Real numbers and Inequalities

 

This is an introduction to some basic concepts, skills and arguments which will be used in Calculus. Inequalities ate introduced to help to describe the meaning of boundary distances with regard to continuity and others.

Syllabus

• Basic set notation, meaning of intersection and union, relative complement.
• Real numbers and intervals. Positive real numbers.
• Properties of real numbers; addition, multiplication of real numbers.
• Solving Inequalities based on properties of real numbers.
• Absolute value - geometrical interpretation and mathematical expression. The triangle inequality.

Learning Objectives
 

• Students should be able to express a subset of real numbers either by listing (for a finite set), describing (eg {x \in R : x^2 + 3x  -1>0} or {y \in R : y is a prime less than or equal to 1003}) or using the interval notation (eg. (-2, 5] for {x \in R : 2< x = < 5}) with basic set notation such as union, intersection, or complement.
• Students should know simple properties of real numbers and use them in solving inequalities or proving some inequalities.
• Students should be able to solve simple inequalities with confidence and able to express the solution set clearly.
• Students should know the geometrical meaning of absolute value |x| and   |x - a| .
• Students should be able to use the triangle inequality.

Reference: NG TB, Calculus, Chapter s 1 and 3.