Question

Examine, whether the following numbers are rational or irrational:

(i) √7

(ii) √4

(iii) 2 + √3

(iv) √3 + √2

(v) √3 + √5

(vi) (√2-2)²

(vii) (2-√2)(2+√2)

(viii) (√3 + √2)²

(ix) √5 - 2

(x) √23

(xi) √225

(xii) 0.3796

(xiii) 7.478478.....

(xiv) 1.101001000100001.....

Answer 1

(i) √7 

Not a perfect square root, so it is an irrational number.

(ii) √4

A perfect square root of 2.

We can express 2 in the form of 2/1, so it is a rational number. 

(iii) 2 + √3

Here, 2 is a rational number but √3 is an irrational number 

Therefore, the sum of a rational and irrational number is an irrational number. 

(iv) √3 + √2 

√3 is not a perfect square thus an irrational number. 

√2 is not a perfect square, thus an irrational number. 

Therefore, sum of √2 and √3 gives an irrational number. 

(v) √3 + √5

√3 is not a perfect square and hence, it is an irrational number

Similarly, √5 is not a perfect square and also an irrational number.

Since, sum of two irrational number, is an irrational number, therefore √3 + √5 is an irrational number.

(vi) (√2 – 2)² 

(√2 – 2)² 

= 2 + 4 – 4√2 

= 6 + 4√2 

Here, 6 is a rational number but 4√2 is an irrational number. 

Since, the sum of a rational and an irrational number is an irrational number, therefore, (√2 – 2)² is an irrational number.

(vii) (2 – √2)(2 + √2)

We can write the given expression as; 

(2 – √2)(2 + √2) = ((2)² − (√2)²) 

[Since, (a + b)(a – b) = a² – b²] 

= 4 – 2

= 2 or 2/1

Since, 2 is a rational number, therefore, (2 – √2)(2 + √2) is a rational number.

(viii) (√3 + √2)² 

We can write the given expression as; 

(√3 + √2)² 

= (√3)² + (√2)² + 2√3 x √2 

= 3 + 2 + 2√6 

= 5 + 2√6 

[using identity, (a+b)² = a² + 2ab + b²] 

Since, the sum of a rational number and an irrational number is an irrational number, therefore, (√3 + √2)² is an irrational number. 

(ix) √5 – 2 

√5 is an irrational number whereas 2 is a rational number. 

The difference of an irrational number and a rational number is an irrational number. 

Therefore, √5 – 2 is an irrational number. 

(x) √23 

Since, √23 = 4.795831352331… 

As decimal expansion of this number is non-terminating and non-recurring therefore, it is an irrational number. 

(xi) √225 

√225 = 15 or 15/1 

√225 is rational number as it can be represented in the form of p/q and q not equal to zero.

(xii) 0.3796

As the decimal expansion of the given number is terminating, therefore, it is a rational number. 

(xiii) 7.478478…… 

As the decimal expansion of this number is non-terminating recurring decimal, therefore, it is a rational number. 

(xiv) 1.101001000100001……

As the decimal expansion of given number is non-terminating and non-recurring, therefore, it is an irrational number.