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 0.25• 2 ½ can be written as 2.5• -5 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.

• {. . . . -3, -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– 0.14114111411114…

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

IrrationalNumbers

REAL NUMBER SYSTEM

Integers

All Positive Numbers and their opposites

including 0.WholeNumbers

All positive numbers plus 0.

NaturalNumbers

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.