MTC-1011 :
Algebra and Calculus I
Unit 1: Integers
1.1 Well Ordering Principle and Principle of Mathematical Induction (First Principle).
1.2 Divisibility in integers (Z) -Definition and elementary properties, Division algorithm, Greatest Common Divisor (GCD), Least Common Multiple (LCM) of integers, basic properties of GCD, Euclidean Algorithm, relatively prime integers.
1.3 Prime numbers- Definition, fundamental theorem of Arithmetic, Euclid’s lemma, Theory of Congruences, basic properties, Fermat’s theorem, Euler’s phi function, Euler’s theorem.
Unit 2: Polynomials
2.1 Definition of a polynomial, degree of a polynomial, algebra of polynomials, division algorithm (Statement only) and examples, Greatest Common Divisor (GCD) of two polynomials (Definition and examples).
2.2 Synthetic division, Remainder theorem, Factor theorem.
2.3 Relation between roots and co-efficient of a polynomial.
For Notes
Reference Books:
1. Elementary Number Theory, David M. Burton, Tata McGraw Hill, Seventh Edition.
Chapter 1: Sec. 1.1, Chapter 2: Sec. 2.2, 2.3,2.4, Chapter 3: Sec. 3.1, Chapter 4:Sec. 4.2,
Chapter 5: Sec. 5.2 up to corollary on Theorem 5.1, Chapter 7: Sec. 7.2 only definition, Section 7.3, lemma and Theorem 7.5.
2. Theory of Equations, J. V. Uspensky, McGraw Hill Book Company.
Chapter 2, Chapter 3: Sec. 5
3. Textbook of Algebra, S. K. Shah and S. C. Garg, Vikas Publishing House Pvt. Ltd. Edition 2017.
Unit 3: Real Numbers
3.1 Number system - N,Z,Q,R, Algebraic and Order properties of R.
3.2 Absolute Value of a real number, geometrical meaning, Absolute value properties of R , triangle inequality, examples on absolute value of R.
3.3 Boundedness of R -Neighborhood of a point on real line, Intervals, Lower bound, Upper bound and examples, Well Ordering Principle of N, Supremum and Infimum of a subset of R and examples, Completeness property of R.
Unit 4: Limits and Countinuity
4.1 Limit of Real valued function-Definitions and examples, Algebra of limits and examples.
4.2 Limit theorems- Squeeze theorem and some results, one sided limits and limits at infinity and examples.
4.3 Continuity - Definition of deleted neighborhood of a point, Continuity of a function at a point - Definitions and examples, Algebra of continuous functions, properties, Continuity on an interval - Definition and examples, Bounded function, Boundedness theorem (Statement only), Absolute maximum and minimum of a function - definition, Maximum-Minimum theorem (statement only), Location of roots theorem statement only), Bolzano?s theorem (statement only) the intermediate value theorem
For Notes
Reference Books:
1. Introduction to Real Analysis - R. G. Bartle and D. R. Sherbert, Third Edition, John Wily and Sons, Inc.
(a) Chapter 1: Section 1.2 - 1.2.1, 1.2.2, 1.2.3.
(b) Chapter 2: Section 2.1: 2.1.1, 2.1.2, 2.1.3, 2.1.4, 2.1.5, 2.1.6, 2.1.7 Theorem), 2.1.8
(Theorem), 2.1.9 (Statement only), 2.1.10 (Theorem), 2.1.11, 2.1.12, 2.1.13. Section 2.3: 2.3.1, 2.3.2, 2.3.3, 2.3.6, 2.4.3, 2.4.8, 2.4.9.
2. Differential Calculus- Shantinarayan Tenth Revised Edition
3. Introduction to Real Analysis - William F. Trench, Free Edition, 2010.
4. Calculus of single Variable - Ron Larson, Bruce Edwards, Tenth Edition.
5. Elementary analysis: the theory of Calculus - Kenneth A. Ross, Second Edition, Springer Publication.
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