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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, 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 2 between them). 1 4 3 • The integer part of a positive rational number t is denoted by t 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. s What about the following power? v • 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.