Both rational and irrational numbers are
real
numbers.

This Venn Diagram shows the
relationships between sets of numbers. Notice that rational
and irrational numbers are contained in the large blue rectangle
representing the set of Real Numbers. 
A
rational number is a number that can be expressed as a fraction or
ratio.
The
numerator and the denominator of the fraction are both integers.
When the fraction is divided out, it
becomes a terminating or repeating decimal.
(The repeating decimal portion may be one number or a billion numbers.)
Rational numbers can be ordered on a number line.
Examples of
rational numbers
are :
6 or

can also be written as

6.0

2
or

can also be written as 
2.0 

can also be written as 
0.5 

can also be written as 
1.25 

can also be written as 
0.666666666...


can
also be written as 
0.38181818...


can
also be written as 
0.62855421687...
the decimals will repeat
after 41 digits 
Be careful when using your
calculator to determine if a decimal number is irrational.
The calculator may not be displaying enough digits to show you
the repeating decimals, as was seen in the last example above. 
Hint:
When given a rational number in decimal form and asked to write it as a
fraction, it is often helpful to "say" the decimal
out loud using the place values to help form the
fraction.
2 
. 
3 
4 
5 
6 
o 
a 
t 
h 
t 
ten 
n 
n 
e 
u 
h 
t 
e 
d 
n 
n 
o 
h 
s 

t 
d 
u 
o 


h 
r 
s 
u 


s 
e 
a 
s 



d 
n 
a 



t 
d 
n 



h 
t 
d 



s 
h 
t 




s 
h 





s 
Examples: Write
each rational number as a fraction:
Rational number in decimal form 
Rational number in fractional form 
1. 0.3 

2. 0.007 

3. 5.9 

Hint:
When checking to see which
fraction is larger, change the fractions to decimals by dividing and
compare their decimal values.
Examples:

Which of the given
numbers is greater? 
Using full
calculator display to compare the numbers. 
1. 

.6666666667 > .25 
2. 

2.333333333 > 3.666666667 
An irrational number
cannot be expressed as a fraction.
Irrational numbers cannot be represented as terminating or repeating
decimals.
Irrational numbers are nonterminating, nonrepeating
decimals.
Examples of irrational numbers are:

= 3.141592654….. 

= 1.414213562….. 
and
0.12122122212… 

Note: Many students think that
is
the terminating decimal, 3.14, but it is not. Yes, certain math
problems ask you to use
as 3.14, but that problem is rounding the value of
to make your calculations easier.
is actually a nonending decimal and is an irrational number.
