Question

Show how $\surd 5$can be represented on the number line.

Answer 1

To represent $\surd 5$ on number line let us follow the given procedure:

Step 1: Let us assume line $AB$ be of $2$unit on a number line.

Step 2: At $B$, draw a perpendicular line $BC$ of length $1$ unit.

Step 3: Join $CA$

Step 4: Now, we get $ABC$ as a right-angled triangle.

On applying Pythagoras theorem,

$$\begin{gathered} A{B}^2+B{C}^2=C{A}^2=2^2+1^2 \\ \Rightarrow C{A}^2=5 \\ \Rightarrow CA=\surd 5. \end{gathered}$$

Thus, $CA$ is a line of length $\surd 5$ unit.

Step 5: Considering $CA$ as a radius and $A$ as a centre draw an arc touching the number line.

The point at which number line get intersected by arc is at $\surd 5$ distance from $0$ because it is a radius of the circle whose centre was $A$.

Hence, $\surd 5$ is represented on the number line as shown in the figure.