MA2108: Mathematical Analysis I
AY2012/2013, Semester 2, Lecturer: Lee Soo Teck
Course Coverage:
1. The Real Numbers
2. Sequences
3. Infinite Series
4. Limits of Functions
5. Continuous Functions
The module is the first of three parts to real analysis. In this part, the focus is on the properties of real numbers, their sequences and series and finally, an introduction to basic properties of functions.
The module begins with the system of real numbers, and their analytical properties. Some important results include the completeness and orderness of the real numbers, the supremum, infimum and Archimedean properties, the triangle inequalities and the Density Theorem of real numbers.
The next two chapter focus on limiting properties of constant-term sequences and their series at infinity. These two chapter lays the foundation for subsequent analysis on sequences and series of functions in the second course of real analysis. They also pave the way of more formal introductions to subsequent topics, also known as the sequential criteria of continuity, differentiability and integrability. Two important results from the chapters are the Monotone Convergence Theorem, Bolzano-Weiestrass and Heine-Borel Theorems.
Next, limits of functions, as taught in MA1102R, are introduced formally for analysis purposes, and then applied to the derivation and analysis of continuous functions. The most prominent result in this chapter is the Intermediate Value Theorem.
Mathematical analysis focus on the preservation and implications of theorems, which places mathematics in a very different perspective than what is being taught in the many years of pre-university mathematics. It delves further in to the language of mathematics, asking the why theorems are true and what implications do the theorems hold, instead of how to differentiate, integrate or obtain numerical/closed-form solutions. A highly recommended module for those who wish to truly understand mathematics. (I would also recommend MA2202: Algebra I, but I have not taken the course and hence cannot give an overview about it.)
Workload: Moderate
Difficulty: Moderate
Grade: B-
Course Coverage:
1. The Real Numbers
2. Sequences
3. Infinite Series
4. Limits of Functions
5. Continuous Functions
The module is the first of three parts to real analysis. In this part, the focus is on the properties of real numbers, their sequences and series and finally, an introduction to basic properties of functions.
The module begins with the system of real numbers, and their analytical properties. Some important results include the completeness and orderness of the real numbers, the supremum, infimum and Archimedean properties, the triangle inequalities and the Density Theorem of real numbers.
The next two chapter focus on limiting properties of constant-term sequences and their series at infinity. These two chapter lays the foundation for subsequent analysis on sequences and series of functions in the second course of real analysis. They also pave the way of more formal introductions to subsequent topics, also known as the sequential criteria of continuity, differentiability and integrability. Two important results from the chapters are the Monotone Convergence Theorem, Bolzano-Weiestrass and Heine-Borel Theorems.
Next, limits of functions, as taught in MA1102R, are introduced formally for analysis purposes, and then applied to the derivation and analysis of continuous functions. The most prominent result in this chapter is the Intermediate Value Theorem.
Mathematical analysis focus on the preservation and implications of theorems, which places mathematics in a very different perspective than what is being taught in the many years of pre-university mathematics. It delves further in to the language of mathematics, asking the why theorems are true and what implications do the theorems hold, instead of how to differentiate, integrate or obtain numerical/closed-form solutions. A highly recommended module for those who wish to truly understand mathematics. (I would also recommend MA2202: Algebra I, but I have not taken the course and hence cannot give an overview about it.)
Workload: Moderate
Difficulty: Moderate
Grade: B-