An irrational number^{2.4} is one that *cannot be written* as a ratio of two integers e.g. . Most proofs
that any given number is irrational involve assuming that it can be so
written and showing that this leads to a contradiction.

It is quite easy to show that e.g. is irrational using this method, although there are other numbers of ``interest'' in physics and science where the property of irrationality is not so easy to prove. Two irrational numbers that are of great importance in physics are and .

Whenever we *compute* a number answer we *must* use rational
numbers to do it, most generally a finite-precision decimal
representation. For example, 3.14159 may *look* like , an
irrational number, but it is really
, a rational
number that *approximates* to six significant figures.

For that reason we will often carry important irrationals along with us
in computations as *symbols* and only evaluate them numerically at
the end. This often yields more satisfactory answers. Consider
in a decimal
representation. This answer is not *exactly* 2.

Also, we work quite often with functions that yield a rational number
when an irrational number is used as an argument, e.g.
. If we did finite-precision arithmetic we might get instead
which is not *exactly* -1.

These computational errors aren't usually terribly important for small one or two step calculations, but in a long-running computer program they can easily add up! However, unless we can recognize a rational answer in an expression that contains irrational numbers, we will more or less have to work with decimal (rational) approximations and do our best to control these ``round-off'' errors. Fortunately, we can make the difference between an irrational number and a rational approximation to it as small as we like by just adding more digits to the latter.

There are lots of nifty truths regarding irrational and irrational
numbers. For example, in between any two rational numbers lie an *infinite* number of *irrational* numbers. This is a ``bigger
infinity''^{2.5} than *just* the countably infinite
number of integers or rational numbers, which actually has some
important consequences in physics. Mostly, however, we will be pretty
happy working with a truncated rational decimal representation with a
finite number of ``significant digits''.