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.
Upon successful completion of this unit, you will be able to:
- find a pair of integers whose quotient is a given repeating decimal;
- prove basic propositions about rational and irrational numbers;
- recognize that the set of rational numbers is an ordered field, and use the properties of ordered fields to prove statements concerning rational numbers; and
- prove that the set of rational numbers is not complete, but the set of real numbers is complete.
4.1: Fractions and Rational Numbers Are Not the Same
Read this article. Pay special attention to the five problems on rational numbers at the beginning of the paper. Problem 10 will enable you to appreciate the different between the value of a number and the numeral used to express it. Pay special attention to Simpson's Paradox in the paper. Try the practice problems at the end of the reading. After you have attempted these problems, please check your answers here.
4.2: Representing Rational Numbers as Decimals
Watch this video to learn about the difference between rational and irrational numbers.
Watch this video on the conversion of repeating decimals into fractions of the form
with 
and 
integers.
4.3: The Existence of Irrational Numbers
4.3.1: The Square Roots of 2, 3, and 6 are All Irrational Numbers
Watch this video, which shows a proof of the irrationality of the square roots of 2. Can you see how to use these ideas to prove that the square root of 3 and of 6 are also irrational?
4.3.2: Density of Irrational Numbers
- Read pages 7 through 9, from "Density of Rational Numbers" through "Density of Irrational Numbers."
4.3.3: Algebraic versus Transcendental Numbers
Read this page, which describes transcendental numbers. Note that both
and 
are transcendental.
4.4: The Field of Rational Numbers
- Read pages 1-7 of the text. The first 6 pages discuss the field and order axioms for real numbers. The Completeness Axiom on page 6 is what distinguishes the rational numbers from the real numbers - the latter is COMPLETE, while the former is not.
