Question

Comment on the given statements:
Statement I: $\sqrt{12}$ cannot be precisely represented on a number line.
Statement I: $\sqrt{12}$ is a non-terminating and non-repeating decimal number.

• A. Both statement I and statement II are correct, and statement II is a correct reason of statement I
• B. Statement II is correct, but statement I is wrong.
• C. Both the statements are wrong.

Answer 1

The correct option is B Statement II is correct, but statement I is wrong.

$\sqrt{12}$ is a quadratic surd hence it is irrational number or non-terminating non-repeating decimal number, but it can be represented on a number line with the help of geometry.

From pythagorean theorem, ${\text{hypotenuse}}^2={\text{base}}^2+{\text{perpendicular}}^2$
$\Rightarrow (\sqrt{12}{)}^2=12=6+6=(\sqrt{6}{)}^2+(\sqrt{6}{)}^2$

To represent $\sqrt{12}$ on a number line, we perform the following steps:

1. Draw a line of length $\sqrt{6}$ on a number line with 0 on its left end.
2. Draw a perpendicular line of length $\sqrt{6}$ on the number line from the right end of the first line.
3. Connect the other endpoints of both the lines to draw the hypotenuse.
4. Measure the length of the hypotenuse using a compass and mark it on the number line to represent $\sqrt{12}.$