How does the 8008 processor work?

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The historic 8008 processor is 45 years old: It was released on the 13th of March 1972. Ken Shirriff has a blog post that explains features of the CPU, from a die photo.
Compared to modern CPUs, the 8008 is weird in multiple ways: The storage for the stack is on chip instead of in memory, and it is not accessed in linear order for weird transistor saving reasons. Things have changed since then a lot, but back then, a central processing unit was usually not a single die, but a bunch of discrete electronics on a board.
Having enough transistors on a single die to make up a CPU inside a chip was a novel concept.
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