Download The Real Number System

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
This is an ellipse
Which means it
Continues.
• Always begin with 1
• {1, 2, 3, 4, 5, 6, 7, . . . .}
• Sometimes referred to as Counting
Numbers
Real Numbers
• {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
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
Irrational
Numbers
All
Natural
positive
All Positive
Numbers
Numbers
numbers
Any number
that
can be
and their
opposites
plus
0.
written asincluding
a fraction0.
a whereWhole
be can not equal 0.
Numbers
b
Integers
Rational Numbers
REAL NUMBER SYSTEM
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.