Questions:

a) How many bits are there in a gigabyte?

b) What is the name given to a half byte? (If you don't know, try guessing!)

c) How few bits are needed to represent two octal digits?

a) The simple answer here is, as stated above, that there are approximately eight thousand million bits in a gigabyte. But, although this number is generally accepted to be an acceptable answer for most general purposes, from the point of view of a computer programmer it is not accurate.

There are really 1024 bytes in a kilobyte, and 1024 kilobytes in a megabyte.
A gigabyte is 1024 megabytes (1,073,741,824 bytes) or 8,589,934,592 bits.

How has this discrepancy between 'common knowledge' and the facts come about, and does it matter?

The prefix names kilo-, mega- and giga- were adopted as being convenient names by programmers who were using binary and octal arithmetic in their programming. The fact that the prefixes did not accurately describe the actual numbers of bits, and bytes did not seem to matter and was misleading only to non-programmers. The discrepancy is now generally ignored - except by programmers who are very conscious of the true numbers.

The 2 values of gigabyte are both commonly used.
The 1,073,741,824 number (bytes) is used in programming and also used when referring to memory (RAM, ROM, etc.). the 1,000,000,000 value is used, for example, when talking about disk drive sizes.