The Real Number System Created by Mrs. Gray 2010

What is the Real Number System? The set of all rational and irrational numbers. { } indicates a set. (braces) All numbers can be classified as rational or irrational.

FYI……For Your Information …(ellipsis)—continues without end { } (set)—a collection of objects or numbers. Sets are notated by using braces { }. Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.

Real Numbers can be classified as: Rational –Fractions (proper, improper and mixed) –Integers (positive and negative numbers) –Whole Numbers –Natural Numbers Irrational

Natural Numbers Always begin with 1 {1, 2, 3, 4, 5, 6, 7,....} Sometimes referred to as Counting Numbers This is an ellipse Which means it Continues.

{x | x can be written as a decimal number.} Read as all numbers x, such that x is a decimal. –Examples 3 can be written 3.0 ¼ can be written ½ can be written as can be written as -5.0 Real Numbers

Whole Numbers Always begin with 0 { 0, 1, 2, 3, 4, 5,.....} The set of Whole Numbers is the same as Natural except that it includes 0. The way to remember it is think “0” in “whole”

Integers The set of all natural numbers and their additive inverses (opposites) and 0. { , -2, -1, 0, 1, 2, 3,....} Does not include fractions or decimals

Rational Numbers Numbers that can be expressed as the ratio (fraction) of two integers, a/b where b ≠ 0. Decimal representations of rational numbers either terminate or repeat. Examples: – 2.375, can be read as 2 and 375 thousandths and written as 2 375/1000, (terminating decimal) – 4, can be written as 16/4, 4/1, 8/2 – −0.25, can be read as negative 25 one-hundredths and written as - 25/100 – 0.14, repeating decimal and can be written as 14/99

Irrational Numbers Numbers that cannot be expressed as a ratio (fraction) of two integers. Their decimal representations neither terminate nor repeat. Decimals that go on forever without repeating a pattern. Examples: –– – 3– 3 – …

Real Number System Irrational Numbers Rational Numbers Integers - + Whole Numbers “0” Natural Numbers “Counting” Fractions

Rational Numbers Any number that can be written as a fraction a where be can not equal 0. b Irrational Numbers REAL NUMBER SYSTEM Integers All Positive Numbers and their opposites including 0. Whole Numbers All positive numbers plus 0. Natural Numbers

Questions Determine if the following statements are true or false and give a short reason why: –Every integer is a rational number. –Every rational number is an irrational number. –Every natural number is an integer. –Every integer is a natural number.