sme1 wks3-8
TRANSCRIPT
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8/2/2019 SME1 Wks3-8
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Week 3
Term 1
2012
Theoretical Components
1. Read through Chapter 9 (9A & 9B). Go through
all the terminologies and examples. Make a list
of all new terms you come across. Chapter 9 is
under the Learning Brief Folder in cLc.2. Watch this only after you have studied examples from
Chapter 9:
http://www.youtube.com/watch?v=Wnc3_AekOno
3. Basic Concepts of Propositional Logic:
http://www.youtube.com/watch?v=qV4htTfow-E
4. Modus Tollens:
http://www.youtube.com/watch?v=fLlkSDb0UFk&feature
=channel
5. Modus Ponens:
http://www.youtube.com/watch?v=vtXksnrMtog
Tollens & Ponnen:
http://www.youtube.com/watch?v=s5sbEcGrdS4
6. Read through this site on what is a Logic Puzzle, how to
solve such puzzles and attempt to solve at least 2 puzzles.
Keep a record of the puzzles you have solved in your
portfolio.
http://www.logic-puzzles.org/
You may find more puzzles here:
http://www.puzzlersparadise.com/page1034.html
Practical Components1. Complete Exercises 9A and 9B from Chapter 9.
2. Attempt as many as you can, but nothing lessthan 5 exercises from here:
http://www.math.csusb.edu/notes/quizzes/tablequiz
/tablepractice.html
3. Do all the exercises from here:
http://dsearls.org/courses/M120Concepts/ClassNot
es/Logic/130B_exercises.htm
A:
Of Messi, Figo and Cantona one is honest (always tells the
truth), one is a liar (always lies), and one is ordinary (sometimes
tells the truth and sometimes lies). Deduce who is what from
the statements they make as shown below:
Messi: I am a liar.
Figo: I am ordinary.
Cantona: I am honest.
B: Investigate DeMorgans Law of negating AND and OR.
Use De Morgans laws to determine whether
the two statements are equivalent:
p q, (p q).
By the end of this week, you should be able to:
Understand the terminologies associated with the study of Propositional
Logic and Truth Tables
Use notations to represent arguments of various forms (injunction,conjunction, inverse, converse, negation, contrapositive, implication, bi-
conditional)
Determine the truth value of a statement using truth tables
Use inductive and deductive reasoning to solve logic puzzles.
Goals
Learning BriefSME1: Number
Theory, Graphs and
Networks
Negation Conjunction
Injunction Implication
Inverse Converse
Contrapositive
Syllogism Ponens
Tollens Premise
Conclusion Truth
Tables
http://www.youtube.com/watch?v=Wnc3_AekOnohttp://www.youtube.com/watch?v=Wnc3_AekOnohttp://www.youtube.com/watch?v=qV4htTfow-Ehttp://www.youtube.com/watch?v=qV4htTfow-Ehttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=vtXksnrMtoghttp://www.youtube.com/watch?v=vtXksnrMtoghttp://www.youtube.com/watch?v=s5sbEcGrdS4http://www.youtube.com/watch?v=s5sbEcGrdS4http://www.logic-puzzles.org/http://www.logic-puzzles.org/http://www.puzzlersparadise.com/page1034.htmlhttp://www.puzzlersparadise.com/page1034.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.puzzlersparadise.com/page1034.htmlhttp://www.logic-puzzles.org/http://www.youtube.com/watch?v=s5sbEcGrdS4http://www.youtube.com/watch?v=vtXksnrMtoghttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=qV4htTfow-Ehttp://www.youtube.com/watch?v=Wnc3_AekOno -
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ADDITIONAL
READING:http://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htm
QuizTo be completed by midnight, 26
thFeb. Quiz is available on cLc.
ForumNext Week.
http://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htmhttp://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htmhttp://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htmhttp://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htm -
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Week 4
Term 1
2012
Theoretical Components
1. By now, it is assumed that you have completed all
requirements of Learning Briefs for Weeks 2 & 3. If you
havent, you got to see Sheikh asap.
