NEKG33 Mathematical methods for economists
Chapter 2 : Day 02: Powers, inequalities and sign diagrams
By Lund University
You also did square root-stuff in high school. Come on in if you need to review. Focus today will be on powers, inequalities and sign diagrams.
Powers with rational exponents
This section focuses on powers when the exponents is not necessarily an integer. The main focus is on the case when the exponent is one half which is the square root. We will also look at cube roots and extensions to an arbitrary nth root. From this, we can define a power with an arbitrary rational exponent. It is also important to know which combinations of a base and an exponent is allowed (for example, you cannot take the square root of a negative number if you are working with the real numbers).
Square root
Cube root
Nth root
Powers with rational and real exponent
General powers: Problems
Exercises on square roots and powers
Problem: Square roots
Problem: Square roots
Problem: Square roots and equations
Problem: n’th roots
Problem: Power rules with rational exponents
Problem: Rationalize denominator
Problem: Power functions
Problem: Proving fraction rules
Problem: Fractions, true or false
Inequalities and sign diagrams
We begin this section by looking at some terminology related to inequalities, namely reflexivity, anti-symmetry, transitivity and totalness. We then look at sign diagrams, a helpful method for figuring out the sign of an expression containing a variable. The next topic is double inequalities which are useful for specifying a range of real numbers. We also cover the absolute value of a number in this section, mostly because the definition of an absolute value uses inequalities. Finally, we look at intervals, and concepts related to intervals, such as open and closed intervals, unbounded intervals, boundary points and the interior and the closure of an interval.
Inequalities, Terminology
Sign diagram
Double Inequalities
Absolute Values
Intervals
Inequalities: Problems
Exercises on inequalities and sign diagrams