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 ∈ Z and q≠0.the number √16, 3.7,4 are rational numbers. √16 can be reduced to the form p/q where p, q ∈Z and q≠0 because √16=4=4/1.rational numbers denoted by Q.

Q= {x|x=p/q, where p, q ∈ Z and q≠0.}

Irrational numbers

Those numbers which   cannot be put into the form p/q where p, q ∈ Z and q≠0.the numbers

√2, √3    7/ (√5) are irrational numbers. Irrational number denoted by Q’.

Q’ = {x| x ≠p/q where p, q ∈ Z and q≠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 1.4235……, 0.013246….   , 2.33333333…, are examples of non-terminating decimal.

Non- recurring decimals

In general a non –recurring decimals is a decimal in which do not repeat digits in decimal part. Thus 1.5673….. 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. 0.3333….=1/3 is recurring decimal, it is rational number
3. 0.142857142857…..=1/7 is a recurring decimal is a rational number.
4. 1.4142135… is an irrational number.
5. 7.3205080…..is an irrational number.
6. 3.141592……. 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.,

π =circumference of any circle /length of its diameter.

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

Prove that √2 is irrational number

Proof: suppose that √2 is rational number.

Than,

 √2=p/q → (a)

 Where p, q ∈ Z and q≠0. Also let p/q is lowest form.

Squaring both sides equation (a) we get,

2=p²/q²

2q²= p2→ (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 → (2)

Using (2) in (1)

2q2= (2m) ²

2q²=4m²

q²=2m² → (3)

This implies that q² is an even number.

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

 q=2n → (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,√2  is not rational number .it is irrational number.

Using the same method we can prove the irrationality of √3, √5, √7,…….√n where n is any prime number,

Prime number is denoted by P.

P= {2, 3, 5, 7, 11,…….}

Theorem:

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

Proof: suppose that

               rational + irrational =rational 

                 P/q +x=m/n →(1)

                                 ∴    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 ∈z          

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.

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