When a n 'n' digit number is squared then the number of digit in the square thus obtained is ______.

Answer

To determine the number of digits in the square of an n-digit number, we can follow these steps:

### Step 1: Understand the range of an n-digit number
An n-digit number can be represented as:
- The smallest n-digit number is \(10^{(n-1)}\).
- The largest n-digit number is \(10^n - 1\).

### Step 2: Square the smallest and largest n-digit numbers
Now, we will square both the smallest and largest n-digit numbers:
- Squaring the smallest n-digit number:
\[ (10^{(n-1)})^2 = 10^{(2(n-1))} = 10^{(2n-2)} \]
- Squaring the largest n-digit number:
\[ (10^n - 1)^2 \approx (10^n)^2 = 10^{(2n)} \]

### Step 3: Determine the number of digits in the squares
The number of digits in a number \(x\) can be found using the formula:
\[ \text{Number of digits} = \lfloor \log_{10} x \rfloor + 1 \]
- For \(10^{(2n-2)}\):
\[ \text{Number of digits} = \lfloor \log_{10}(10^{(2n-2)}) \rfloor + 1 = \lfloor 2n - 2 \rfloor + 1 = 2n - 1 \]
- For \(10^{(2n)}\):
\[ \text{Number of digits} = \lfloor \log_{10}(10^{(2n)}) \rfloor + 1 = \lfloor 2n \rfloor + 1 = 2n + 1 \]

### Step 4: Conclusion
From the above calculations, we can conclude that when an n-digit number is squared, the number of digits in the square can either be \(2n - 1\) or \(2n\), depending on whether the squared number is less than \(10^{(2n)}\) or not. Thus, the number of digits in the square of an n-digit number is:
\[ \text{Number of digits} = 2n \text{ or } 2n + 1 \]

### Final Answer
When an n-digit number is squared, the number of digits in the square thus obtained is \(2n\) or \(2n + 1\).

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