MMaatthhemeamtics a2 tics Number Theory Prepared by Paul Cheqe African Virtual university Université Virtuelle Africaine Universidade Virtual Africana African Virtual University NOTICE This document is published under the conditions of the Creative Commons http://en.wikipedia.org/wiki/Creative_Commons Attribution http://creativecommons.org/licenses/by/2.5/ License (abbreviated “cc-by”), Version 2.5. African Virtual University Table of ConTenTs I. Mathematics 2, Number Theory _______________________________ 3 II. Prerequisite Course or Knowledge _____________________________ 3 III. Time ____________________________________________________ 3 IV. Materials _________________________________________________ 3 V. Module Rationale __________________________________________ 3 VI. Overview _________________________________________________ 3 6.1 Outline _____________________________________________ 4 6.2 Graphic Organizer _____________________________________ 4 VII. General Objective(s) ________________________________________ 5 VIII. Specific Learning Objectives __________________________________ 5 IX. Pre-Assessment ___________________________________________ 7 9.1 Rationale : This pretest aims to: __________________________ 7 9.2 Answer Key __________________________________________ 9 9.3 Pedagogical Comment For The Learners ___________________ 9 X. Key Concepts (Glossary) ____________________________________ 10 XI. Compulsory Readings ______________________________________ 11 XII. Compulsory Ressources ____________________________________ 12 XIII. Useful Links _____________________________________________ 13 XIV. Learning Activities _________________________________________ 16 XV. Synthesis Of The Module ___________________________________ 53 XIV. Summative Evaluation ______________________________________ 54 XVII. References ______________________________________________ 55 XVIII. Student Records _________________________________________ 56 XIX. Main Author of the Module _________________________________ 57 XX. File Structure _____________________________________________ 58 African Virtual University I. Mathematics 2, number Theory By Paul Cheqe, Amoud University II. Prerequisite Course or Knowledge Basic Mathematics III. Time 120 hours IV. Materials Study book online or on CD; ICT activity files online or on CD, References online, Pre-assessment materials,Necessary freely distributed software V. Module Rationale Number theory is an essential module that assists teachers in understanding and interpreting the properties of numbers. It forms a background to the numerous proofs and solutions of various mathematical equations. It forms the backbone of theory crucial for the teaching of secondary school mathematics and is an important building block in the study of higher-level mathematics. VI. Overview The number theory module consists of two units. It pre-supposes the teacher trainee is conversant with Basic mathematics. The first unit deals with properties of integers and linear diophantine equations. It progresses from properties of in- tegers through divisibility with remainder, prime numbers and their distribution, Euclid’s proof of infinitely many primes and Euclid’s algorithm and its application in solving linear diophantine equations. The unit is concluded with Pythagorean triplets and Fermat’s last theorem for the vitas powers and the proof of Wiles African Virtual University The second unit assumes unit one as a prerequisite for the trainees. It introduces the field of integers( mod p), squares and quadratic residues, Alar’s criterion, Legendre symbol, Gauss lemma and quadratic reciprocity law, Euclid’s algorithm and unique factorisation of Gaussian integers, arithmetic of quadratic fields and application of diophantine equations, and is concluded with Fermat’s last theorem for cubes, Pell’s equation and units in real quadratic fields. 6.1 Outline Unit 1: Properties of integers and linear Diophantine equations Level 1. Priority B. Basic Mathematics 2 is prerequisite. Properties of integers. Divisibility with remainder. Prime numbers and their distribution. Euclid’s proof of infinitely many primes. Euclid’s algorithm. Conse- quences, residue classes, the integers (mod n). The case of prime n. Primitive roots and indices. Use of Euclid’s algorithm in solving linear Diophantine equations. Pythagorean triplets and Fermat’s last theorem for the vitas powers. Unit 2: Theory of congruences and quadratic fields Level 2. Priority B. Number Theory 1 is prerequisite. The field of integers (mod p). Squares and quadratic residues. Alar’s criterion. Legendre symbol. Gauss lemma and quadratic reciprocity law. Evaluation of quadratic character by the reciprocity law. Quadratic fields. Norm and trace. Euclid’s algorithm and unique factorisation of Gaussian integers. Arithmetic of quadratic fields and application of Diophantine equations. Fermat’is last theorem for cubes. Pell’s equation and units in real quadratic fields. African Virtual University 6.2 Graphic Organizer African