How to used the alphabet binary symbols

Question

I was reading an article on binary numbers and it had some practice problems at the end but it didn't give the solutions to the problems. The last is "How many bits are required to represent the alphabet?". Can tell me the answer to that question and briefly explain why? Thanks.

Answer 1

You would only need 5 bits because you are counting to 26 (if we take only upper or lowercase letters). 5 bits will count up to 31, so you've actually got more space than you need. You can't use 4 because that only counts to 15.

If you want both upper and lowercase then 6 bits is your answer - 6 bits will happily count to 63, while your double alphabet has (2 * 24 = 48) characters, again leaving plenty of headroom.

Answer 2

It depends on your definition of alphabet. If you want to represent one character from the 26-letter Roman alphabet (A-Z), then you need log2(26) = 4.7 bits. Obviously, in practice, you'll need 5 bits.

However, given an infinite stream of characters, you could theoretically come up with an encoding scheme that got close to 4.7 bits (there just won't be a one-to-one mapping between individual characters and bit vectors any more).

If you're talking about representing actual human language, then you can get away with a far lower number than this (in the region of 1.5 bits/character), due to redundancy. But that's too complicated to get into in a single post here... (Google keywords are "entropy", and "information content").

Answer 3

There are 26 letters in the alphabet so you 2^5 = 32 is the minimum word length than contain all the letters.

Answer 4

How direct does the representation need to be? If you need 1:1 with no translation layer, then 5 bits will do. But if a translation layer is an option, then you can get away with less. Morse code, for example, can do it in 3 bits. :)