There are two ways (in common use) of denoting orders of magnitude to make large numbers easier to read, first you can use a power of 10.
10⁰ = 1
10¹ = 10
10² = 100
10³ = 1000
Or powers of two
2⁰ = 1
2¹ = 2
2² = 4
2³ = 8
Using these series as a base we arrive at the numbers 1000 and 1024 (10³ and 2¹⁰) for a kilo.
There are eight bits to a byte. So one kilobyte is 8×10³ = 8000 bits. Hard drive manufacturers use this method. In computer science, people usually use powers of two, so one kibibyte is 8×2¹⁰ = 8192 bits.
The difference only gets larger as the numbers get larger. Some have even mixed those two systems to get nice numbers to put on their packaging. This is why a 1.44MB floppy disk has neither 1.44 megabytes nor 1.44 mebibytes (they use 1024×1000).
The logic behind the i is that the terms are derived from the original si prefixes, kilo, mega, giga, but with the word binary put in in. So the i is the second letter of binary. The mnemonic for the kibibyte is "kilo binary byte", and "KiB" is pronounced "Kibibyte".
Keep in mind that, very often, the term kilobyte is used when the author means kibibyte. The binary unit was only introduced around 1999, as Randy Orrison points out.
Since no one actually answered this: "MiB" stands for "megabinary byte", which can be abbreviated to "mebibyte" (though this sounds kind of stupid, and I'd rather just stick with saying "megabinary"). See the NIST explanation.
So the "i" comes from the word "binary".
There were other proposals to abbreviate these units in the past, but they all failed to gain traction:
κ = 1024, κ2 = 1024², κ3 = 1024³, ... (Greek letter kappa, hard to type)
KKB = 1024, MMB = 1024², GGB = 1024³, ... (could be misinterpreted as megamegabyte = TB)
bK = 1024, bK² = 1024², bK³ = 1024³, ... (when proposed, many computers didn't even have lowercase)
There are two ways (in common use) of denoting orders of magnitude to make large numbers easier to read, first you can use a power of 10.
Or powers of two
Using these series as a base we arrive at the numbers 1000 and 1024 (10³ and 2¹⁰) for a kilo.
There are eight bits to a byte. So one kilobyte is 8×10³ = 8000 bits. Hard drive manufacturers use this method. In computer science, people usually use powers of two, so one kibibyte is 8×2¹⁰ = 8192 bits.
The difference only gets larger as the numbers get larger. Some have even mixed those two systems to get nice numbers to put on their packaging. This is why a 1.44MB floppy disk has neither 1.44 megabytes nor 1.44 mebibytes (they use 1024×1000).
The logic behind the i is that the terms are derived from the original si prefixes, kilo, mega, giga, but with the word binary put in in. So the i is the second letter of binary. The mnemonic for the kibibyte is "kilo binary byte", and "KiB" is pronounced "Kibibyte".
All of this is defined in the IEC_80000 Standard.
Note that a mebibyte is not defined as 2²⁰, but as (210)2, although they are equal. A gibibyte is (210)3, a tebibyte is (210)4 and so on.
Keep in mind that, very often, the term kilobyte is used when the author means kibibyte. The binary unit was only introduced around 1999, as Randy Orrison points out.
As nealmcb found out in the comments, there is an official policy on this:
https://wiki.ubuntu.com/UnitsPolicy
In summary, this policy reminds developers to either use SI or IEC prefixes, but to never mix them. It goes on to say:
Since no one actually answered this: "MiB" stands for "megabinary byte", which can be abbreviated to "mebibyte" (though this sounds kind of stupid, and I'd rather just stick with saying "megabinary"). See the NIST explanation.
So the "i" comes from the word "binary".
There were other proposals to abbreviate these units in the past, but they all failed to gain traction:
it's an IEC standard prefix it means "by power of two"
2^10 = 1024 = Ki-
2^20 = 1048576 = Mi-
more details on
http://en.wikipedia.org/wiki/Kibi-#IEC_standard_prefixes
http://en.wikipedia.org/wiki/Mebibyte
They are binary prefixes, corresponding to the more common decimal prefixes. (That's a good article which explains the rationale well.)