"Real numbers" may be thought of informally as numers with infinite precision, i.e. as numbers which, if we could write them
down as a decimal might have an infinite decimal expansion.
These are needed to get the mathematics to work out nicely.
The mathematics in question includes the theory of analysis which in turn includes the differential and integral calculus
and more, and is widely applied in engineering and science.
When it comes to using computers to do calculations, real numbers pose serious provlems, because of the fact that they involve
infinite amounts of information.
Digital computers can store and compute with only finite amounts of information.
Consequently, real number computation normally uses finite precision representations, most often "floating point" representations
of various precisions.
The results of each step in the computation are obtained as an approximation within the relevant limits of precision.
The results of computations involving many steps then suffer from the cumulative effects of many such approximations, and
there is no limit to how widely they may diverge from the correct value.
Though techniques are available for estimating the accuracy of such floating point computations, these do not usually yield
definite upper bounds on the error.