Step-by-Step Calculation for the Number of Times Digit '5' Appears from 1 to 1000:

From 1 to 100: The numbers containing the digit '5' from 1 to 100 are: 5, 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 65, 75, 85, 95. Total count: 20 numbers. Note: The number 55 contains two 5's, so it contributes an additional occurrence of the digit '5'.

Pattern for every set of 100 numbers (100 to 200, 200 to 300, etc.): The same pattern repeats for every set of 100 numbers up to 1000: For each block of 100 numbers (e.g., 100-199, 200-299, 300-399, ..., 900-999), there are 20 numbers containing the digit '5'. Therefore, the total number of times the digit '5' appears in each block of 100 numbers is 20.

Special Case for 500 to 599: The block from 500 to 599 is unique because every number in this block contains at least one '5' in the hundreds place. Total count for this block: 100 numbers (500 to 599).

Total Count: From 1 to 100: 20 occurrences. For the next 9 blocks (100-199, 200-299, ..., 900-999): Each contributes 20 occurrences. Total for 1 to 900 = $20 \times 9 = 180$ For the block 500 to 599: 100 occurrences.

Total number of occurrences of the digit '5' from 1 to 1000: Total = 20 (from 1 to 100) + 180 (from 100 to 900) + 100 (from 500 to 599) Total = $20 + 180 + 100 = 300$

Conclusion: The digit '5' appears 300 times when writing the integers from 1 to 1000.