2. Read through Chapter 9C. Read through notes
and examples on techniques of proofs. Make
your notes. Chapter 9 is under the Learning
Brief Folder in cLc.3. Examples of Direct Method of Proof:
http://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htm
4. Examples of Indirect Proof (or Proof by Contradiction):
http://www.personal.kent.edu/~rmuhamma/Philosophy/
Logic/ProofTheory/proof_by_contradictionExamples.htm
5. Proof by Mathematical Induction:
http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=4
1373&ResourceID=155339
6. Watch these examples on methods of proof and make
your notes:
http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726
&ResourceID=155334
http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726
&ResourceID=155335
Practical Components1. Complete Exercises 9C from Chapter 9.
2. Exercise on Proof by Mathematical Induction:http://clc.act.edu.au/GroupDownloadFile.asp?Gr
oupId=41373&ResourceID=155340
Investigate how you can prove the truth of the statement: There
are exactly four prime numbers between 1 and 10, using
Exhaustion.
Find another example that can be proved by using this method.
By the end of this week, you should be able to:
Understand the terminologies associated with the study of PropositionalLogic and Truth Tables
Use notations to represent arguments of various forms (injunction,conjunction, inverse, converse, negation, contrapositive, implication, bi-
conditional)
Determine the truth value of a statement using truth tables
Use truth tables to determine the validity of an argument
Use various methods of proof to establish validity of arguments
Goals
Learning BriefSME1: Number
Theory, Graphs and
Networks
Implications
Converse
ContrapositiveInverse
Tautology
ContradictionCounter Examples
Mathematical Induction
http://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htm -
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Additional Reading:
http://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdf
QuizNext Week
ForumDiscuss which method of proof do you like, and why.
http://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdf -
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Week 5
Term 1
2012
Theoretical Components
1. Go through the notes and examples on Number
Theory (pdf version available on cLc under
Learning Brief folder.
2. Review proof by mathematical induction
3. Sharpen your skills on Logic Puzzles you find on
this link:
http://www.printable-puzzles.com/printable-
logic-puzzles.php
Additional Reading:
http://school.maths.uwa.edu.au/~gregg/Academy/200
5/inductionprobswsoln.pdf
8th
March (LINE2): I will go through Power Point Presentation
on Number Theory.so dont miss it!
There is no forum or quiz this week, so your attendance to
this class is compulsory.
If anyone has issues on Proof by Induction can ask me asmany question during this time. But, you can also come and
meet me in the Maths Staff room anytime during the week.
Practical ComponentsAttempt the questions from Exercises 4.1 to 4.5.
Practice on Logic Puzzles.Practice on Proof by Mathematical Induction.
IFYOUHAVEANYPROBLEMS(ABOUTA
NYCONCEPTSWEHAVESTUDIEDSOFA
R),YOUNEEDTOSEESHEIKHASAP.
NO INVESTIGATION FOR THIS WEEK.
PREPARE YOURSELF FOR THE IN-CLASS ON LOGIC
PUZZLES AND PROOF BY INDUCTION.
Quiz Next Week
By the end of this week, you should be able to:
Use integers and their properties to understand basic principles ondivisibility, greatest common divisors, least common multiples and
modular arithmetic
Review logic puzzles and proof by mathematical induction for theassessment.
NOTE: We are meeting inRoom 23 on Thursday 8
th
March (LINE 2) to discuss
the assessment task for
next week. Your
attendance to this meeting
is compulsory!
Goals
Learning BriefSME1: Number
Theory, Graphs and
Networks
For
um
Next Week.
Prime NumbersDivisibility
Euclidean Algorithm
Useful LINK:
http://www.onlinemathlearnin
g.com/mathematical-induction-examples.html
http://www.onlinemathlearnin
g.com/divisibility-rules-
explained.html
http://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.php -
8/2/2019 SME1 Wks3-8
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Week 6
Term 1
2012
Theoretical Components
1. Go through the notes and examples on Number
Theory (pdf version available on cLc under
Learning Brief folder.
