Finding The 2022nd Digit In A Decimal Expansion A Step By Step Guide
Introduction
In this article, we will delve into a fascinating problem involving the decimal representation of a number formed by concatenating integers. Specifically, we are given the number x = 0.123456789101112...998999, where the digits are obtained by writing the integers from 1 to 999 in order. Our task is to determine the 2022nd digit to the right of the decimal point. This problem requires a careful analysis of the number of digits contributed by different ranges of integers and a methodical approach to pinpoint the digit at the desired position. The problem falls under the category of number theory and digit manipulation, often encountered in mathematical competitions and recreational mathematics. Understanding the structure of decimal expansions and the distribution of digits within concatenated sequences is crucial for solving this problem efficiently.
Problem Statement
Let's restate the problem clearly. We are given a decimal number x formed by concatenating the integers from 1 to 999 after the decimal point:
x = 0.123456789101112...998999
The question asks us to find the 2022nd digit in this decimal expansion. This means we need to identify which integer contributes the 2022nd digit and which position within that integer the digit occupies. To solve this, we need to analyze how many digits are contributed by single-digit, double-digit, and triple-digit numbers in the sequence. This involves counting the number of integers in each category and multiplying by the number of digits each integer contributes. The cumulative digit count will help us locate the integer containing the 2022nd digit. Once we identify the relevant integer, we can determine the specific digit at the 2022nd position.
Methodical Approach to Solving the Problem
To solve this problem efficiently, we will break it down into several steps. The first step is to determine the number of digits contributed by each group of integers: single-digit, double-digit, and triple-digit numbers. Then, we will calculate the cumulative number of digits as we move through these groups. This cumulative count will help us pinpoint which group of integers contains the 2022nd digit. Next, we'll identify the specific integer within that group that contributes the 2022nd digit. Finally, we'll determine the exact digit within that integer that corresponds to the 2022nd position. This step-by-step approach will allow us to systematically work through the problem and arrive at the correct answer. Let's begin by counting the digits contributed by each group of integers.
Counting Digits from Single-Digit Numbers
First, we consider the single-digit numbers (1 to 9). There are 9 such numbers, and each contributes 1 digit to the decimal expansion. Therefore, the total number of digits contributed by single-digit numbers is:
9 integers × 1 digit/integer = 9 digits
These are the first 9 digits after the decimal point in our number x. This initial count is crucial as it sets the baseline for our subsequent calculations. Understanding how many digits these numbers contribute is essential for determining the range within which the 2022nd digit falls. This simple calculation lays the groundwork for analyzing the contribution of double-digit and triple-digit numbers, which will be significantly larger and require more detailed analysis. The key here is to be methodical and account for every group of integers systematically to avoid errors in the final result.
Counting Digits from Double-Digit Numbers
Next, we consider the double-digit numbers (10 to 99). To find the number of double-digit numbers, we subtract the smallest (10) from the largest (99) and add 1:
99 - 10 + 1 = 90 integers
Each double-digit number contributes 2 digits to the decimal expansion. Therefore, the total number of digits contributed by double-digit numbers is:
90 integers × 2 digits/integer = 180 digits
These 180 digits follow the 9 digits contributed by the single-digit numbers. Adding these, we find that the numbers from 1 to 99 contribute a total of:
9 digits (single-digit) + 180 digits (double-digit) = 189 digits
This cumulative count is important because it tells us how far into the decimal expansion we have reached. The 2022nd digit is beyond this point, so we now need to consider the triple-digit numbers. The systematic addition of digits from each group helps us to narrow down the range in which the target digit lies, making the problem more manageable.
Counting Digits from Triple-Digit Numbers
Now, let's consider the triple-digit numbers (100 to 999). To find the number of triple-digit numbers, we subtract the smallest (100) from the largest (999) and add 1:
999 - 100 + 1 = 900 integers
Each triple-digit number contributes 3 digits to the decimal expansion. Therefore, the total number of digits contributed by triple-digit numbers is:
900 integers × 3 digits/integer = 2700 digits
These 2700 digits are added to the 189 digits from the single and double-digit numbers. So, the numbers from 1 to 999 contribute a total of:
189 digits (single and double-digit) + 2700 digits (triple-digit) = 2889 digits
Since 2022 is less than 2889, we know that the 2022nd digit falls within the triple-digit numbers. This is a crucial finding as it allows us to focus our attention on the range of integers from 100 to 999. Understanding this range helps in refining our search and calculating the exact position of the digit we are looking for. The systematic accumulation of digit counts has guided us to this critical point in the problem-solving process.
Locating the 2022nd Digit
We know that the 2022nd digit falls within the triple-digit numbers. To find out which specific number contains this digit, we subtract the number of digits contributed by single and double-digit numbers from 2022:
2022 - 189 = 1833
This means that the 2022nd digit is the 1833rd digit contributed by the triple-digit numbers. Since each triple-digit number contributes 3 digits, we divide 1833 by 3 to find out how many triple-digit numbers we need to count:
1833 ÷ 3 = 611
This result indicates that the 2022nd digit falls within the 611th triple-digit number. The first triple-digit number is 100. Therefore, the 611th triple-digit number is:
100 + 611 - 1 = 710
However, the division 1833 ÷ 3 gives us a whole number, which means the 1833rd digit is the last digit of the 611th triple-digit number. Therefore, we have reached exactly the end of the number 710. Thus, we know the 1833rd digit contributed by triple-digit numbers is the last digit of 710. The step-by-step approach of subtracting and dividing has allowed us to pinpoint the number containing the 2022nd digit, bringing us closer to the final solution.
Determining the Specific Digit
Since 1833 is perfectly divisible by 3, it means the 2022nd digit is the last digit of the 611th triple-digit number, which we found to be 710. The last digit of 710 is 0. Therefore, the 2022nd digit to the right of the decimal point is 0. This final step confirms our methodical approach has led us to the correct answer. The key to solving this problem was breaking it down into smaller, manageable steps and systematically accounting for the contribution of each group of digits.
Final Answer
The 2022nd digit to the right of the decimal point in the number x = 0.123456789101112...998999 is 0. This was determined by calculating the cumulative number of digits contributed by single, double, and triple-digit numbers, and then pinpointing the specific number and digit within that number that corresponds to the 2022nd position. The final answer is:
(2) 0
Conclusion
In conclusion, finding a specific digit in a concatenated sequence of integers requires a systematic and methodical approach. By breaking down the problem into steps—counting digits from each group of numbers, calculating cumulative digit counts, and pinpointing the relevant number and digit—we can efficiently arrive at the correct answer. This problem highlights the importance of number theory and digit manipulation skills, and it demonstrates how careful analysis and step-by-step reasoning can solve complex mathematical problems. The method used here can be applied to similar problems involving concatenated sequences and digit patterns, providing a valuable framework for problem-solving in mathematics.