How many numbers can a typical computer represent?

Question

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.

Answer 1

There are 8-byte signed or unsigned integers that cannot be represented as a double-precision floating point.

For instance, the range of 64-bit unsigned integers is from 0 to $2^{64}-1$. This is a total of $2^{64}=18446744073709551616$, the maximum possible for a 64-bit value. Signed integers represent a different range, but the total number of integers possible is the same.

Evidently double-precision floating points, due to the NaNs, cannot represent so many numbers. If I'm not mistaken, every double-precision floating (except zero) has a unique representation. This means that we need exclude only the redundant NaNs. If the exponent is 7FF, then the value represented must be one of the infinities or NaN. There are $2^{53}=9007199254740992$ such numbers, of which only three are unique (NaN, $+\infty$, and $-\infty$).

If $0$ and $-0$ are considered distinct, there are then: $$2^64-2^53+3=18437736874454810627$$

such numbers. If $0$ and $-0$ are considered equal, then there is one fewer. Your number is almost correct, but you double-count zeros (once as a subnormal number, and once as a special value).

(Also, it's worth noting that this limit is pretty artificial. Some [rare] computers support quadruple-precision floating points natively, for example. And of course, there are BigNumber and BigDecimal software implementations that support numbers of any size that will fit in memory.)

Answer 2

The eight byte signed integers are not a subset of the double precision numbers if by double precision you mean $64$ bits. The eight bit signed integers have $63$ bits of mantissa plus a sign bit, while the double precision floats only have $52$ bits of precision. To compute the overlap is not so easy. Let us focus on positive values. Of the $63$ bits, we need at least $11$ zeros between the two ends to make it representable in double precision. There are $11\cdot 2^{50}$ numbers with exactly $52$ bits of precision-that have a $1$ in the leading and trailing places. The factor $11$ comes because the highest order bit can be in $11$ places. Similarly there are $12 \cdot 2^{49}$ numbers with $51$ bits of precision, $13 \cdot 2^{48}$ with exactly $50$ bits, ... $61$ with $2$ bits of precision and $62$ with a single bit of precision. Alpha tells me this is $54043195528445950$ including the negatives. Adding your count of floating point numbers to $2^{64}$ signed integers and deducting the double count gives $36830437752635916294$ floats and signed integers. To this we have to add the positive signed integers that cannot be otherwise represented. They have to have a $1$ in the most significant bit-otherwise they fit in a signed integer. This would give $2^{63}$ of them, but we have to deduct the ones that have twelve trailing zeros because they can be represented by floats. That gives $2^{63}-2^{51}$ of them. Adding these in to the previous count gives $46051557989677006854$ This compares with $5.534E19$ if we could use all $3\cdot2^{64}$ bit patterns times formats. We are only $9E18$ short of that. Of course, some languages have other ways of representing numbers that have many more possibilities.

Answer 3

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.