Irrational Numbers

An irrational number^2.4 is one that cannot be written as a ratio of two integers e.g. $a/b$. 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. $\sqrt{2}$ 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 $e = 2.718281828...$ and $\pi = 3.141592654...$.

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 $\pi$, an irrational number, but it is really $\frac{314159}{100000}$, a rational number that approximates $\pi$ 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 $\sqrt{2}*\sqrt{2} = 1.414*1.414 = 1.999396$ 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. $\cos(\pi) = -1$. If we did finite-precision arithmetic we might get instead $\cos(3.14) = -0.999999$ 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''<sup>2.5</sup> 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''.