I'm currently studying Numerical Analysis with the book "Numerical Analysis: Mathematics of Scientific Computing" by Kincaid.
In this book, the authors have introduced a computer called "Marc-32" which is a 32-bits computer representing a nonzero real number with the form: x = ±q * 2^m
with the allocation:
- sign of the real number x: 1 bit
- biased exponent (integer e): 8 bits
- mantissa part (real number f): 23 bits
Now a problem in the book asks to consider different given machine numbers and whether they are contained in Marc-32. For example the number 2^-1 + 2^-26.
My problem is that I honestly do not know how to determine this. I've tried reading the section about "marc-32" multiple times, but the closest I've got was that in the book, the authors write that "...23 bits means that our machine numbers have a limited precision of rougly six decimal places".
Therefore I thought that 2^-1 + 2^-26 = 0.10000000000000000000000001 being a number with 26 decimal places would not be contained in marc-32, but I'm really not sure if this is correct at all, and if it is the method I'm supposed to be using.
Therefore I would like to reach out to the math community here for any pieces of advice on how to tackle this problem. It would be much appreciated.