Thinking Mathematically Number Theory and the Real Number System 5.5 Real Numbers and Their Properties

Real Numbers Real numbers are the union of the rational numbers and the irrational numbers.

Subsets of the Real Numbers Natural Numbers: {1, 2, 3, 4, 5, …} These numbers are used for counting. Whole Numbers: {0, 1, 2, 3, 4, 5, …} The whole numbers add 0 to the set of natural numbers. Integers: {…, -3, -2, -1, 0, 1, 2, 3,…} The integers add the negatives of the natural numbers to the set of whole numbers.

Subsets of the Real Numbers Rational Numbers: These numbers can be expressed as an integer divided by a nonzero integer: a/b: a and b are integers: b does not equal zero. Rational numbers can be expressed as terminating or repeating decimals. Irrational Numbers: This is the set of numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers.

Examples: Classifying Numbers Exercise Set 5.5 #3 For the set Which elements of this set are: Natural numbers Whole numbers Integers Rational numbers Irrational numbers Real numbers

Properties of Real Numbers Closure under addition, multiplication Addition and multiplication are commutative. Addition and multiplication are associative Multiplication distributes over addition

“Closure” of the Real Numbers The sum or the difference of any two real numbers is another real number. This is called the “closure” property of addition. The product or the quotient of any two real numbers (the denominator cannot be zero) is another real number. This is called the “closure” property of multiplication.

“Commutative” Property of Addition Two real numbers can be added in any order. This is called the “commutative” property of addition. a + b = b + a Example: = 3 + 2

“Commutative” Property of Multiplication Two real numbers can be multiplied in any order. This is called the “commutative” property of multiplication. a x b = b x a Example:12 x 5 = 5 x 12 = 60

“Associative” Property of Addition When three real numbers are added, it makes no difference which two are added first. This is called the “associative” property of addition. (a + b) + c = a + (b + c) Example: (12 + 5) + 3 = 12 + (5 + 3) = = = 20

“Associative” Property of Multiplication When three real numbers are multiplied, it makes no difference which two are multiplied first. This is called the “associative” property of multiplication. (a x b) x c = a x (b x c) Example: (12 x 5) x 3 = 12 x (5 x 3) = 60 x 3 = 12 x 15 = 180

Multiplication “Distributes” over Addition The product of a number with a sum is the sum of the individual products. This is called the “distributive” property of multiplication over addition. a x (b + c) = (a x b) + (a x c ) Example: 12 x (5 + 3) = (12 x 5) + (12 x 3) = 12 x 8 = = 96

Exercises – Identifying Properties Exercise Set 5.5 #31, 33, 35 Name the property illustrated 6 + (2 + 7) = (6 + 2) + 7 (2 + 3) + (4 + 5) = (4 + 5) + (2 + 3) 2 (-8 + 6) =

Thinking Mathematically Number Theory and the Real Number System 5.5 Real Numbers and Their Properties