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Special Sets of Numbers Remember to Silence Your Cell Phone and Put It In Your Bag! Mathematics was invented. Numbers vs. numerals The Set of Counting Numbers or Natural Numbers N = {1, 2, 3, 4, 5, . . . } The Counting Process Say the names of the counting numbers Name the numerals Write the numerals Count a number of objects The Whole Numbers W = {0, 1, 2, 3, 4, 5, . . . } A whole number is the unique characteristic embodied in each finite set and all the sets equivalent to it. 2.1 p. 65 The Set of Integers I = { . . . -3, -2, -1, 0, 1, 2, 3, . . . } For every natural number n, there is a unique number the opposite of n, denoted by –n, such that n + -n = 0. The set of integers, I, is the union of the set of natural numbers, the set of the opposites of the natural numbers, and the set that contains zero. I = {1, 2, 3, …} {-1, -2, -3 ...} {0} 5.1 p. 249 The Set of Rational Numbers Q = { a | a, b, ϵ I, b ≠ 0} b This textbook calls a b a fraction. Fractions are Rational Numbers! Integers are Rational Numbers! Whole Numbers are Rational Numbers! 6.1 p. 302 The Set of Rational Numbers (cont.) A decimal is a symbol that uses a base-ten place-value system with tenths and powers of tenths to represent a number A decimal is a rational number! 6.1 p. 207 Relationships Among these Sets of Numbers NWIQ Q I W N 6.5 p. 362 What numbers are not Rational Numbers? Every rational number can be expressed as a terminating or repeating decimal. Numbers which cannot be expressed as either repeating or terminating decimals are not rational numbers. 6.1 p. 310 & 6.5 pp. 359-362 The Set of Irrational Numbers Real numbers which cannot be expressed as either repeating or terminating decimals. Examples: 6.5 pp. 361-363 The Set of Real Numbers R = {Rational Numbers} ⋃ {Irrational Numbers} Note – The set of rational numbers and the set of irrational numbers are disjoint sets. (They have no elements in common.) 6.5 pp. 361-363 What numbers are not Real numbers? _____________ numbers are not real numbers. Examples: