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64bitOS takes/uses more memory?

is it documented that 64 bit os uses more GB RAM than 32 bit OS for the same kind of operation? if yes, what will be the technical reason for that?
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>>
an integer uses 4 bytes of memory while a 64-bit integer uses 8 bytes of memory.
There isn't much difference.
maybe 100MB of extra memory space.
<<
for the same operation, if all it takes extra is 100MB, then that is a easy choice indeed.
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>>
In a 64-bit operating system the video card is mapped into memory space that is not used by real memory, so all memory installed is usable.

This is phenomenal, too.

So, if we had only 4GB RAM for a machine, then it is always better to go to 64 bit, since you will have more memory to play with.
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But could it be ever that a 64bit machine gets overwhelmed at 4GB compared to a 32 but machine, for any other or some other reason, not covered already in this post (assuming the same operation only will be performed- in this case for example: SQL 2008R2)?
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