Square Root Digital Expansion

It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all.
The square root of two is $1.41421356237309504880\cdots$, and the digital sum of the first one hundred decimal digits is $475$.
For the first one hundred natural numbers, find the total of the digital sums of the first one hundred decimal digits for all the irrational square roots.

To find the total of the digital sums of the first one hundred decimal digits for all the irrational square roots of the first one hundred natural numbers, you can follow these steps:

1. Import the math module in Python to access mathematical functions and constants.
“`python
import math
“`

2. Initialize a variable `total_sum` to keep track of the total sum of the digital sums.
“`python
total_sum = 0
“`

3. Iterate over the range of natural numbers from 1 to 100 using a for loop.
“`python
for number in range(1, 101):
“`

4. Use the `math.sqrt()` function to calculate the square root of `number`.
“`python
square_root = math.sqrt(number)
“`

5. Check if the square root is an integer using the `is_integer()` method of a float. If it is an integer, continue to the next iteration of the loop.
“`python
if square_root.is_integer():
continue
“`

6. Convert the square root to a string and remove the decimal point.
“`python
decimal_digits = str(square_root).replace(“.”, “”)
“`

7. Extract the first one hundred decimal digits by slicing the string.
“`python
decimal_digits = decimal_digits[:100]
“`

8. Calculate the sum of the individual digits by converting each character back to an integer using the `int()` function and summing them using the `sum()` function.
“`python
digit_sum = sum(int(digit) for digit in decimal_digits)
“`

9. Add the digit sum to the `total_sum` variable.
“`python
total_sum += digit_sum
“`

10. After the loop completes, print the final `total_sum`.
“`python
print(total_sum)
“`

Here is the complete code:

“`python
import math

total_sum = 0

for number in range(1, 101):
square_root = math.sqrt(number)
if square_root.is_integer():
continue

decimal_digits = str(square_root).replace(“.”, “”)
decimal_digits = decimal_digits[:100]

digit_sum = sum(int(digit) for digit in decimal_digits)
total_sum += digit_sum

print(total_sum)
“`

The output of this code will be the total of the digital sums of the first one hundred decimal digits for all the irrational square roots of the first one hundred natural numbers.

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