Download Rational and Irrational Numbers

The Real Number System
Real Numbers
• Real numbers consist of all the rational
and irrational numbers.
• The real number system has many
subsets:
– Natural Numbers
– Whole Numbers
– Integers
Natural Numbers
• Natural numbers are the set of counting
numbers.
{1, 2, 3,…}
Whole Numbers
• Whole numbers are the set of numbers
that include 0 plus the set of natural
numbers.
0, 1, 2, 3, 4, 5,…}
{
Integers
• Integers are the set of whole numbers and
their opposites.
{…,-3, -2, -1, 0, 1, 2, 3,…}
Rational Numbers
• Rational numbers are any numbers that
can be expressed in the form of a , where
a and b are integers, and b ≠ 0. b
• They can always be expressed by using
terminating decimals or repeating
decimals.
Terminating Decimals
• Terminating decimals are decimals that
contain a finite number of digits.
• Examples:
 36.8
 0.125
 4.5
Repeating Decimals
• Repeating decimals are decimals that contain a
infinite number of digits.
• Examples:
 0.333…
 1.9
 7.689689…
FYI…The line above the decimals indicate that number
repeats.
Irrational Numbers
• Irrational numbers are any numbers that cannot
be expressed a as .
b
• They are expressed as non-terminating, nonrepeating decimals; decimals that go on forever
without repeating a pattern.
• Examples of irrational numbers:
– 0.34334333433334…
– 45.86745893…
–  (pi)
– 2
MGSE9–12.N.RN.3 Explain why the sum or
product of rational numbers is rational; why
the sum of a rational number and an
irrational number is irrational; and why the
product of a nonzero rational number and
an irrational number is irrational.
the sum and product of any
two rational numbers is
rational
True or False
Experiment with sum and
product of two numbers from
the following:
5, ½, -1/3, -7, 10, 5/4
the sum of a rational number and an irrational
number is irrational
True or False
• Experiment with
sums and products
of two numbers
from the following:
Rational:
5, ½, -1/3, -7
Irrational:
2

 5
the product of a nonzero rational number and
an irrational number is irrational
True or False
Experiment with sums
and products of two
numbers from the
following:
Rational:
5, ½, -1/3, -7
Irrational:
2

 5