Question

Where would the number $\sqrt{2}$ lie on the number line?

• A. between $0$ and $1$
• B. between $1$ and $2$
• C. between $2$ and $3$
• D. None of the above

Answer 1

The correct option is B

between $1$ and $2$

We know that when $a<b<c$, then $a^2<b^2<c^2$ .
Thus, the converse must be true. i.e., when $a^2<b^2<c^2$, then $a<b<c$.
Now, we know that $1<2<4$. Hence, $\sqrt{1}<\sqrt{2}<\sqrt{4}$ (When we consider the positive values of square roots).
Thus, $1<\sqrt{2}<2$. Hence, $\sqrt{2}$ lies between $1$ and $2$.

We have also seen that $\sqrt{2}$ lies between $1$ and $2$ by Pythagoras theorem. When we take the line $OA=1$ unit, where $O$ represents the point $0$ on the number line, and draw a perpendicular to $OA$ at $l$, and call it $AP$, such that $AP=1$ unit, then $OP=\sqrt{2}$. If we draw an arc with centre $O$ and radius $OP$, the point at which the arc cuts the number line will represent the value $\sqrt{2}$.