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A rational number is any number that can
be represented by the ratio of two integers.
Examples:
,
,
, 
The horizontal bar is called a vinculum and acts
both as a grouping symbol (serves as parenthesis) and to indicate division.
These numbers are also called fractions,
where
the number above the vinculum is called the numerator
and
the number below the vinculum is called the denominator. Many times we'll say '6' over
'4', meaning, .
Properties:
A. Unlimited
Representation
Every rational
number can be represented in an unlimited number of ways.
Example:  = =  =  ...
Example: 1 =  =  =  =  ...
(The previous example is used heavily to manipulate rational numbers, more on
this later.)
B. Perfect Rationals are the Integers
A perfect rational is a fraction whose
denominator is 1.
Example: ,  ,  ,  ...
which are integers 1, 2, 3, 4, ...
C.
Changing forms
Recall
that multiplying any number by 1 (the multiplicative identity) does
not change that number. This result holds for rational numbers as
well. In other words recall that
1 = = = = ... =  =  = ...
Examples:
Write so that
the denominator is 16
Well, 4 * 4 = 16, and multiplying any number by
1 does not change that number, the version of 1 we need is .
So, we have: * = 
Write  so that
the denominator is 21
3 * 7 = 21, we have *  = 
Write  so that
the numerator is -10
5 * -2 is -10, we have *= 
D. Writing rationals in their simplest form
Factor the numerator and the denominator into
their prime factors, then use the previous result to simplify. This operation
is usually called canceling common factors.
Examples:
=  = * * 
= 1 * 1 *
=
=  = * *
=
1 * 1 *
=
Operations:
B.
Multiplication (first since addition requires multiplication)
When multiplying
two or more rational numbers together, the result is
the product of the numerators over the product of the
denominators.
Example:
Since
a rational has an unlimited number of forms, we usually write them in their
simplest form, that is, the form with the smallest possible numerator and
smallest possible denominator. A systematic way to do this will follow shortly.
We would write  as  for the
answer, although, technically, both are correct.
Example:
*  *  =  = 
Example:
*  *  =  =  = 
Example:
3
*
=
*
=
Example:
*
5 = *
 =
C. Division
To divide rational numbers (fractions) you invert the
denominator then multiply.
Example: Here we invert 2/9 to
9/2 first
Example: In this example
we recall 12 is 12/1, then we invert
D. Adding
Rational Numbers
There are two ways to add rational numbers, the
first is to convert them so that they all have the same denominators, the
second is using multiplication and addition.
Example: using same denominators
+
 = * + *
= + 
=
= 
Example: Same as above but using multiplication
and addition
+
=  = =
In the first case we use the fact that
multiplying any number by 1 does not change that number, and in particular,
only changes the form of the rational number.
= 1 as
does
= 1.
Our goal is to make
the denominators the same. Once the denominators are
the same, we add the numerators together. The common
denominator we seek is the lcm of the denominators,
and in this example ,the lcm of 3 and 7 is 21.
The second case is really the same as the first
but it does not necessarily use the lcm of the denominators. In this second
case, notice that the second step, , can
be split into two fractions,  and  which
are and respectively.
(This decomposition, using the distributive law of multiplication over
addition (which is used heavily in algebra to
factor expressions), will be explained later. In a sense we are combining steps.
Example:
+ +
method 1: lcm = 20 we have then,
*
 + * + *
= 
method 2:
= * =
More on method 2:
Method 2 is how addition is defined for rational
numbers in later courses covering the theory of mathematics. Sometimes we refer
to this method as cross multiplying when
two fractions are involved, since we we multiply the numerator
of the first fraction with the denominator of the second fraction then add the
product of the second fraction's numerator and the first fraction's
denominator. We then write this sum over the product of the denominators.
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-
multiply a by d
-
multiply b by c
-
add these two products
-
write over the
product b
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F.
Mixed Numbers
A mixed number is a number represented by an
integer and a proper fraction. A
proper fraction is a fraction where the numerator is less
than the denominator.
Example: 2 read,
two and one third. This form looks like multiplication but is NOT
multiplication, it is actually: ( 2 +), with
the + sign and the parenthesis omitted, but understood
to be there.
To write this number as a fraction (improper
fraction), remember that 2 is the same as
then
combine this with
. We need a denominator equal to 3.
1 =, * =  .
So, we have
+ =  = 
2 =
Using cross multiplication we have ( 2 * 3 + 1 )
divided by 3.
Note: the
vinculum is a grouping symbol, it groups the sum 6 + 1.
This fraction could be
written as  = * (6 +
1), and using the distributive law of multiplication over addition we have
*
(6 + 1) = * 6 + * 1 = +
=  +
= 2 * +
= 2 + = 2
It is important to note that the denominator 3
in this last example divides each term of the sum in the numerator. Again, this
is nothing more than the distributive law of multiplication over addition.
However, not understanding this causes numerous
errors later in algebra simplifying expressions.
To write an improper fraction as a mixed number,
divide the numerator by the denominator; the integer
is the quotient, the proper fraction will have the
remainder as its numerator.
Example:
27 divided by 6 is 4 with remainder 3
, so
we have 4  ,
3 over 6 can be reduced to 1 over 2 so we have
the answer: 4
= 4
Example: 
75 divided by 25 = 3 (no remainder),
so the answer is 3.
G. Negative mixed
numbers
Consider: — 3
Remember the implied
+ sign and parenthesis?
- 4 + 1 
+ 
+ 
NOW, careful here,
-31 divided by 10 is -3 remainder -1, so
we have -3 as the
answer. --> -1 *( 3 +)
In other words,
momentarily ignore the – sign, think of the result as a positive improper
fraction, change it into a mixed number, then put the – sign back.
Operations with mixed
numbers.
Addition,
Subtraction:
Two methods:
1) convert all numbers to improper fractions
first.
2) separate the numbers into integer and
fractional parts first.
Example:
Multiplication,
Division:
First change every mixed number into an improper
fraction, then carry out the multiplication and/or
division as shown earlier.
Example:
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