The File Allocation Table (FAT) is the cornerstone of DOS, or Disk Operating System, the basic operating system that PCs have used for many years. While DOS handles other tasks, such as orchestrating printers, keyboards, and monitors, the bulk of the effort is managing how a single file is stored (and sometimes scattered) on the magnetic bits of a floppy or hard drive. This same concept is still in use in modern file systems, which organize the manner in which bits are stored on disk.
An analogy of FAT is in a library that uses a card system to look for a book. To find a book, you search the card system for the book title. When the correct card is found, it tells you where in the library to look for the book. Taking the concept one stage further, visualize that the individual chapters of the book are scattered through different parts of the library. The index card would tell you firstly that the book exists, and secondly where to find the different chapters in the library. This is how FAT works.
If you then remove the book from the library because, for example, the book was out of date, you could insert the chapters of a new book into the spaces the old book occupied. Over time, the books within the library would become quite scattered, and it would take a progressively longer amount of time to collate the data for one entire book. In a computer file system, this is known as fragmentation, and it slows down the computer in the same way it would slow down a person who had to visit multiple locations within the library to retrieve all the parts of a book.
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