A mebibyte is a unit of measure. It represents 1024 * 1024 bytes.
The mebibyte is closely related to the megabyte (MB). This can either be a synonym for mebibyte, or refer to 106 bytes = 1,000,000 bytes, depending on context (see binary prefix for background). The two numbers are relatively close together, but mistaking the two has nevertheless led to problems.
Mibibyte is a common misspelling, presumably from the "i" in the abbreviation or confusion about how the surrounding prefixes (i.e. "kibi" and "gibi") were actually formed. The "i" is actually from the end of the prefix. Tebi- and pebi- are similarly abbreviated Ti- and Pi-, but they are not usually misspelled, as storage sizes (and corresponding terms) currently are only this large in specialized situations.
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