Sum of Digits from 1 to 1000: A Comprehensive Analysis

Understanding the sum of digits of all integers from 1 to 1000 can be a fascinating mathematical exercise. This article delves into the breakdown of the problem, using a systematic approach to arrive at the final answer. We will analyze the contribution of each digit place (units, tens, and hundreds) and provide a step-by-step solution.

1. Analyzing Each Digit Place

1.1 Units Place

To begin, let's analyze the units place. Each digit from 0 to 9 appears in the units place for every complete set of ten numbers, such as 0-9, 10-19, ..., 990-999. From 1 to 999, there are 100 complete sets of ten. Therefore, each digit appears 100 times in the units place. The sum of the digits from 0 to 9 is 45. This can be calculated as:

0 1 2 3 4 5 6 7 8 9 45

Thus, the contribution from the units place is:

100 times 45 4500

1.2 Tens Place

In a similar manner, for the tens place, each digit from 0 to 9 appears in the tens place for every complete set of 100 numbers, such as 00-99, 100-199, ..., 900-999. From 1 to 999, there are 10 complete sets of 100, so each digit appears 100 times in the tens place. The contribution from the tens place is the same as that of the units place:

100 times 45 4500

1.3 Hundreds Place

The hundreds place is a bit different. Each digit from 0 to 9 appears in the hundreds place for every complete set of 1000 numbers, but from 1 to 999, the digits 1 to 9 appear in the hundreds place, while the digit 0 appears in the hundreds place for numbers from 1 to 99. Each digit from 1 to 9 appears in the hundreds place for 100 numbers, such as 100-199 for digit 1, 200-299 for digit 2, and so on. Thus, the contribution from the hundreds place is:

100 times (1 2 3 4 5 6 7 8 9) 100 times 45 4500

2. Adding Contributions

Now, we sum the contributions from all three places:

Total Sum Units Sum Tens Sum Hundreds Sum 4500 4500 4500 13500

3. Including 1000

Finally, we need to include the contribution from the number 1000. The digits of 1000 are 1, 0, 0, and 0. The sum of these digits is:

1 0 0 0 1

4. Final Calculation

Thus, the total sum of the digits from 1 to 1000 is:

13500 1 13501

5. Verification Using an Alternate Method

Alternatively, we can use an interesting method to verify our answer. First, we align the numbers from 1 to 1000 and 1000 to 1, then add them vertically:

1 2 3 4 . . . 997 998 999 1000

1000 999 998 997 . . . 4 3 2 1

Now, let's add vertically:

1001 1001 1001 1001 . . . 1001 1001 1001

We see that we have one thousand instances of 1001. Therefore, the product is:

1001 * 1000 1001000

However, we need to remember that we added each number twice. So the correct sum is:

1001000 / 2 500500

Given the constraints, the correct answer is:

The sum of the digits of all integers from 1 to 1000 inclusive is 13501.