What are Rational Numbers ?


In this chapter we will learn about the concept of rational numbers with properties and examples.

Rational Number Definition


The numbers which can be arranged in the form of \mathtt{\frac{p}{q}} are called rational numbers.

Where; \mathtt{q\ \cancel{=} \ 0}


Remember that word “Rational number” are derived from the word ratio. So if you see any two different integers in the form of ratio, you can note the number as rational number.


Examples of rational numbers;

\mathtt{\frac{2}{3} ,\ \frac{-7}{6} ,\ \frac{9}{4}} are all examples of rational numbers.

Are simple integer numbers also part of rational numbers ?

Yes!!

Consider the simple integer 2.

We can write this integer in the form of p / q as \mathtt{\frac{2}{1}} .

Hence, the number 2 can be termed as rational number.

Similarly consider the integer -5.

The integer can be expressed in form of p / q as \mathtt{\frac{-5}{1}} .

Hence, the simple integer -5 can be termed as rational number.

Are decimal numbers part of rational numbers ?


Well, it depends.

There are three possibilities in this case;

(a) Terminating Decimals that can be represented as ratio.

(b) Non Terminating repeating decimals that can be represented as ratio.

(c) Decimals that cannot be represented as ratio.


Here point (a) & (b) belongs to rational numbers.

Let us understand each of the points with examples.


(a) Terminating decimals represented as ratio

Most of the decimals that we deal in math are terminating decimals.

These numbers can be easily represented in the form of ratio p / q and hence are part of rational numbers.


For example;

(i) 0.025

The above decimal can be converted into ratio as;

\mathtt{0.025\ \Longrightarrow \ \frac{25}{1000}}

Hence, the number is rational number.



(ii) 7.8921

In the ratio form, the number can be written as;

\mathtt{7.8921\ \Longrightarrow \ \frac{78921}{10000}}

Hence, the decimal is a rational number.

(b) Non terminating repeating decimals

These are decimals whose numbers repeat again and again.

These numbers can be represented in the form of ratio p / q, hence are non-terminating decimals.

Examples of non terminating repeating decimals;

(i) 0.11111 . . .

The number can be represented in the form of p / q as;

0.1111 . . ⟹ \mathtt{\ \frac{1}{9}}


Hence, the above decimal is a rational number.



(ii) 0.333333 . . .

The number can be represented in the form of ratio as;

0.3333 . . ⟹ \mathtt{\ \frac{1}{3}}

Hence, the above decimal is a rational number.

Note;
Whenever you see a repeating non terminating decimal, make a note that it is a rational number.

(c) Decimals that can’t be expressed as ratios


There are various decimal values which cannot be converted in form of ratio p / q and are not part of rational numbers.

For example;

(i) \mathtt{\sqrt{2}} = 1.414. . . .

The square root of 2 or any other number is a decimal which cannot be converted into ratio.

Hence, it is not a rational number.



(ii) Value of 𝜋

we know that 𝜋 = 3.14 . . .

This decimal value cannot be written in form of p / q. Hence, it is a irrational number.

Note:
Some say that fraction form of 𝜋 = 22 / 7.
I want to correct that this is just an approximate value and cannot replace the actual decimal number.

How Fraction number is different from rational number ?

Since both fraction and rational numbers are represented in the form of p / q, there develops a confusion between the two.

Remember that rational number consist of number which are fractions or can be represented as fraction.

For example;

\mathtt{\frac{5}{3}} ⟹ fraction value

2.5 ⟹ decimal value but can be represented as fraction

Both the above two values are part of rational numbers.

Is 0 a rational number ?

Yes !!

The number 0 can be written in the form of ratio such as \mathtt{\frac{0}{1} ,\ \frac{0}{5} ,\ \frac{0}{17}} etc.

Hence, we count 0 as a rational number.

Does fraction with denominator 0 is considered as rational number ?


NO !!

The value of fraction with 0 denominator is not defined.

\mathtt{\frac{5}{0} \ =\ \infty }

Hence, it is not a rational number.

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