# Rational Numbers

Fractional numbers are a subset of Rational Numbers. The set of Rational Numbers consist of both positive fractions, negative fractions and zero. These numbers are numbers of the form $\frac{p}{q}$ where p, q - I and q ? 0. I is the set of integers.

I = {0, ±1, ±2, ±3,...}

The set of Rational numbers is represented by the letter Q. The terminating decimal numbers and the recurring decimal numbers can be written in the form of a fraction. Hence, they are also rational numbers. The rational numbers can be represented on a number line. Some examples of rational numbers are $\frac{2}{3}$, $\frac{-3}{5}$, 0 .

In general we write whole numbers, integers, fractions and decimal numbers in numerator over denominator form, that is $\frac{p}{q}$,q $\neq$ 0 where p and q are integers. Positive rational numbers are represented to the right of 0. Negative rational numbers are represented to the left of 0. The rational number zero is neither a negative rational number nor a positive rational number.

The numbers of the form $\frac{h}{k}$ , where h and k are integers and k ? 0 are called rational numbers. The sets of rational number is denoted by "Q".

Definition of classical rational numbers: Q={$\frac{h}{k}$ : h, k Z, k ? 0},

where Z= integer.

In general we says that every integer is a rational number.

For example $\frac{5}{8}$ ,$\frac{(-3)}{14}$ , $\frac{7}{(-15)}$ , $\frac{(-6)}{-11}$

### Positive rationals:

A rational number is said to be positive if its numerator and denominator are either both positive or negative.Thus, $\frac{5}{7}$ and $\frac{(-2)}{-3}$ are both positive rationals.

### Negative rationals :

A rational number is said to be negative if its numerator and denominator are of opposite signs.

Thus, $\frac{(-4)}{9}$ and $\frac{5}{-12}$ are both negative rationals.

All problems on rational numbers are solved by the " PEDMAS " rule.

**Observe the following:**

Numbers |
As a fraction |
Rational (Yes or No) |

2 | $\frac{1}{2}$ |
Yes |

$\sqrt{3}$ | $\frac{3}{\sqrt{3}}$ |
No |

0.25 | $\frac{1}{4}$ |
Yes |

0.40 | $\frac{2}{5}$ |
Yes |

-0.35 | $\frac{(-7)}{20}$ |
Yes |

1. $\frac{10}{0}$ this is not a rational number because here the denominator =0 (as per the definition of a rational number)

2. Express $\frac{(-3)}{5}$ as a rational number with the denominator 20.

Since we need to make the denominator 20 we need to divide 20 by 5. Now by multiplying the numerator and the denominator by 4 we get $\frac{(-3)}{5}$ × $\frac{4}{4}$ = $\frac{(-12)}{20}$ [here we can observe that the denominator is 20 as per the question.

3. Express $\frac{48}{60}$ as a rational number with the denominator 5.

Since the denominator needs to be made 5, we divide 60 by 5 to get 12. Now by dividing the numerator and denominator by 12 we get, $\frac{(48\div12)}{(60\div12)}$ = $\frac{4}{5}$ [here we can observe that the denominator is 5 as per the question].

Notes: In the above two examples we multiplied as well as divided by same number in the numerator and the denominator the fraction looks different but the numerical value of the fraction remains the same.If we have two integers ‘m’ and ‘n’ that can be written in the form of rational number $\frac{m}{n}$ provided ‘n’ is not equal to zero.

We cannot have a rational form in which the denominator is equal to zero.

If the numerator of a rational number is not zero and the denominator of the expression comes very close to zero then the value of the fraction becomes very large and may approach infinity as the denominator approaches zero.

The rational expression of having both the numerator and the denominator equaling zero would be the same as any other number.

$\frac{0}{0}$ = 2 or 0 x 2 = 0 and we can also see that $\frac{0}{0}$ = 9 or 0 x 9 = 0

Now from the above explanation it is very clear that zero is rational.

**If an expression terminates it is definitely rational.**

If an expression repeats it is definitely rational.

If it does not terminate or repeat then it is considered as irrational.

If an expression repeats it is definitely rational.

If it does not terminate or repeat then it is considered as irrational.

All rational numbers can be expressed as a ratio of integers, but pi (p) cannot be expressed as a ratio of integers and there fore pi (p) is not a rational number.

The value of the number pi (p ) begins with 3.1415 and is a mathematical constant and whose value is equal to the ratio of circumference of a circle and its diameter.

**Pi (p) is an irrational number and consequently its decimal representation would never end or repeat.**

All rational numbers can be expressed as a ratio integers.

Pi (p) is cannot be expressed as a ratio of integers.

Pi (p) is not a rational number.

Pi (p ) is considered as a number.

There exists at the least one non-rational number.

The list of rational numbers would comprise repeating and terminating expressions in an sequential manner where the maximum value of a denominator would be 9.

