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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
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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 Numbersor 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, b, ϵ I, b ≠ 0} • This textbook calls 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 N W I Q 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: