An irrational number^{8} is one that *cannot be written* as a ratio of two integers e.g.
. It is not
immediately obvious that numbers like this exist at all. When rational
numbers were discovered (or invented, as you prefer) by the
Pythagoreans, they were thought to have nearly mystical properties -
the Pythagoreans quite literally worshipped numbers and thought that
everything in the Universe could be understood in terms of the ratios of
natural numbers. Then *Hippasus*, one of their members,
demonstrated that for an isoceles right triangle, if one assumes that
the hypotenuse and arm are commensurable (one can be expressed as an
integer ratio of the other) that the hypotenuse had to be even, but the
legs had to be both even and odd, a contradiction. Consequently, it was
certain that they could *not* be placed in a commensurable ratio -
the lengths are related by an *irrational* number.

According to the (possibly apocryphal) story, Hippasus made this
discovery on a long sea voyage he was making, accompanied by a group of
fellow Pythagoreans, and they were so annoyed at his *blasphemous*
discovery that their religious beliefs in the rationality of the
Universe (so to speak) were false that they *threw him overboard* to
drown! Folks took their mathematics quite seriously, back then...

As we've seen, all digital representation of finite precision or digital
representations where the digits eventually cycle correspond to rational
numbers. Consequently its digits in a decimal representation of an
irrational number *never* reach a point where they cyclically
repeat or truncate (are terminated by an infinite sequence of **0**
's).

Many numbers that are of great importance in physics, especially and are irrational, and we'll spend some time discussing both of them below. When working in coordinate systems, many of the trigonometric ratios for ``simple'' right triangles (such as an isoceles right triangle) involve numbers such as , which are also irrational - this was the basis for the earliest proofs of the existence of irrational numbers, and was arguably the first irrational number discovered.

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.

Because we *cannot* precisely represent them in digital form, in
physics (and mathematics and other disciplines where precision matters)
we will often carry important irrationals along with us in computations
as *symbols* and only evaluate them numerically at the end. It is
important to do this because we work quite often with functions that
yield a rational number or even an integer when an irrational number is
used as an argument, e.g.
. If we did finite-precision
arithmetic prematurely (on computer or calculator) we might well end up
with an *approximation* of -1, such as -0.999998 and could not be
sure if it was *supposed* to be -1 or really was supposed to be a
bit less.

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''^{9} than *just* the countably infinite
number of integers or rational numbers, which actually has some
important consequences in physics - it is one of the origins of the
theory of deterministic chaos.