Unit 4: Rational Numbers
In this unit, you will learn to prove some basic properties of rational numbers. For example, the set of rational numbers is dense in the set of real numbers. That means that strictly between any two real numbers, you can always find a rational number. The distinction between a fraction and a rational number will also be discussed. There is an easy way to tell whether a number given in decimal form is rational: if the digits of the representation regularly repeat in blocks, then the number is rational. If this is the case, you can find a pair of integers whose quotient is the given decimal. The unit discusses the mediant of a pair of rational fractions, and why the mediant does not depend on the values of its components, but instead on the way they are represented.
Completing this unit should take you approximately 9 hours.
4.1: Fractions and Rational Numbers Are Not the Same
4.2: Representing Rational Numbers as Decimals
4.3: The Existence of Irrational Numbers
4.3.1: The Square Roots of 2, 3, and 6 are All Irrational Numbers
4.3.2: Density of Irrational Numbers
4.3.3: Algebraic versus Transcendental Numbers
4.4: The Field of Rational Numbers