In any number line, zero occupies the middle place. The positive rational numbers lie to the right of zero and the negative rational numbers lie to the left of zero. Mark zero and name that point as O. Now mark +1 to the right and -1 to the left of zero equally spaced from zero. The mid-point between O and +1 is $\frac{1}{2}$ and the mid-point between 0 and -1 is $\frac{-1}{2}$. Similarly we can mark as many rational numbers on the number line as possible.

To mark a rational number on a number line, divide each of the unit lengths on it into as many divisions as in the denominator of the rational number and then move as many steps as is mentioned in the numerator of the rational number. Let us try to plot $\frac{4}{7}$ on the number line.

Observe that each unit is divided into 7 equal parts.

### Examples for Rational Numbers on a Number Line

Given below are the some examples for a rational number on a number line

**1.** Represent $\frac{5}{3}$ and $\frac{-5}{3}$ on the number line.

**Solution: **

$\frac{5}{3}$ = 1 + $\frac{2}{3}$ Hence $\frac{5}{3}$ lies between 1 and 2.

$\frac{-5}{3}$ = -(1 + $\frac{2}{3}$) Hence $\frac{-5}{3}$ lies between -1 and -2.

In order to represent these on the number line, we shall divide the unit between -1 and -2 and the unit between 1 and 2 into three equal parts.

To plot $\frac{5}{3}$ move 2 steps to the right of 1 and to plot $\frac{-5}{3}$ move 2 steps to the left of -1 on the number line

Observe that both $\frac{5}{3}$ and $\frac{-5}{3}$ are at equal distances from 0 but in opposite directions.

**2.** Write the rational number, $\frac{15}{21}$ in its simplest form.

**Solution: **

A rational number is said to be in its simplest form, if there are no common factor between the numerator and denominator other than 1.

We can write 15 as the product, 3 and 5 and 27 can be written as the product of 3 and 7.

15 = 3 x 5

21 = 3 x 7

Now we can now rewrite $\frac{15}{21}$ as

$\frac{15}{21}$ = $\frac{(3\times5)}{(3\times7)}$ = $\frac{5}{7}$

Hence the simplest form of the rational number, $\frac{15}{21}$ is $\frac{5}{7}$

The numbers of the form $\frac{h}{k}$ , where h and k are integers and k $\neq$ 0 are called rational numbers.

Each of the numbers $\frac{5}{8}$, $\frac{(-13)}{14}$, $\frac{7}{(-15)}$ and $\frac{(-16)}{11}$ is a rational number.

**There are three properties of rational numbers.**

**Property 1:** If $\frac{h}{k}$ is a rational number and m is a nonzero integer then $\frac{h}{k}$ = $\frac{(h\times m)}{(k\times m)}$.

**For examples:**

$\frac{(-3)}{4}$ = $\frac{((-3)\times2)}{(4\times2)}$ = $\frac{((-3)\times3)}{(4\times3)}$ = $\frac{((-3)\times4)}{(4\times4)}$ .....

or, $\frac{(-3)}{4}$ = $\frac{(-6)}{8}$ = $\frac{(-9)}{12}$ = $\frac{(-12)}{16}$ =......

Such rational numbers are called **equivalent rational numbers.**

**Property 2:** If $\frac{h}{k}$ is a rational number and m is a common divisor of h and k, then $\frac{h}{k}$ = $\frac{(h\div m)}{(k\div m)}$.

Thus, we can write, $\frac{(-32)}{40}$ = $\frac{(-32\div8)}{(40\div8)}$ [Here in the fraction $\frac{(-32)}{40}$ we can observe that both the numerator and

denominator are divisible by 8 so, -32 ÷ 8 =-4 and 40 ÷ 8 =5]

= $\frac{(-4)}{5}$ .

Standard form of rational number

A rational number $\frac{h}{k}$ is said to be in standard form if h and k are integers having no common divisor other than 1 and k is positive.

**For example:** Express $\frac{33}{(-44)}$ in standard form.

$\frac{33}{(-44)}$ = $\frac{(33\times(-1))}{(-44\times(-1))}$ = $\frac{(-33)}{44}$ [Here 33 × (-1)=-33 and = -44 × (-1) = 4 and as we know two negative becomes positive]

The greatest common divisor of 33 and 44 is 11

Therefore, $\frac{(-33)}{44}$ = $\frac{((-33)\div11)}{(44\div11)}$ = $\frac{(-3)}{4}$.

Hence, $\frac{(-33)}{44}$ = $\frac{(-3)}{4}$ (in standard form)

**Property 3 :** Let $\frac{h}{k}$ and $\frac{l}{o}$ be two rational numbers.

Then, $\frac{h}{k}$ = $\frac{l}{o}$

or, (h×o) = (k×l).[here you need to do cross multiply]

or, ho = kl

**Is zero a rational number ?**

Yes, zero is a rational number, because zero divided by anything is zero.**Is pi a rational number ?**

No, $\pi$ is an irrational number, because $\pi$ cannot be expressed as a fraction.
If two rational numbers are to be added we should convert each of them into a rational number with a positive denominator.

