FAQ - Number Systems

Q1: What are the differences between notations?

Q2: How should I convert notation between power of two bases and decimal?

Q3: For floating point representations, how can I determine the smallest
and largest values?

Q1:
I just want to know what the difference between notations and
representations. I know notations contain binary, octal, hexadecimal,
and decimal. But I just don't know what's the difference between these
two!

Notation defines the symbols (base2, base10, etc.) used to list
values. Representations define the "meaning" of a string of symbols.
So 1011011 (in binary) could be transformed to 133 in octal, 91 in
decimal, etc. but this does change (or imply) meaning. Stating it is
a seven bit unsigned integer or an seven bit signed two's complement
integer, or an eight bit signed two's complement integer, or and ASCII
character changes what the string represents.

Q2:
If I get stuck on how to convert from hex to oct, decimal to hex, or
vice versa for those, the general thing I can do is to convert to
binary and then do the final conversion? That will still obtain a
valid answer?

Yes, power of two conversion are best performed by first converting to binary
and then regrouping. Conversion between binary and decimal are also easier
with binary than other powers of two.

Q3:
I have a few questions about the floating point representation. How
can I determine the smallest and largest values from a given mantissa
size and exponent size? As I recall the mantissa determines the amount
of significant figures and the exponent shifts the point. However, I
am still puzzled in solving problems such as the one in exam 2. Can I
get some help?

Yes, mantissa does determine the precision (number of significant
figures) for a floating point value. It ranges between 0 and 1. This
is then scaled by 2 raised to the exponent (a signed integer). This
can make a very large or very small number.For example, an eight bit
exponent (-128 to +127) can produce a huge number 2127 or a tiny
number 2-128. The mantissa have no effect on this range, just its
sigfigs.