Classify the following numbers as rational or irrational.

i.

ii.

iii.

iv.

v. 2π

vi.

vii.

i. 5 - √3

Here, 5 is a rational number and √3 is an irrational number.

We know that subtraction of a rational number and an irrational number always gives an irrational number.

∴ 5 - √3 is an irrational number.

ii. √3 + √2

Here, both √3 and √2 are irrational numbers.

We know that sum of two irrational number is always an irrational number.

∴ √3 + √2 is an irrational number.

iii. (√2 – 2)²

We know that (√a - b)² = a - 2√a(b) + b².

Comparing with the given expression,

⇒ a = 2; b = 2

On simplification, we get

⇒ (√2 – 2)² = 2 – 2 (√2) (2) + 2²

= 2 – 4√2 + 4

= 6 – 4√2

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

We know that subtraction of a rational number and an irrational number always gives an irrational number.

∴ 6 - 4√2 is an irrational number.

iv.

On simplification, we get

⇒

which is a rational number.

v. 2π

Here, 2 is a rational number and π is an irrational number.

We know that the product of a rational number and an irrational number is always an irrational number.

∴ 2π is an irrational number.

vi.

Here, 1 is a rational number and √3 is an irrational number.

We know that division of a rational number and an irrational number gives an irrational number.

∴ is an irrational number.

vii. (2 + √2) (2 - √2)

We know that (a + √b) (a - √b) = a² – b.

Comparing with the given expression,

⇒ a = 2; b = 2

∴ (2 + √2) (2 - √2) = 2² – (√2)²

= 4 – 2 = 2

Here, 2 is a rational number.

∴ (2 + √2) (2 - √2) is rational.