# Fixed decimal point numbers binary representation

I saw this formula that I cannot understand how it applies (the proof).

Q: say we have a $N$ bit register (imagine a binary number with $N$ digits) for representing fixed decimal point numbers. In two special cases define the range in which the (whole?) "content" of the register changes:

a) decimal point is placed at the end of the register after LSB (meaning 0 digits after decimal point) (As in this picture)

b) decimal point is placed between Sign Bit and MSB (As In This Picture)

I cannot understand what does the question refer to as "Content". Either way the answer follows:

Let $N$ be the content value, $n$ the register bits answer to (a): $0 \le \lvert N \rvert \le 2^{n-1}$

and answer to (b): $0 \le \lvert N \rvert \le 1 - 2^{n-1}$

Other info: In a $n$ bit register the Sign bit if "0" means the binary content is showing a positive number otherwise if "1" means a negative number is being represented.

MSB = most significant bit

LSB = least significant bit

For example: 6 equals (110) in binary representation with MSB=1 and LSB=0, hence (0110) represents +6 and (1110) represents -6 in a 4 bit register.

How can those formulas be applied? thanks.

• I believe the "content" is the binary number in the register – vonbrand Mar 25 '13 at 18:49