Irrational Numbers

An irrational number⁸ is one that cannot be written as a ratio of two integers e.g. $a/b$ . 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 $e = 2.718281828...$ and $\pi = 3.141592654...$ 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 $\sqrt{2}$ , which are also irrational - this was the basis for the earliest proofs of the existence of irrational numbers, and $\sqrt{2}$ 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 $\pi$ , an irrational number, but it is really $\frac{314159}{100000}$ , a rational number that approximates $\pi$ 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. $\cos(\pi) = -1$ . 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.