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

Classify the following numbers as rational or irrational:

(i) $2-\sqrt{5}$

(ii) $(3+\sqrt{23})-\sqrt{23}$

(iii) $\frac{2 \sqrt{7}}{7 \sqrt{7}}$

(iv) $\frac{1}{\sqrt{2}}$

(v) $2 \pi$

To classify the given numbers as rational or irrational, we need to understand that a rational number can be expressed as the quotient of two integers, while an irrational number cannot be expressed in such a way.

(i) $2-\sqrt{5}$: This is irrational because $\sqrt{5}$ cannot be expressed as a ratio of two integers, and subtracting it from an integer doesn't change its irrational nature.

(ii) $(3+\sqrt{23})-\sqrt{23}$: This simplifies to $3$, which is rational because it can be expressed as $3/1$.

(iii) $\frac{2 \sqrt{7}}{7 \sqrt{7}}$: This simplifies to $\frac{2}{7}$, which is rational since it can be expressed as a ratio of the integers $2$ and $7$.

(iv) $\frac{1}{\sqrt{2}}$: This is irrational. Multiplying the numerator and denominator by $\sqrt{2}$ gives $\frac{\sqrt{2}}{2}$, which cannot be expressed as a ratio of two integers since $\sqrt{2}$ is irrational.

(v) $2 \pi$: This is irrational because $\pi$ is a well-known irrational number, and multiplying it by an integer ($2$ in this case) does not change its irrational nature.

So, summarizing:

• (i) is irrational.

• (ii) is rational.

• (iii) is rational.

• (iv) is irrational.

• (v) is irrational.