1. Understand the Problem: We are given that when a certain number is squared, it results in a 24-digit number. We need to find out how many digits are in the original number.
2. Relationship Between the Number of Digits in a Number and Its Square: - If a number [tex]\( n \)[/tex] has [tex]\( d \)[/tex] digits, then when [tex]\( n \)[/tex] is squared, the resulting number will have approximately [tex]\( 2d \)[/tex] digits (it could be slightly less, but this estimation is usually sufficient for this type of problem).
3. Formula Application: - Let [tex]\( d \)[/tex] be the number of digits in the original number. - The number [tex]\( n^2 \)[/tex], which is the square of [tex]\( n \)[/tex], has 24 digits.
4. Set up the Equation: - Since squaring a number roughly doubles the number of digits, we can write: [tex]\[
2d \approx 24
\][/tex]
5. Solve for [tex]\( d \)[/tex]: - Divide both sides of the equation by 2 to isolate [tex]\( d \)[/tex]: [tex]\[
d \approx \frac{24}{2}
\][/tex] - Simplifying this, we get: [tex]\[
d = 12
\][/tex]
6. Conclusion: - Thus, the certain number has 12 digits.
So, when a number is squared to obtain a 24-digit number, the original number has 12 digits.