Virtual University VII. General objective(s) The trainee is equipped with knowledge of the properties of numbers and the relationships among numbers necessary to effectively teaching mathematics in secondary schools. VIII. Specific learning objectives (Instructional objectives) Unit 1: By the end of this unit, the learners should be able to: - Have a knowledge and understanding of the basic concepts related to and the properties of numbers - Have knowledge and understanding of the relationships and recurring patterns among numbers. - Illustrate the properties of Integers and divisibility with remainders - Compute the greatest common divisor and least common multiple by factorisation - Compute the greatest common divisor using Euclidean algorithm - Illustrate properties of prime numbers and their distribution - Illustrate Euclid’s proof of infinitely many primes - Evaluate integers (mod n), the case of prime n, primitive roots and indi- ces - Use and apply Euclid’s algorithm in solving linear Diophantine equations of the nature - Illustrate Pythagorean triplets and Fermat’s last theorem - Analise pythagorean triplets and Fermat’s last theorem. Unit 2: By the end of this unit, the learners should be able to: - Illustrate the field of integers mod p, squares and quadratic residues - Outline Euler’s criterion - Use Legendre symbol, Gauss Lemma and quadratic reciprocity law - Evaluate quadratic character by the quadratic reciprocity law - Define the Norm - Apply Eucliud’s algorithm in the factorisation of Gaussian integers - Explore arithmetic of quadratic fieldsand application of Diophantine equations - Investigate Pell’s equation and units in real quadratic fields. African Virtual University IX. Pre-assessment Title of Pre-assessment : Review of Basic Mathematics 9.1 Rationale: Basic mathematics is the pre-requisite for Number theory QUESTIONS 1. Find the value of x in 2(2x + 2) = 64 a) 3 b) 5 c) 1 d) 2 2. Solve the simultaneous 3x + 2y = 22 x + y = 9 a) 7, 2 b) 1, 8 c) 4, 5 d) 6, 2 3. Solve the quadratic x2 – 3x – 10 = 0 a) -5, 2 b) 5, -2 c) -5, -2 d) 5, 2 4. Find the inverse of the function g(x) = 2x – 3 a) g-1(x) = b) g-1(x) = c) g-1(x) = d) g-1(x) = 5. Find the greatest common divisor(gcd) of 986 and 289 a) 17 b) 58 c) 9 d) 3 6. Solve the equation a) -2 b) 4 c) 2 d) 3 7. Work out (2 – í)(4 + 3 í) a) 8 + í b) 13 c) 11 + 2í d) 10í - 3 í2 African Virtual University 8. Find ( 4i + 2) a) 6 b) 26 c) 84 d) 97 8 9. Evaluate C2 a) 20 b) 28 c) 16 d) 4 10. The third term of a geometric sequence is equal to 1 and the 5th term is 16. Find the seventh term. a) 4 b) 128 c) 256 d) 4096 11. If s= ut + ½ at², evaluate s when u = –3, a = 10 and t = 5 ma) 30 b) 60 c) 110 d) 140 12. Given the equation y = x2 + 5x – 14, find the turning point. a) -2, 7 b) -7, 2 c) -2½, 14 1/8 d) -2½, -20 1/4 13. When factorised 36j – 48 becomes: a) 12(3j – 4) b) 12(24j – 36) c) 9(4j – 7) d) 8(4j – 6) 14. The solution to - 11 = - 2 is a) 56 b) 64 c) 72 d) 96 15. Solving the equation 6(7+y)-2(5y-1)=12(3y+5)-16(y-5) yields a) -2 b) -4 c) -3 d) 2 16. A 20 cm long straw is the longest that can fit into a cylindrical can with radius of 6 cm. The height of the can in centimetres is closest to: a) 8 b) 15 c) 16 d) 9 17. Given the equation y = - x² + 2x + 8, find the x-intercepts. a) - 2, 4 b) 2, - 4 c) 2, 4 d) -2, - 4 18. African Virtual University 19. For the sequence 7, 16, 25, 34, …. ……… the 56th term is a) 495 b) 640 c) 55 d) 502 20. Each interior angle of a regular polygon is 1400 .How many sides does it have a) 5 b) 9 c) 11 d) 7 9.2 Answer Key 1. d 11. c 2. c 12. d 3. b 13. a 4. a 14. c 5. a 15. b 6. a 16. c 7. c 17. a 8. d 18. b 9. b 19. d 10. c 20. b 9.3 Pedagogical Comment For the Learners A learner’s entering behaviour determines whether he / she will have grasp of the Number theory module. Number theory module builds on basic mathematics. Thus, the pre-assessment is a gauge of how well you are conversant with basic mathematics. It indicates your level of preparedness. Learners should revise basic mathematics if they have problems in scoring the pre-assessment. It is highly recommended that learners revise basic mathematics before and after the pre-testing so as to improve on their perfomance in this module. Number theory is a branch of abstract mathematics that utilises many mathematical notations. Before starting the module, revise all the mathematical notations encountered from secondary schools and in basic mathematics. African Virtual University 0 X. Key Concepts (Glossary) 1. ALGORITHM: An algorithm is a procedure for solving a problem in a finite number of steps 2. INTEGER: Integers mean any element of the set { …-3, -2, -1, 0, 1,2,3,…} 3. PRIME NUMBERS: A prime number is a number that has only two divisors: 1 and itself. 4. EVEN NUMBERS: Even numbers are numbers that can be divided by 2 without remainders. 5. ODD NUMBERS: Odd numbers are numbers that have remainders when divided by 2 6. EUCLIDEAN ALGORITHM: Is a systematic procedureí for finding the greatest common divisor of two integers.