How to consider binary numbers?
LEWIS CARROLL (Charles L. Dodgson), Through the Looking-Glass, chapter 6, p. 205 (1934). First published in 1872.

I your case you are Humpty Dumpty and can decide what you want a digital word to mean. 4-bit logic would be rather unusual these days so let's use 8-bit for our example. 8 bits gives us 28 = 256 combinations including zero. Generally these would be used as follows:

Type                 Range
-------------------- ----------
Unsigned integer:       0 - 255
Signed integer:      -128 - 127
BCD:                   00 -  99


You also mention the possibility of using the first bit as sign and the remainder as the absolute value. This doesn't work out so well as the logic gets complicated.


There are two values for zero, 00000000 and 10000000.
The count sequence has to reverse when you cross zero.


In comparison the standard two's compliment handles the negative transition quite simply by rolling over from 00000000 (0) to 11111111 (-1), etc., in much the same way that the old mechanical mileometers in cars would roll over if you ran them backwards or forwards for long enough.

Some programming languages force you to define the data type before use. This cab be a good thing as it forces the user to consider exactly what they want and how much memory to set aside for each variable. Others are more flexible and adapt on the fly.

Lewis Carroll was a mathematician and would have appreciated the possibilities that this gives us.