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The original IBM PC could only address 1 MB of system memory, and the original versions of DOS created to work on it were designed with this in mind. DOS is by its nature a single-tasking operating system, meaning it can only handle one program running at a time. The decisions made in these early days have carried forward until now, and in each new processor, care had to be taken to be able to put the processor in a mode that would be compatible with the original Intel 8088 chip. This is called real mode.
When a processor is running in real mode, it acts like an "8088 on steroids". What this means is that it has the advantage of speed, but it otherwise accesses memory with the same restrictions of the original 8088: a limit of 1 MB of addressable RAM, and slow memory access that doesn't take advantage of the full 32-bit processing of modern CPUs. All processors have this real mode available, and in fact the computer normally starts up in real mode.
Real mode is of course used by DOS and "standard" DOS applications. In fact, today there are relatively few simple DOS programs that just use the standard 640K that DOS makes available. Even within DOS now there are special programs available that will "extend" DOS to allow access to extended memory (over 1 MB) and faster 32-bit access. These are sometimes called DOS extenders. The protocol that describes how to make DOS work in protected mode is called DPMI (DOS protected mode interface). DOS extenders are used by most DOS games (since the standard DOS 640 KB limit has gone from inflexible to downright laughable in recent years as games have become very large).
Note: The first 64 KB of
extended memory is actually accessible to the PC in real mode, despite the fact that it is
not supposed to be possible. This is the result of a bug in the original IBM AT. This area
of memory is called the high memory area (HMA).
Next: Protected Mode