A general purpose register in computing is a very small area of memory stored in the CPU that has an extremely fast access time, generally single cycle or close. Each register is named by a number, in CPSC 213/SM213, 0-7.
The size of each register is of the machine architecture’s common bits (either 32 or 64 - in other words the word size).
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In sm213, the register size is always 4 bytes. Since the underlying machine that runs it is virtual, the hardware is irrelevant. Also later on into the course, registers 0, 5, and 6 will become not really general-purpose anymore, but more by convention rather than implementation.
General purpose registers, as the name implies is used for general computing purposes that can store temporary data, memory addresses, and intermediate results. This is in opposition to special purpose registers.