Classify the following numbers as rational or irrational: (i) 2 -√5 (ii) (3 + √23) - √23 (iii) 2√7/7√7 (iv) 1/√2 (v) 2

(i) 2 –√5 

As √5 = 2.2360678… 

This is non-terminating and non-recurring. It is an irrational number.

On substituting the value of √5 in equation 2 –√5, we get,

2-√5 = 2-2.2360678… 

2-√5 = -0.2360678

Since the number, – 0.2360678…, is a non-terminating and non-recurring,

Therefore, 2 –√5 is an irrational number.

(ii) (3 +√23)- √23

(3 +√23) –√23 = 3+√23–√23

= 3

Since, the number 3 is rational number

Therefore, (3 +√23)- √23 is rational.

(iii) 2√7 / 7√7

2√7 / 7√7 = (2/7)× (√7/√7)

2√7 / 7√7 = (2/7)× (√7/√7) 

= (2/7)×1 [As (√7/√7) = 1]

= 2/7 

Since the number, 2/7 is in p/q form

Therefore, 2√7/7√7 is rational.

(iv) 1/√2

As, √2 = 1.41421… 

This is non-terminating and non-recurring. It is a rational number.

On dividing 1/√2 we get,

1/√2 = 1/1.41421…

=0.70710…

Since the number, 0.7071..is a non-terminating and non-recurring,

Therefore, 1/√2 is an irrational number.

(v) 2π

The value of π is 3.1415…

On substituting the value of π in equation 2π, we get,

2π = 2 × 3.1415… = 6.2831…

Since the number, 6.2831…, is non-terminating non-recurring, 

Therefore, 2π is an irrational number.

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