Pages in a book are numbered in typical fashion, starting with 1. The folio for page 10 will contain the tenth and eleventh digits necessary to paginate the book. On what page will the 2009th digit occur?
To find the page on which the 2009th digit occurs, we need to determine the pattern of how the digits are distributed in the book.
Let's break it down step by step:
1. Each page in the book has two digits, except for some potential exceptions at the beginning and end.
2. The first page of the book, page 1, contains the digits 1 and 2.
3. The second page, page 2, contains the digits 3 and 4, and so on.
4. As we can see, the digit arrangement follows a pattern where the page number determines the starting digit for that page. For example, on page 1, the starting digit is 1, on page 2, it is 3, on page 10, it is 19, and so on.
5. The folio for page 10 will contain the digits from 19 through 20 (tenth and eleventh digits necessary to paginate the book).
Now, to find the page on which the 2009th digit occurs, we can calculate it using the following steps:
Step 1: Determine the page range where the 2009th digit lies:
Since each page contains two digits, we need to find the range of pages that include the 2009th digit. We can do this by dividing 2009 by 2:
2009 / 2 = 1004.5
This means that the 2009th digit lies between page 1004 and page 1005.
Step 2: Calculate the starting digit of page 1004:
To determine the starting digit, we subtract 1 from the page number and multiply it by 2:
(1004 - 1) * 2 = 2006
This means that the starting digit on page 1004 is 2006.
Step 3: Determine the position of the 2009th digit within page 1004:
To find the position of the 2009th digit within the page, we subtract the starting digit of the page (2006) from 2009:
2009 - 2006 = 3
This means that the 2009th digit is the third digit on page 1004.
Therefore, the 2009th digit occurs on page 1004, specifically as the third digit on that page.