### Theory:

In the division process, we used bars to find the square root. The number of digits in the square root of a perfect square number identified using these labelling bars. Take a look at the following cases (with bars shown as if we compute square root by division procedure).

 Square root of numbers No. of bars No. of digits in the square root $$\sqrt{\overline{49}} = 7$$ $$1$$ $$1$$ $$\sqrt{\overline{1} \ \ \overline{69}} = 13$$ $$2$$ $$2$$ $$\sqrt{\overline{6} \ \ \overline{55} \ \ \overline{36}} = 256$$ $$3$$ $$3$$ $$\sqrt{\overline{10} \ \ \overline{53} \ \ \overline{00} \ \ \overline{25}} = 3245$$ $$4$$ $$4$$

From the above table, we can conclude that number of bars is equal to the number of digits in the square root.

To find the number of digits in the square root without calculation:
Case I: If $$n$$ is the number of digits of a number and $$n$$ is odd:

$$\text{Number of digits in the square root of the number}$$ $$=$$ $\frac{n+1}{2}$

Case II: If $$n$$ is the number of digits of a number and $$n$$ is even:

$$\text{Number of digits in the square root of the number}$$ $$=$$ $\frac{n}{2}$
Example:
1. Find the number of digits in the square root of $$196$$.

Solution:

Number of digits, $$(n) = 3$$

Since $$n$$ is odd:

$$\text{Number of digits in the square root of the number}$$ $$=$$ $\frac{n+1}{2}$

$$=$$ $\frac{3+1}{2}$

$$=$$ $\frac{4}{2}$

$$=$$ $$2$$

Therefore, the number of digits in the square root of $$196$$ is $$2$$.

2. Find the number of digits in the square root of $$123201$$.

Solution:

Number of digits, $$(n) = 6$$

Since $$n$$ is even:

$$\text{Number of digits in the square root of the number}$$ $$=$$ $\frac{n}{2}$

$$=$$ $\frac{6}{2}$

$$=$$ $$3$$

Therefore, the number of digits in the square root of $$123201$$ is $$3$$.