When you say 32-bit and 64-bit computers do you mean the IA-32, IA-64, AMD-64,
x86??? No matter anyways.
A true 32-bit machine has 32-bit wide CPU registers,
and also for true 64-bit machines their registers are 64-bits wide.
The x86 line is backwards compatible and:
Lower 16-bits are used in Real Mode and V8086 mode
Lower (or) 32-bits are used in 32-bit Protected Mode
Whole 64-bits of the register is used in an x86-64 native 64-bit environment.
AX - Accumulator Carry Over From 8008 8080
EAX - Extended Accumulator
RAX - Re-Extended Accumulator
[ A WORD equals 16-bits in Intel terminology. ]
From Low-WORD to Hi-WORD is shown the arrangement of RAX
[ 0000 | 0000 | 0000 | 0000 ] __RAX
[ 0000 | 0000 ] ________________EAX
[ 0000 ] ________________________AX
Segment register's of the x86-32 appear to be 16-bit but have
an invisible high 16-bits used by the privileging mechanism:
WORD 0_________________________________________WORD 1
[14bit Segment Selector | 2bit Table Selector] [ RPL -> Transparent ]
Registers are typically a multiple of a byte in length,
a byte usually being the smallest container and addressable
data item. Thus, all data item's lengths can be expressed in
a number of requisite bytes.
A 32-bit register consists of 4 bytes for instance,
on a machine with an 8-bit byte.