Question

Which among the following irrational numbers represents a point lying between 3 and 4 on the number line?

• A. $$\sqrt{7}$$
• B. $$\sqrt{8}$$
• C. $$\sqrt{10}$$
• D. $$\sqrt{5}$$

Answer 1

Answer:

$$\sqrt{10}$$

Understanding Irrational Numbers and Number Lines

An irrational number is a number that cannot be expressed as a simple fraction (a/b), where 'a' and 'b' are integers and 'b' is not zero. Examples include π and e. On a number line, these numbers fall between integers but do not have a terminating or repeating decimal representation.

Locating $\sqrt{10}$ on the Number Line

• To determine if $\sqrt{10}$ lies between 3 and 4, we can compare its square to the squares of 3 and 4.

• The square of 3 is $3^2 = 9$.

• The square of 4 is $4^2 = 16$.

• Since 10 is greater than 9 and less than 16 ($9 < 10 < 16$), it follows that the square root of 10 must be greater than the square root of 9 and less than the square root of 16.

• Therefore, $\sqrt{9} < \sqrt{10} < \sqrt{16}$.

• This simplifies to $3 < \sqrt{10} < 4$.