2. Divisibility RULES:
http://www.youtube.com/watch?v=i16N01IdIh
k&feature=topics
http://www.youtube.com/watch?v=y1rVfa1nhj
w
http://www.youtube.com/watch?v=P5oHmgB4
Nfs&feature=channel
3. Review Truth Tables & proof by mathematical
induction
4. Sharpen your skills on Logic Puzzles you find on
this link:
http://www.printable-puzzles.com/printable-
logic-puzzles.php
Additional Reading:
INDUCTION:
http://school.maths.uwa.edu.au/~gregg/Academy/200
5/inductionprobswsoln.pdf
INTEGER DIVISIBILITY:
http://www.cs.cmu.edu/~adamchik/21-
127/lectures/divisibility_1_print.pdf
Practical ComponentsAttempt the questions from Exercises 4.1 to 4.5.
Practice on Logic Puzzles.Practice on Proof by Mathematical Induction.
IFYOUHAVEANYPROBLEMS(ABOUTA
NYCONCEPTSWEHAVESTUDIEDSOFA
R),YOUNEEDTOSEESHEIKHASAP.
Construct modulo 6 addition and multiplication tables.
Quiz No Quiz this week, so prepare for yourassessment.
By the end of this week, you should be able to:
Use integers and their properties to understand basic principles ondivisibility, greatest common divisors, least common multiples and
modular arithmetic
Review logic puzzles and proof by mathematical induction for theassessment.
NOTE: In-Class Assessment
is on Thursday, 15th
March.
This will be during LINE 2 @
8.45 a.m. in Room 23.
You can bring handwritten
notes. Be prompt.
Goals
Learning BriefSME1: Number
Theory, Graphs and
Networks
For
um
No Forum this week, so
prepare for your assessment.
Useful LINK:
http://www.onlinemathlearnin
g.com/mathematical-induction-examples.html
http://www.onlinemathlearnin
g.com/divisibility-rules-
explained.html
http://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topics -
8/2/2019 SME1 Wks3-8
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Week 7
Term 1
2012
Theoretical Components
1. Go through the notes and examples on Number
Theory (pdf version available on cLc under
Learning Brief folder.
2. Divisibility RULES:
http://www.youtube.com/watch?v=i16N01IdIh
k&feature=topics
http://www.youtube.com/watch?v=y1rVfa1nhj
w
http://www.youtube.com/watch?v=P5oHmgB4
Nfs&feature=channel
3. Euclidean Algorithm for gcd:
http://www.youtube.com/watch?v=fwuj4yzoX1o
4. Modular Arithmetic Notes & Examples & Exercises:
http://www.dsert.kar.nic.in/textbooksonline/Text%20bo
ok/English/class%20x/maths/English-Class%20X-Maths-
Chapter06.pdf
Practical Components
Attempt the questions from Exercises in the pdffile available under 4.
Read the notes below, and answer the questions
that follow.
Quiz No Quiz this week.
By the end of this week, you should be able to:
Use integers and their properties to understand basic principles on
divisibility, greatest common divisors (gcd), least common multiples (lcm)
and modular arithmetic
Use Euclidean Algorithm to find the gcd of a pair of numbers
Recognise congruent elements Find the residues in a modular system
Construct Cayleys table and appreciate the importance the modular
system
USE CLASSPAD TO COMPUTE GCD, AND PERFORM MODULAR
ARITHMETIC
This is a busy week,
manage your timewisely, and dont
wait but see me if
you are stuck on any
concept.
Goals
Learning BriefSME1: Number
Theory, Graphs and
Networks
http://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=fwuj4yzoX1ohttp://www.youtube.com/watch?v=fwuj4yzoX1ohttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.youtube.com/watch?v=fwuj4yzoX1ohttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topics -
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The Sieve of EratosthenesHow many prime numbers are there from 1 to 100? The following procedure is a relatively simple way of
identifying the primes to 100.
1. Put a single slash (/) through the 1 block with a blue crayon or colored pencil. One is special:It is the unit.
2. Circle 2 in orange. Then cross out with red all other numbers in the chart divisible by 2.
(Another way of saying this is: Cross out all numbers in the chart that are multiples of 2. It may help some
children to have them count by twos.)
3. Circle in orange the next prime number: 3. With red, cross out any other multiple of 3 that has not
already been crossed out. (It may help some children to count by threes and cross out any of these
numbers not already crossed out.)
4. Circle in orange the next prime number: 5.
With red, cross out any other multiple of 5 that has not already been crossed out. It may help some
children to encourage them to count by fives and cross out any numbers not already crossed out.)
5. Continue in this manner until all numbers are circled in orange (the primes) or crossed out in red (the
composites). Note for what prime number you did not have to cross out any multiples to 100. Why was itunnecessary to cross out any numbers for this prime? Will it be necessary to check for multiples of the
remaining primes or can you simply circle in orange all remaining numbers at this point?
Forum On cLc.The Math Book from Hell had the following rather unrealistic fraction question:
What is the sum of 1/54 and 1/72 of a kilometre? Rodney sensed that 54 x 72
(3888) was not the lowest common denominator and that using it would mean a
lot of extra work.(a) How could the lowest common denominator for 1/54 and 1/72 be found?
(b)What is the LCM of these fractions?
-
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Modular Arithmetic and Congruence
CONGRUENCE
We say that two integers aand b are congruent modulo n if and only if
n|(a-b), where n is a positive integer.
This is written as: ()
Example 1
Are 25 and 17 congruent modulo 4?
25 17 = 8. 4|8, 25 17( 4)
Properties:
, ,, ,, , +:
1. () =
2. () (
0).
3. (). This is known as the Reflexive Property
4. () () . This is known as the
Symmetric Property
5. () () (). This is
known as the Transitive Property.
6. () + + () ()
7. () () +
+ () ()
8. () ()
Example 2
Prove congruence Property 5. () () ()
() (), |( )|( )From the division identity, we have:( ) = 1( ) = 2 1,2 , = 0
( ) + ( ) = 1 + 2 = (1 + 2) = 1 + 2
|( ), ()Q.E.D.
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The basic congruency properties can be regarded as rules for modular
arithmetic, which operates with equality (=) replaced by congruence ().Properties 6 & 7 show that multiplication preserve congruency, but the
operation is not reversible. This is obvious because division does not
always give an integral answer. These properties allow some problems tobe solved quite easily.
Example 3
Find the remainder when 330
is divided by 7.
33 = 27 1( 7)
2710 (1)10( 7) 8
330 1( 7)
From Property 2, the remainder when 3
30
is divided by 7 is 1.
Exercises:
1. Prove Property 6
2. Prove Property 7.
3. Find the remainder when:
i) 230 is divided by 7
ii) 516 is divided by 24
iii) 9120 is divided by 40
iv) 220 is divided by 41
v) 2316 is divided by 7
4. Use the remainder when 330 is divided by 7 to show that 330-1 is
divisible by 7.
5. Show that 7|(36 1) +
6. Show that 24|(52 1) +
7. Show that 41|(220 1) +
8. Show that if () and m|n, then ()
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Week 8
Term 1
2012
Theoretical Components
You will find the reading/examples/exercises for this
week below.
Practical ComponentsRead the notes and examples and attempt a few
questions.
Quiz On cLc.
By the end of this week, you should be able to:
Appreciate the use of Diophantine Equations to find integer solutions to
linear equations
Determine which of the linear equations are solvable with integer solution
Solve a Diophantine Equation using Euclidean Algorithm to find particular
solution. Solve any equation of the form + = , given | =
gcd(, ) to find all solutions (if they exist).
By now you should
now these:o gcd, lcm using prime factors
o Cayleys Table
o Euclidean Algorithm
o Congruence
Use this as checklist!
Got a question = see Sheikh
Goals
Learning BriefSME1: Number
Theory, Graphs and
Networks
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Investigate the use of Euclidean Algorithm to find a particular solution to a Diophantine Equation.
This video may be helpful:
http://www.youtube.com/watch?v=FjliV5u2IVw
Remember, this video shows how to use Euclidean Algorithm to find a particular solution.
Do a detailed write-up of this technique, with a different example (there is no limit, so you can write as
many examples).
Forum Next week.
http://www.youtube.com/watch?v=FjliV5u2IVwhttp://www.youtube.com/watch?v=FjliV5u2IVwhttp://www.youtube.com/watch?v=FjliV5u2IVw -
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