
Rational Numbers



Definition 
Comparing rational numbers

Equivalent fractions 
Reducing or simplifying fractions 
Adding and subtracting rational
numbers or fractions

Multiplication of
rational numbers 
Reciprocal fractions, multiplicative inverse 
Division of
rational numbers 
Simplifying complex or compound
fractions 





Definition 
A
rational number is a ratio or quotient of two integers, usually written
as the vulgar fraction a/b, where
b
is
not zero. The set of all rational numbers is denoted as 
Q
= { a/b
 a,
b
Î
Z,
b is
not 0 }. 
A rational number or a fraction
a/b
denotes the result of dividing a
by b, i.e.,
a/b
= a ¸
b, b
is
not 0. 
All integers are rational numbers as they can be written as a fraction with a denominator of 1. 
Terminating decimals and recurring decimals are rational numbers as they can be written
as fractions. 
Not all decimals are rational numbers, since not all decimals can be written as fractions. 

A rational number is positive if its numerator and denominator are both positive integers or both
negative integers. A rational number is negative if its numerator and
denominator have different signs, that is,



Examples: 




Comparing rational numbers

A positive rational number
a/b =
r corresponds to a point
P(r) of
the real number line, symmetrically regarding the origin O, an
opposite number −
a/b corresponds to a point
P(−r). A rational
number r_{1}
is less than a rational number
r_{2} if lies to the left of
r_{2} on
the number line, written r_{1
}< r_{2}. 



Two
rational numbers, 

and 

are
equal, i.e., 

if 
a
· d
= b
· c. 


If the numerators
a and
c
both are integers and the denominators b
and d
both are natural numbers then; 


if 
a
· d
< b
· c 
or 

if 
a
· d
> b
· c. 


Examples: 


since
(4)
· (10)
= 5
· 8 


since
3 · 35 = (7)
· (15) 



since
5 · 7 > 8 · 4 


since
(9)
· 3 < 5
· (5) 


Equivalent fractions 
Convert a rational number or fraction to an equivalent fraction by multiplying the numerator and denominator by the same nonzero
number. 
Equivalent
fractions are different fractions that represent the same
number. 

Examples: 




Reducing
or simplifying fractions 
To reduce or simplify a fraction to lowest terms, divide the
numerator and denominator by their greatest common divisor (gcd)
or (greatest common factor  gcf). 


Examples: 




Adding and subtracting rational
numbers or fractions

Find the least common denominator, write equivalent fractions, then
add or subtract the fractions. Reduce if necessary. 

Examples: 




12 = 2 · 2 · 3, 36 = 2 ·
2 · 3 · 3, 48 = 2 · 2 ·
2 · 2 · 3 


LCD(12, 36, 48) = 2 · 2 · 2 · 2 · 3 · 3 = 144 





Multiplication of
rational numbers 
Rational numbers are multiplied by multiplying
numerators and multiplying denominators. Change any mixed numbers to improper fractions. Reduce fractions before
multiplication. 


Examples: 




Reciprocal fractions, multiplicative inverse 
Two rational numbers different from zero are reciprocal
or multiplicative inverse if their product is unity. 
Thus, the reciprocal of a fraction 



Division of
rational numbers 
Dividing a rational number or fraction by another
fraction is equivalent to multiplying the dividend with the reciprocal of the divisor.



Examples: 







Simplifying complex or compound
fractions 
A complex or compound fraction is a fraction whose numerator
and/or denominator are also a fraction or mixed number.
Divide the numerator by the denominator.



Examples: 











Beginning
Algebra Contents A 



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