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 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
R = {Rational Numbers} ⋃ {Irrational Numbers} 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? Examples: