Rational numbers

1. RATIONAL NUMBERS
Different kinds of Numbers
We use numbers to solve problems. A simple problem such as "If Ellen has 3 walls on which to watch the
paint dry, and she watches 1 of them, how many walls are left to watch?" has a simple answer. As
problems get more interesting, as surely they must, so do the numbers we need in order to solve them.
Natural numbers, Whole numbers, Integers, Rational Numbers can be represented in a simple diagram as
below:
Integers
Whol e numbers
Natural numbers
Rational numbers
Natural Numbers are also known as counting numbers. These are, as the second name implies, the
numbers 1, 2, 3, 4, etc. These numbers are very useful for counting things.
Whole Numbers natural numbers including zero.
Integers Natural numbers, their negatives and zero.
So, the sum, difference and product of two integers is again an integer. But the result of dividing an
integer may not always be an integer, it can be a fraction.

2. For example,
4
2
= 2, but what about
5
2
?
More formally we would say:
A rational number is a number that can be in the form p/q
where p and q are integers and q is not equal to ozero.
So, a rational number can be:
p
q
Where q is not zero
Examples:
p q p / q =
1 1 1/1 1
1 2 1/2 0.5
55 100 55/100 0.55
1 1000 1/1000 0.001
253 10 253/10 25.3
Rational Form and Various Forms
 Rational numbers arise from the attempt to measure all quantities with a common unit of
measure.
The pursuit of infinity begins with an examination of the idea of number, or quantity. Numbers
originally were tools used to quantify groups of objects or to measure real things, such as the
length of a pole or the weight of a piece of cheese. Measuring something requires some
fundamental unit that can be used as a basis of comparison. Some things, such as rope or time,
can be measured, or quantified, using a variety of units. In measuring a length of rope, for
example, we might express the result as either "5 feet" or "60 inches." The length of time from
one Monday to the next Monday is commonly called a "week," but we could just as easily—and
correctly—call it seven "days," 168 "hours," 10,080 "minutes," or 604,800 "seconds." These
number expressions all represent the same length of time and, thus, are interchangeable.
Converting any one of these equivalent values into another simply requires multiplying or
dividing by some whole number. For example, 10,080 minutes is 168 hours times 60. In fact,

3. every one of the above measurements could be converted into seconds by multiplying by
appropriate whole number values.
Now, we can take any two quantities and ask a similar question: can we find a common unit of
measurement that fits a whole number of times into both? Take 8 and 6, for example. This is
straightforward; both 8 and 6 are whole numbers, so we can use whole units to measure them
both. What about and ? Here, each of the quantities uses different base units, namely halves
and eighths, and comparing them would make little sense, because they are different things. We
can find a common unit of comparison, however, by recognizing that is the same as .
So, we found a common denominator of 8, which implies that both of the fractions could be
expressed as multiples of the same unit, "one eighth." In some sense, we have redefined our basic
unit of measurement, or fundamental piece, to be of the original piece, thereby transforming
the original fractions into easily-compared multiples of the same fundamental unit.
If two numbers can be expressed as whole-number multiples of some common unit, of whatever
size, they are in some sense "co-measurable" in that we can measure both using the same ruler.
The proper mathematical term for "comeasurable" is "commensurable." One way to think about
this is that two lengths are commensurable if there is a basic unit of measure that fits into both of
them a whole number of times. If we were to cut some length of rope into two pieces of lengths a
and b, there would be some third length, c, such that a = mc and b = nc. In other words, these
two numbers could be expressed as multiples of some common unit.

4. Using a little algebra, we can confirm that the ratio of magnitudes of our two commensurable
quantities is equal to a ratio of whole numbers:
Canceling the common factor of c yields the equation:
Numbers that can be expressed as ratios of two whole numbers are called rational numbers. This
idea of "creating" numbers that may not relate to any observable value in the real world is fairly
modern. Although today we are comfortable speaking of a number such as as a concept that
"exists" in its own right, the ancient Greeks generally took care to phrase things only in terms of
geometric quantities—those that exist in the physical world. For instance, they might have
spoken of two lengths of rope, one that could be described as 13 measures of a certain unit, and
the other 25 units of that same measure, but they would not necessarily speak of the shorter
length being of the other.

5. On the other hand, today we are comfortable saying, for example, that a length of string is of
an inch long. In doing so, we are saying that it is commensurable with a piece of string one inch
long, the fundamental unit of comparison being .
The modern and ancient views of rational numbers are intimately linked, but it is important to
remember that the Greeks thought of commensurability in terms of whole units. The
Pythagoreans, the followers of Pythagoras of Samos in 6th century BC Greece, held sacred the
idea that the first principle underlying everything is "arithmos," the intrinsic properties of whole
numbers and their ratios. It is certainly a tidy idea that whole numbers, or ratios of them, are all
that is required to describe the world mathematically. It is thought that this belief had origins in
both the study of figurate numbers and the recognition that strings or hammers of
commensurable length sounded harmonious when played or struck together.
To simplify a rational expression:
1. Factor numerator as much as possible.
2. Factor denominator as much as possible
3. Cancel common factors.
Numerator and denominator are linear functions
Example 1
Simplify the follow ing rational expression:

6. Solution
1: Factor numerator:
2: Factor denominator:
3: Cancel common factors:
Example 2
Simplify the follow ing rational expression:
Solution
1: Factor numerator:
2: Factor denominator:
3: Cancel common factors:
The factors 2 - x and x - 2 are almost the same, but not quite, so they can't be cancelled. Remember to sw itch the sign out f ront: 2 -
x = -(x - 2)
 Simplify the following rational expression:
How nice! This one is already factored for me! However (warning!), you will usually need to do the
factorization yourself, so make sure you are comfortable with the process!
The only common factor here is "x + 3", so I'll cancel that off and get:

7. Then the simplified form is:
Exercise -1
Simplify the expressions given below

푥푢+푦푢
푥푣+푦푣

3푥+6
푥+2
 2x2 + 6
5xy +15y

4−8푥
2푥−6
Addition and Subtraction
Adding and subtracting rational expressions is identical to adding and subtracting with integers.
Recall that when adding with a common denominator we add the numerators and keep the
denominator. This is the same process used with rational expressions. Remember to reduce, if
possible, your final answer. Subtraction with common denominator follows the same pattern,
though the subtraction can cause problems if we are not careful with it. To avoid sign errors we
will first distribute the subtraction through the numerator. Then we can treat it like an addition
problem.
Wi th the Same De nominator
Add the nume r a tor s toge the r a nd the de nominat o r is ma int a ine d.
Wi th a Di f fe re nt De nominator
Fir s t , r e duc e the de nomina tor s to a c ommon de nomina tor a nd a dd or
s ubt r a c t the nume r a tor s of the e quiva le nt f r a c t ions obt a ine d.

8. To Add or Subtract Rational Expressions with common factor
1. Add or subtract the numerators.
2. Place the sum or difference of the numerators found in step 1 over
the common denominator.
3. Simplify the fraction if possible.
Example 1
Add
ퟔ
풙−ퟓ
+
풙+ퟐ
풙−ퟓ
6
푥−5
+
푥+2
푥−5
=
6+(푥+2)
푥−5
=
풙+ퟖ
풙−ퟓ
Example 2
Simplify
Then the answer is:
Exercise -2

2
2푥+1
+
3
2푥−1
 x +
푥
푥−1

1
푥+2
-
1
푥−2

9.  1 −
2
푥+1
Multiplication And Division
With regular fractions, multiplying and dividing is fairly simple, and is much easier than adding and
subtracting. The situation is much the same with rational expressions (that is, with polynomial fractions).
how you multiply regular fractions: You multiply across the top and bottom. For instance:
And you need to simplify, whenever possible:
While the above simplification is perfectly valid, it is generally simpler to cancel first and then do the
multiplication, since you'll be dealing with smaller numbers that way. In the above example, the 3 in the
numerator of the first fraction duplicates a factor of 3 in the denominator of the second fraction, and the 5
in the denominator of the first fraction duplicates a factor of 5 in the numerator of the second fraction.
Since anything divided by itself is just 1, we can "cancel out" these common factors (that is, we can
ignore these forms of 1) to find a simpler form of the fraction:
This process (cancelling first, then multiplying) works with rational expressions, too.
 Simplify the following expression:
Simplify by cancelling off duplicate factors:
Then the answer is:

10. For dividing rational expressions, you will use the same method as you used for dividing
numerical fractions: when dividing by a fraction, you flip-n- multiply. For instance:
 Perform the indicated operation:
To simplify this division, I'll convert it to multiplication by flipping what I'm dividing by; that is, I'll
switch from dividing by a fraction to multiplying by that fraction's reciprocal. Then I'll simplify as
usual:
Can the 2's cancel off from the 20's? No! This is as simplified as the fraction gets.
Division works the same way with rational expressions
Example-1
x2 - y2
is a rational expression.
(x - y)2
To simplify, we just factor and cancel:
(x - y)(x + y) x + y
=
(x - y)2 x - y
Example-2
Divide and simplify.
Rewrite the division problem as multiplication by the reciprocal of the divisor
Division is same as multiplying by the reciprocal of the divisor.
Here, the reciprocal of the divisor is,
So, the given expression becomes:

11. To multiply rational expressions, multiply the numerators and multiply the denominators.
So,
cancel the common factors.
Therefore,
Exercise-3
1. x ÷
푥
푥−1
2.
푥
푥+1
÷
푥
푦−1
3.

12. EQUAL FRACTIONS
How can we check whether two fractions are equal?
A simple illustration is given below
First draw a rectangle, cut it into even size pieces using vertical lines and color in some of the pieces.
The image below shows a rectangle cut into 4 pieces. One of the pieces has been colored in, giving the
fraction 1/4.
Second draw a horizontal line through the centre of the rectangle. This cuts the rectangle into twice
as many pieces as before:
Again, we count the number of colored pieces and put that over the total number of pieces. It is clear
that nothing has changed in this fraction except the numbers used to describe it. The first and second
rectangles both have the same amount colored in.
We can push this example a little further by using more horizontal lines. In the below example a total
of three horizontal lines have been used. All we need to do is remember to keep all the pieces the
same size, which means space the horizontal lines evenly:

13. These three fractions are all exactly the same. It makes absolutely no difference how many horizontal
lines we add to the original fraction picture. The amount colored in does not change, and so they are
all equal.
So by reducing each fraction to its lowest forms we can check whether two fractions are different
forms of same fraction .But this method is not easy to apply if numerator and denominator are
large.
Lets think in algebraic terms. Suppose
푎
푏
푝
푞
=
Then
푎
푏
divided by
푝
푞
should be 1.
푎
푏
÷
푝
푞
= 1
That is,
푎
푞
x
푏
푝
= 1
That means,
푎푞
푏푝
= 1
So here we get , aq = bp
For the numbers a, b, p, q, if
풂
풃
=
풑
풒
, then aq=bp. On the other hand, if aq=bp and also b≠0, q≠0 then
풂
풃
=
풑
풒
.
For the numbers a, b, p, q, if
풂
풃
=
풑
풒
, then
풂
풑
=
풃
풒

14. Exercise-4
1. If
푥
푦
= 3
4
, then what is
5푥+2푦
5푥−2푦
?
2. Prove that if
푥
푦
= 푢
푣
then each is equal to
2푥+5푢
2푦+5푣
. Is this true for other
numbers, instead of 2 and 5?
DECIMAL FORMS
 Rational numbers can be expressed as decimals that repeat to infinity.
In modern arithmetic, we use a base-10 system to count, or evaluate, things. Large quantities are
generally represented in terms of ones, tens, hundreds and the li ke, whereas small quantities are
more easily represented in terms of tenths, hundredths, thousandths, and so on. Although the
Greeks did not use a base-10, or decimal, number system, it is illuminating to see how rational
numbers behave when expressed as decimals.
For example, we can interpret the number 423 as four 100s, two 10s and 3 units (or 1s), and the
value 0.423 as four s, two s and three s. In such a decimal system it is
necessary to think of all quantities in terms of units of tens, tenths and their powers. Thus, , for
instance, must be interpreted as , to be written as 0.5.
For example,
Find the decimal expansions of
10
3
,
7
8
and
1
7

16. From the above examples it is clear that decimal expansion terminates or ends after a finite no.of
steps or it will go on forever.
The decimal expansion that terminates after a definite no. of steps is called terminating
decimals. Ex:
Exercise – 5
Write the following fractions as decimals
1.
1
9
2.
2
9
3.
1
12
4.
2
11
ퟕ
ퟖ
,
ퟏ
ퟐ
The decimal expansion in which the remainders repeat after a certain no. of steps forcing
the decimal to go on forever is called non-terminating or recurring decimals. Ex:
ퟏ
ퟑ
,
ퟏ
ퟕ