenly with no remainder.
48. 48
Euclidean Algorithm
> Let us look at a graphical interpretation of the Euclidean
algorithm.
> Obviously if p=GCD(a,b) then p|a and p|b, that is to say p
divides both a and b evenly with no remainder.
49. 49
Show the computations for a=210 and b=45.
• Divide 210 by 45, and get the result 4 with remainder
30, so 210=4·45+30.
• Divide 45 by 30, and get the result 1 with remainder
15, so 45=1·30+15.
• Divide 30 by 15, and get the result 2 with remainder
0, so 30=2·15+0.
• The greatest common divisor of 210 and 45 is 15.
50. 50
Show the computations for a=420 and b=55.
• Divide 420 by 55, and get the result 7 with remainder 35,
so 210=7*55+35.
• Divide 55 by 35, and get the result 1 with remainder 20, so
55=1·35+20.
• Divide 35 by 20, and get the result 1 with remainder 15, so
35=1·20+15.
• Divide 20 by 15 and get the result 1 with remainder 5, so
20= 1 * 15 + 5
• Divide 15 by 5 and get the result 3 with remainder 0, so
15= 3 * 5 + 0
• The greatest common divisor of 420 and 55 is 5.