Dr. Nouha Boudjerida
Dr. Nouha Boudjerida is an Assistant Professor in the Department of Foundational Computer Science Training at the National Higher School of Advanced Technologies (ENSTA), Algiers.
Mathematical Analysis 1
ensta_dfbiAbout This Course
In a context where precision and analytical rigor are essential, the Analysis 1 module introduces the fundamental concepts of real analysis. It focuses on the structure of real numbers and the study of real sequences, forming the basis for understanding limits and advanced calculus.
The course develops key notions such as order, bounds, supremum and infimum, as well as convergence and divergence of sequences, while emphasizing logical reasoning and rigorous proof methods.
Structured into Lectures and Tutorial Sessions, this course will enable you:
- To acquire solid theoretical foundations in the structure of real numbers and their completeness properties,
- To master logical tools and proof methods used in mathematical reasoning,
- To analyze and justify the asymptotic behavior of real numerical sequences using fundamental convergence theorems,
- And to build a strong analytical framework necessary for further studies in calculus, differential equations, and advanced mathematical analysis.
Whether you aim to become an engineer, researcher, or scientist, this module will equip you with the rigorous thinking skills and mathematical foundations required to address complex quantitative problems in your future academic and professional career.
Target audience: The Mathematical Analysis 1 module is designed for first-year FB students at ENSTA.
Objectives
By the end of this module, students should be able to:
Understand the Structure of Real Numbers:
- Define real numbers and identify their subsets (ℕ, ℤ, ℚ, ℝ).
- Explain and apply properties such as order, absolute value, and
- inequalities
- Interpret intervals and bounds of subsets of ℝ.
Work with Bounds of Sets:
- Determine maximum and minimum of a set when they exist
- Identify upper and lower bounds of subsets of ℝ.
- Compute and interpret supremum and infimum.
Analyze Real Sequences:
- Define a sequence and interpret its general term.
- Identify bounded, monotonic, and special types of sequences.
- Study the behavior of sequences through examples
Study Convergence of Sequences:
- Determine whether a sequence is convergent or divergent.
- Apply convergence criteria such as the Squeeze Theorem and
- monotone convergence theorem.
- Analyze limits and justify results using appropriate methods.
Develop Problem-Solving and Reasoning Skills:
- Solve problems involving real numbers and sequences.
- Apply appropriate theorems and properties in different contexts.
- Justify mathematical results using logical reasoning and clear arguments.
Prerequisites
Recommended Prerequisites for the Analysis Mathematical 1 Module:
Academic Knowledge:
Basic Mathematics:
- Master algebraic operations (expansion, factorization, fractions).
- Solve first- and second-degree equations and inequalities.
- Understand powers, roots, and remarkable identities.
Real Numbers:
- Know number sets (\mathbb{N},\;\mathbb{Z},\;\mathbb{Q},\;\mathbb{R}).
- Understand absolute value and its properties.
- Be familiar with order, bounds, and intervals.
Basic Functions:
- Know elementary functions (linear and quadratic).
- Be able to interpret and sketch simple graphs.
Practical Skills:
Computation and Reasoning:
- Ability to perform accurate calculations.
- Ability to write clear and logical mathematical arguments.
Sequences:
- Understand the definition of a sequence.
- Be able to compute and work with arithmetic and geometric sequences.
Soft Skills:
- Scientific rigor: accuracy in calculations and reasoning.
- Logical thinking: ability to follow and build mathematical arguments.
- Autonomy: ability to work independently and check results.
Language Prerequisites:
- Mathematical English: understanding of terms such as “sequence”, “limit”, “convergence”, “bound”, etc.