Problem 1235

To find the thousandth digit after the decimal point of $\sqrt{N}$, where $N$ is a number with 1998 digits, all of them being 1, we can proceed as follows:

1. **Express $N$ in a more manageable form:**
$$N = 111\ldots111 \quad \text{(1998 digits)}$$
This can be written as:
$$N = \frac{10^{1998} - 1}{9}$$

2. **Approximate $\sqrt{N}$:**
We need to find $\sqrt{N}$. Using the expression for $N$:
$$\sqrt{N} = \sqrt{\frac{10^{1998} - 1}{9}}$$
For large $n$, $\sqrt{10^{2n} - 1} \approx 10^n$. Therefore:
$$\sqrt{N} \approx \sqrt{\frac{10^{1998}}{9}} = \frac{10^{999}}{\sqrt{9}} = \frac{10^{999}}{3}$$

3. **Decimal expansion of $\frac{10^{999}}{3}$:**
To find the thousandth digit after the decimal point of $\frac{10^{999}}{3}$, we need to understand the decimal expansion of $\frac{1}{3}$:
$$\frac{1}{3} = 0.\overline{3}$$
Therefore:
$$\frac{10^{999}}{3} = 10^{999-1} \times \frac{10}{3} = 10^{998} \times 3.\overline{3} = 3.\overline{3} \times 10^{998}$$

4. Determine the thousandth digit:
The decimal expansion of $3.\overline{3} \times 10^{998}$ will be:
$$3.\underbrace{333\ldots333}_{998 \text{ threes}}333\ldots$$
The thousandth digit after the decimal point is the 1000th digit in the sequence of threes.

Therefore, the thousandth digit after the decimal point of $\sqrt{N}$ is $3$.

The final answer is $\boxed{3}$