Bitstrings

N-bit number:

bN-1 bN-2...b1b0

Can also think of this as a string of digits 0 and 1.

How many possible patterns for N-bit string?

Each bit bi can have 2 possible values

There are N such bits

Therefore, total number of patterns is

2 * 2 * . . . * 2 = 2N

(Important Binary Number Fact #1)

How many different VALUES? (trick question)

It depends on the representation being used.  (We will look at several.)

How many bits does it take to represent a given number N?

With K bits we can represent numbers up to 2K

So, we want to find K such that

2K >= N

Expressing this in terms of logarithm base 2,

lg (2K) >= lg N

K lg 2 >= lg N

K >= lg N

K = ceil (lg N)

(Important Binary Number Fact #2)