**Case 1:** When given numbers have same denominators

In this case, we define ($\frac{a}{b}$ + $\frac{c}{b}$)[here we can observe that the denominators are same i.e., b]

= $\frac{(a+c)}{b}$ [here we can observe that as the denominators are same so we wrote (a+c) together]

**Examples for Adding Rational Numbers **

1. Find the sum of $\frac{5}{9}$ + $\frac{(-11)}{9}$.

We have:

$\frac{5}{9}$ + $\frac{(-11)}{9}$ [here we can observe that the given numbers have the same denominator]

= $\frac{(5+(-11))}{9}$ [as the denominators are the same for both the numbers so we write 5+(-11) together]

= $\frac{(5-11)}{9}$ [we know that the positive sign and the negative sign becomes negative]

= $\frac{(-6)}{9}$ [here 3 is the common factor of 6 and 9 so, we need to divide by 3 in the numerator and the denominator]

=$\frac{(-6)}{9}$ = $\frac{(-2)}{3}$

= $\frac{(-2)}{3}$

2. Find the sum of $\frac{8}{(-11)}$ + $\frac{3}{11}$.

$\frac{8}{(-11)}$ + $\frac{3}{11}$

= $\frac{(-8)}{11}$ + [here $\frac{8}{(-11)}$ = $\frac{(8\times(-1))}{((-11)\times(-1))}$ = $\frac{(-8)}{11}$ ]

= $\frac{(-8)}{11}$ + $\frac{3}{11}$ [here we can observe that the given numbers have the same denominator]

= $\frac{((-8)+3)}{11}$ [here as the denominators are the same for both the numbers so we (-8)+3 are written together]

= $\frac{(-8+3)}{11}$

= $\frac{(-5)}{11}$

**Case 2 :** When denominators of given numbers are unequal.

In this case we take the L.C.M (least common multiple) of their denominators and express each of the given numbers with this L.C.M (least common multiple) as the common denominator. Now, we add these numbers like Case 1.

3. Find the sum of $\frac{(-5)}{6}$ + $\frac{4}{9}$ .

The denominator of the given rational numbers are 6 and 9 respectively.

L.C.M (least common multiple) of 6 and 9 = (2×3×3) = 18

Now, = $\frac{(-5)}{6}$ = $\frac{(-5\times3)}{(6\times3)}$ [here in the fraction $\frac{(-5)}{6}$ we need to divide the L.C.M (least common multiple) =18 with the denominator of this number is 6 so,18 ÷ 6 =3 so, we need multiply by 3, both the numerator as well as in the denominator of this number.]

= $\frac{(-15)}{18}$

and $\frac{4}{9}$ = $\frac{(4\times2)}{(9\times2)}$ [similarly here in the fraction $\frac{4}{9}$ we need to divide the L.C.M (least common multiple) =18 with the denominator of this number is 9 so,18 ÷ 9=2 so, we need multiply by 2 in both the numerator as well as in the denominator of this number.]

= $\frac{8}{18}$

Then, $\frac{(-15)}{18}$ + $\frac{8}{18}$ [now we can observe that both the numbers have same denominators]

= $\frac{((-15)+8)}{18}$

= $\frac{(-15+8)}{18}$

= $\frac{(-7)}{18}$.

**Observe the following**

(iv) We may write, $\frac{(-6)}{(-7)}$ = $\frac{((-6)\times(-1))}{((-7)\times(-1))}$ = $\frac{6}{7}$ .

**Examples for Subtracting Rational Numbers**

For any two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ , we define multiplication as

**Examples for Multiplying Rational Numbers**

2. $\frac{(-7)}{8}$ × $\frac{3}{5}$

3. $\frac{(-15)}{4}$ × $\frac{(-3)}{8}$

4. $\frac{(-3)}{7}$ × $\frac{14}{5}$

$\frac{((-3)\times2)}{(1\times5)}$ [here multiply side * side]

= $\frac{18}{6}$ = $\frac{3}{1}$ [here we observe that = so numerator and denominator both is divisible by 13]

6. $\frac{(-11)}{9}$ × $\frac{(-15)}{44}$

If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers such that $\frac{c}{d}$ ? 0, we define ( $\frac{a}{b}$ ÷ $\frac{c}{d}$ )

**Examples for Dividing Rational Numbers**

= $\frac{9}{10}$

= $\frac{(-2)}{5}$

$\frac{(-72)}{(-120)}$ are divisible by 24]

$\frac{5}{8}$ as $\frac{8}{5}$ as well as division sign change to multiplication sign]

The ordering is done as expected after comparing the numbers and putting the least or most negative first and the greatest or the most positive at the end.

**Let us try an example of ordering numbers:**

Compare $\frac{1}{2}$, $\frac{3}{4}$ and $\frac{2}{3}$

The representation of the following shows:

From the above illustration it is clear that $\frac{3}{4}$ is the greatest of the three and so would be placed at the end.