How many numbers can a typical computer represent? I couldn't find this elsewhere so I thought I'd give it a try to figure out exactly how many numbers a typical desktop computer can represent in memory.  I'm thinking about this in the context of numerical algorithms to solve mathematical problems.  
I'm pretty sure single precision (SP) numbers, along with the typical 4- and 8-byte signed and unsigned integers are all a subset of the representable numbers in double precision (DP), so I think computing the number of representable numbers in DP would answer the question. Let's assume IEEE double precision, a very typical architecture found on most machines : 1 sign bit, 11 exponent ($e$) bits and 52 mantissa bits.
First, the normalized numbers (assumed leading 1 in the mantissa).  With 52 available bits, there are $2^{52}$ different mantissas possible.  For each mantissa there is an exponent associated with it.  Note $e \in [0, 2047]$ for IEEE DP, but $e=0$ and $e=2047$ have special meanings ($0$, $NaN$, subnormalized numbers and $\pm \infty$, depending on the mantissa).  So we actually have $2046$ different exponents available for normalized numbers.  Also for each mantissa there are 2 signs available, $+$ and $-$.
Next, the subnormalized numbers (no leading 1 assumed in mantissa).  Each subnormalized number still has $2^{52}$ bits available, but are characterized by $e = 0$, so only one available value for $e$.  Again for each mantissa there are 2 signs available, $+$ and $-$.
Finally, the four special values $0$, $NaN$, $+\infty$ and $-\infty$ can be represented.
Putting these together, there are total of
$$
2 \cdot 2046 \cdot 2^{52} + 2 \cdot 2^{52} + 4 = 1.8437736874454810628E19
$$
(18.4 quintillion!) numbers representable on a typical computer.
Does this seem correct?  Does anyone know of a good resource to verify it?  I'm afraid I double counted something, or left a significant set of numbers out.  

There is a difference between the numbers with supporting instructions and the numbers that can be represented.
Pi is known to millions of decimal places. It is reasonable to believe any numbers of interest can be represented and manipulated to a very large number of places.
