What major advancement did the Intel 80386 introduce?
The 386 (1985) was the first 32-bit x86 processor — it defined IA-32, widened registers and addresses to 32 bits, and added "flat" protected-mode memory.
The jump from 16 to 32 bits was transformative. Registers grew from 16 to 32 bits (the names gained an "E" for Extended: AX → EAX), and the address space expanded to 4 GB — a vast leap from the 8086's 1 MB. Just as important, the 386 introduced flat addressing and a true protected mode with memory protection and privilege levels, plus paging for virtual memory.
These features are what made the 386 capable of running real multitasking operating systems — it could run Unix, and it became the foundation that Linux and Windows NT were later built on. The architecture it defined is called IA-32 (also "i386" or just "x86"), and remarkably, 32-bit Linux/GCC still restrict themselves to instructions the 386 already had.
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Intel 80386 (Wikipedia) — the 386 as the first 32-bit x86 with protected mode and paging.