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# Rational Numbers and Irrational Numbers – Definition and Example

July 4, 2022
written by Azhar Ejaz

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

1. O.34=34/100 is a rational number.
2. 0.3333â€¦.=1/3 is recurring decimal, it is rational number
3. 0.142857142857â€¦..=1/7 is a recurring decimal is a rational number.
4. 1.4142135â€¦ is an irrational number.
5. 7.3205080â€¦..is an irrational number.
6. 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=p2/q2

2q2= p2â†’ (1)

P2 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

2q2=4m2

q2=2m2 â†’ (3)

This implies that q2 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€¦.}

## 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.

### 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€¦

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