Chapter 2 Real Numbers and Complex Numbers

What is a number? What qualifies a mathematical object to be identified as some type of number? Exactly what basic properties objects called ‘numbers’ should possess can be a subject of debate: is a telephone number a number? One answer comes by introducing the idea of a number system.

What is a number system? A number system is a set of objects, together with operations (+, x, others?) and relations (= and perhaps order) that satisfy some predetermined properties (commutativity, associativity, etc.) Chapter 2 examines the numbers that up the rational, real and complex numbers systems, starting from their most familiar geometric representations: the real number line and the complex plane.

2.1.1 Rational numbers and Irrational numbers Defn: A number is rational if and only if (iff) it can be written as the indicated quotient of two integers: a/b, a ÷ b, Note: A rational number is not the same as a fraction! π/3 or 0.25

What makes rational numbers so nice? Theorem 2.1 a.The set Q of rational numbers is closed under addition, subtraction and multiplication. b.The set Q – {0} of non-zero rational numbers is closed under division. Also, the algorithms we have for operations with fractions make rational numbers easy to add, subtract, multiply and divide.

Estimating rational numbers It is easy to estimate the value of a positive rational number a/b if we write it as a mixed number (the sum of an integer and a fraction between 0 and 1 written with no space between them). The integer part of a positive rational number t is denoted by This is the greatest integer less than or equal to t

Division Algorithm When we divide one integer by another, what guarantees that our quotient and remainder are unique? The Division Algorithm. Theorem 5.3 If a and b are integers with b > 0, then there exist unique integers q and r such that a = bq + r, and 0 ≤ r < b. (or a/b = q + r/b, with 0 ≤ r < b)

Irrational Numbers Defn. An irrational number is a real number which is not a rational number. They show up everywhere—in roots, logarithms, and trig functions to name a few. In fact we will show later that there are more irrational numbers than rational ones!

A classic example of indirect proof Theorem 2.2 Let n be a positive integer. Then the square root of n is either an integer or it is irrational. This theorem is equivalent to asserting that if p is not a perfect square, then x² - p = 0 has no rational solutions. (A special case of the Rational Root Theorem.)

Generating Irrational Numbers Theorem 2.3 Let s be any non-zero rational number and v any irrational number. Then s+v, s-v, sv and s/v are irrational numbers. What about the following power? Sums, differences, products and quotients of irrational numbers may be either rational or irrational, so the set I of irrational numbers is not closed under any of the arithmetic operations.