# Rational Numbers and Irrational Numbers – Definition and Example

Rational numbers include **integers**, **fractions**, or** decimals **that end or repeat. An irrational number is any number that cannot be converted to a fraction**.**

**Rational numbers**

Those numbers which can be put into the form P/q where p, q âˆˆ Z and qâ‰ 0.the number âˆš16, 3.7,4 are rational numbers. âˆš16 can be reduced to the form p/q where p, q âˆˆZ and qâ‰ 0 because âˆš16=4=4/1.rational numbers denoted by Q.

Q= {x|x=p/q, where p, q âˆˆ Z and qâ‰ 0.}

**Irrational numbers**

Those numbers which cannot be put into the form p/q where p, q âˆˆ Z and qâ‰ 0.the numbers

âˆš2, âˆš3 7/ (âˆš5) are irrational numbers. Irrational number denoted by Qâ€™.

Qâ€™ = {x| x â‰ p/q where p, q âˆˆ Z and qâ‰ 0.}

## Decimal representation of rational and irrational Numbers:

The are four **types of Decimal** representation of rational and irrational Numbers:

**Terminating decimals**

A decimal which has only a finite number of digits in its decimal parts, is called a terminating decimal .thus 203.05, 0.00415, 1.456 are examples of terminating decimals.

Since a terminating decimal can be converted into a common fraction, so every terminating decimal represents a rational number.

**Recurring decimals**

This is another type of rational numbers.in general, a recurring or periodic decimal is a decimal in which one or more digits repeat indefinitely.

It well be show that a recurring decimal can be converted into a common fraction.so every recurring decimal represents a rational number.

**Non -terminating decimals**

A decimal which has an infinite number of digits in its decimal parts is called a non-terminating decimals. Thus 1.4235â€¦â€¦, 0.013246â€¦.Â Â , 2.33333333â€¦, are examples of non-terminating decimal.

**Non- recurring decimals**

In general a non â€“recurring decimals is a decimal in which do not repeat digits in decimal part. Thus 1.5673â€¦.. are example of terminating decimal.

A non-terminating, non-recurring decimal is a decimal which neither terminates nor is it recurring.

It is not possible to convert such a decimal into a common fraction.

**Thus a non-terminating and non-recurring decimal represents an irrational number.**

**Example**

- O.34=34/100 is a rational number.
- 0.3333â€¦.=1/3 is recurring decimal, it is rational number
- 0.142857142857â€¦..=1/7 is a recurring decimal is a rational number.
- 1.4142135â€¦ is an irrational number.
- 7.3205080â€¦..is an irrational number.
- 3.141592â€¦â€¦. is an important irrational number called (pi) which denotes the constant ratio of the circumference of any circle to the length of diameter i.e.,

Ï€ =circumference of any circle /length of its diameter.

An approximate value of Ï€ (pi) is 22/7, a better approximation is 355/113 and a still better approximation is 3.14159â€¦ the value of Ï€ correct to 5 lac decimal places has been determined with help of computer .

## Prove that âˆš2 is irrational number

Proof: suppose that âˆš2 is rational number.

Than,

âˆš2=p/q â†’ (a)

Where p, q âˆˆ Z and qâ‰ 0. Also let p/q is lowest form.

Squaring both sides equation (a) we get,

2=p^{2}/q^{2}

2q^{2}= p2â†’ (1)

P^{2} is even number.

P x p is even number.

(Even x even=even)

(Odd x odd=odd)

So, p is an **even number**. Than it can be written as

P=2m â†’ (2)

Using (2) in (1)

2q2= (2m)^{ 2}

2q^{2}=4m^{2}

q^{2}=2m^{2} â†’ (3)

This implies that q^{2} is an even number.

So, q also an even number. We can write it as

q=2n â†’ (4)

From (2) and (4)

P/q=2m/2n

Which is not in lowest form. This is contradiction to our supposition that p/q in lowest form. Therefore,âˆš2 is not rational number .it is irrational number.

Using the same method we can prove the irrationality of âˆš3, âˆš5, âˆš7,â€¦â€¦.âˆšn where n is any **prime number,**

Prime number is denoted by P.

P= {2, 3, 5, 7, 11,â€¦â€¦.}

## Theorem:

## Prove that sum of rational and irrational number is irrational number.

Proof: suppose that

rational + irrational =rational

P/q +x=m/n â†’(1)

âˆ´ by definition rational and irrational number.

From (1)

X=m/n-p/q

X=mq – np/nq

X= (integer-integer)/integer

Where p, q, m, n âˆˆz

X=integer/integer

X must be rational number.

Which is contradiction to our supposition is wrong.

**So, sum of rational and irrational number must be is irrational number.**

## Frequently Asked Question-FAQs

### What are rational numbers? Give Examples.

A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to 0.Some examples of rational numbers include 1/3, 2/4, 1/5, 9/3, and so on.

### Is 0 a rational number?

Yes, **0 is a rational number** because it is an integer that can be written in any form such as 0/1, 0/2, where b is a non-zero integer. It can be written in the form: p/q = 0/1.

Therefore, we can conclude that 0 is a rational number.

### What is the denominator of the rational number?

The denominator of a rational number can be any real number except 0.

### what is an irrational Number?

An irrational number is any real number that cannot be expressed by a rational number. It is a number that cannot be represented as a finite or terminating decimal.

### Is Pi(Ï€) a rational number?

No, **Pi (Ï€) is not a rational number** because its value equals 3.142857â€¦

## Leave a Reply