# Floating-Point Binary to Decimal

I'm having difficulty with the following question:

Given a bit pattern with mantissa $10110000$ and exponent $0111$, what does the bit pattern represent in denary (i.e decimal / base 10)?

I got the right answer(!) but the wrong (or alternative) work:

1. Convert the exponent to denary: $0111$ is $7$
2. Apply this exponent to the mantissa: $10110000\rightarrow1011000$ after shifting $7$ places
3. Convert the mantissa to denary: $1011000$ is $88$
4. Set the sign: $-88$ (which is correct!)

Using a different method, the mantissa $1.0110000$ is somehow deteremined to be $-11/16$ and then $-11/16 \times 2^7 = - 88$ (I understand this shift with the exponent). However, what I don't understand is:

How do you convert 1.0101 (mantissa) to -11/16? Is this a standard way to do it?

• So, $.10110000$ is $-11/16$; supposedly the additional $1$ at the beginning of any mantissa is redundant (because you'll know it will be there), but since you have to add that $1$ back in, I don't know how you would get $-88$ from that. Commented Nov 16, 2012 at 13:31
• I got the answer with a bit of reverse engineering. 1) 0111 is 7 in denary so the exponent is power(2,7) 2) convert 10101000 to a positive number using twos comp 3) you you get: 010110000 which is 1/2 + 1/ 8 + 1/16 using the powers of 2 after the radix point 4) this is 11/16 but we no it is negative from step 2 so it is -11/16! Commented Nov 16, 2012 at 17:59
• This will depend on the exact representation used. The normal IEEE representations for example do not use two's complement
– jk.
Commented Nov 19, 2012 at 12:15
• You may have a point, but the exam board does not use IEEE standard. It is something I will need to look into though. Commented Nov 20, 2012 at 16:16