A. Unlimited
Representation
Every
rational number can be represented in an unlimited number of ways.
Example:
3/2 = 6/4 = 9/6 = 12/8...
Example:
1/1 = 2/2 = 3/3 = 4/4...
(The
previous example is used heavily to manipulate rational numbers.)
B. Perfect
Rationals are also the set of Integers
A perfect
rational is a fraction whose denominator is 1.
Example:
1/1, 2/1, 3/1, 4/1...
which are
the integers 1, 2, 3, 4, ...
if you alter
the numerator you must also alter
the denominator or...
if you alter the denominator you must also
alter the numerator.
A.
Multiplication
When multiplying two
or more rational numbers together, the result is the product
of the numerators over the product of the
denominators.
Examples:
(6/4)*(3/5 ) = (6*3)/(4*5) = 18/20
since
a rational has an unlimited number of forms, we
usually write them in their simplist 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 18/20 as 9/10 for the
answer, (although, technically, both are the correct.)
1/4 * 3/5 *
7/2 = (1*3*7) / (4*5*2) =
21/40
2/-3 * -5/3 * 6/2
= (2*-5*6) / (-3*3*2) = 60/18 = 10/3
B. Changing
forms
Recall
that multiplying any number by 1 (the multiplicative identity) does not alter
that number. This result holds for rational numbers as well. In other words
recall that
1
= 1/1 = 2/2 = 3/3 = 4/4... = -1/-1 = -2/-2 =
-3/-3...
Examples:
Write
3/4 so that the denominator is 16
Well,
4 * 4 = 16, and multiplying any number by one does not change that number, the
version of 1 we're looking for is 4/4.
So,
we have: 3/4 * 4/4 = 12/16
Write
1/3 so that the denominator is 21
3
* 7 = 21, we have 1/3 * 7/7 = 7/21
Write
5/2 so that the numerator is -10
5
* -2 is -10, we have 5/2 * -2/-2 = -10/-4
C.
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.
Example:
using same denominators
1/3
+ 6/7
=
1/3 * 7/7 + 3/3 * 6/7
=
7/21 + 18/21
=
(7+18)/21
=
25/21
The
second is using direct multiplication and addition.
Example:
Same as above but using multiplication and addition
1/3
+ 6/7
= ((7*1)
+ (3*6)) / (3*7)
=
(7 + 18 ) / 21
=
25/21
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.
7/7 = 1 as does 3/3 = 1. Our goal is to make the denominators the same. Once
the denominators are the same, we merely have to add the numerators together.
The common denominator we seek is the lcm (...or least common
multiple), 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,
((7*1) + (3*6)) / (3*7), can be split into two fractions,
(7*1) / (3*7) and (3*6) / (3*7)
which are 7/21 and 18/21 respectively. (This decomposition will be explained
later.) In one sense we are combining steps.
Example:
1/4 + 3/5 + 7/2
method 1:
lcm = 20 we have then,
1/4
* 5/5 + 3/5 * 4/4 + 7/2 * 10/10
(5
+ 12 + 70) / 20 = 87/20
method 2:
(1*5*2 + 3*4*2 + 7*4*5) / (4*5*2)
(10 + 24 + 140) / 40
174/40 = 2/2 * 87/20 =
87/20