Ordinarily any quantity can be measured in terms of any units, to an arbitrary accuracy. Pounds or kilograms work equally well for weights; feet or light-years are both fine for distances; and so on.
But in the pure realm of math, some quantities are incommensurable --- they don't ever relate, not to the absolute perfection that mathematicians demand of their ideals. No ratio of integers can match pi (3.14159265..., the circumference of a circle with diameter of one), though some like 355/113 come pretty close. Even the square root of two (sqrt(2) = 1.41421356..., the diagonal of a unit square) never can be computed as a fraction made of natural numbers.
There's a neat way to see that sqrt(2) is irrational, a clever proof by contradiction that the ancient Greeks geometers found. Here's a quick summary of the argument (skip ahead if you're not interested in the details). Suppose there were a magic fraction, call it A/B, that gave sqrt(2). Squaring that fraction implies A*A = 2*B*B. Now A can't be an odd number, since the product of two odds is still odd, yet the right-hand-side of this equation is clearly even. So A is even; call it A = 2*C. Then plug that into the equation and simplify to get B*B = 2*C*C. By the same logic as we just saw, B has to be an even number; call it B = 2*D.
Zooks! We seem to have proved that both A and B were even numbers in our mythical fraction A/B = sqrt(2). So that fraction could have been simplified by dividing both A and B by 2, reducing A/B to the smaller fraction C/D. And nothing prevents us from applying the same method to that fraction in turn, to get a ratio of still-smaller numbers E/F, and so on, and so on. This is no good! Eventually we've gotta reach a smallest fraction. We can't descend forever, dividing both sides by 2 without end and still always having even numbers. So our original assumption, that the fraction A/B = sqrt(2) existed, must have been wrong. There are no such numbers A and B.
Where was the sleight-of-hand in this proof? Magic crept in on an infinite number of little cat's feet, via the never-ending non-repeating decimal that represents the square root of two. Infinity is counter-intuitive. Some gears never quite mesh; some patterns never exactly recur; some things can never be precisely measured in terms of other things. Infinite variety....
Wednesday, January 26, 2000 at 13:36:40 (EST) = Datetag20000126