# Binary Floating Point Range/Precision for 4-bit Mantissa and 4-bit Exponent

I'm trying to understand binary floating point and using just a 4-bit mantissa and a 4-bit exponent (both 2s compliment) to keep things simple.

As far as I can tell, the largest denary number I can represent is 112: 0111 0111

So why do I have trouble representing denary 11 (eleven) in this system? I get that the largest (positive) mantissa is 7 in denary (0111). So I'm inferring that I can't represent consecutive positive integers between 0 and the maximum value of 112.

Is that correct? If so, can someone help me to see why please? Which numbers in this range can I represent?

Yes, that is correct. Your mantissa has only three magnitude bits, so you can only represent numbers from -8 to 7, multiplied of course by any power of 2 from $$2^{-8}$$ to $$2^7$$. 11 is impossible to represent, because any positive exponent gets you multiples of 2, any negative exponent gets you numbers less or equal to $$7*2^{-1}=3.5$$, and exponent 0 is just numbers from -8 to 7.