# Infinities and NaN in C

A common application of programming languages such as C is to perform numerical calculations, however some mathematical operations which are possible to perform have infinite or undefined results. For example, $$1/0$$ results in a positive infinity, $$\log(0)$$ results in a negative infinity and $$\sqrt{-1}$$ is undefined for real numbers.

Fortunately, C provides a sensible way of storing and returning the result of such operations - it does not simply crash the program. This allows us to handle such outcomes sensibly in our program. Returned infinite results are called just that, whereas undefined results are called “not a number”, or NaN.

## Storage

Infinities and NaNs are stored in a unique way in the floating point data structure that does not conflict with other possible values.

Floating point numbers are stored as three integers - a sign bit $$s$$, a coefficient $$c$$, and an exponent $$q$$. The value of a decimal number is then given by:

$(-1)^s \times c \times 10^q$

For example, the number 10.3452 would be represented as $$s=0$$, $$c=103452$$, $$q=-4$$, giving:

$(-1)^0 \times 103452 \times 10^{-4} = 10.34524$

The special values of infinity and NaN are represented by filling the exponent field bits with 1s. Positive and negative infinity have the coefficient part equal to zero, while the sign bit indicates the sign of the infinity. NaNs have the same exponent part, but have the coefficient as non-zero. The storage and behaviour of these values is set out in IEEE standard 754.

## Logical Tests

An interesting feature of NaN is that, by design, it is unordered, meaning that it is not equal to, less than or greater than any value, including itself. All of these statements will be false when x is NaN:

This provides a possible way to test for a NaN in code - it is the only float for which x == x will be false. However, it is recommended to instead use the macros defined in math.h. These include NAN and INFINITY, macros defining these values as constants, and the isfinite and isnan macros, which allow testing for values of these types:

These special values offer an efficient way of storing and handling the infinite or undefined results of numerical computations. Programs which may accidentally perform such calculations can continue on regardless and at least partially complete their task, providing possibly useful results but also allowing better analysis of the code and why it failed.