Incommensurable Numbers

Date: 07/25/2001 at 15:34:11
From: S Fries
Subject: Incommensurable Numbers

I'm not sure what exactly is meant by an incommensurable number.
What is the clear definition?

Date: 07/25/2001 at 16:52:41
From: Doctor Rob
Subject: Re: Incommensurable Numbers

Thanks for writing to Ask Dr. Math.

A single number cannot be incommensurable. Incommensurability is a 
relation between two numbers. (It would be like equality. There is no
such thing as an equal number. Two numbers can be equal to each
other.)

Two numbers are incommensurable with each other if and only if their 
ratio cannot be written as a rational number (i.e., quotient of two 
integers). In other words, their ratio is irrational.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/26/2001 at 09:22:17
From: Doctor Peterson
Subject: Re: Incommensurable Numbers

Hi! I saw your question and wanted to add some extra information that 
might be helpful in understanding the word.

The word "incommensurable" comes from roots and prefixes that mean: 
"not co-measurable." That is, two numbers are incommensurable if they 
can't be "measured with the same ruler." This goes back to the Greeks, 
who at first assumed that, given any two line segments, you could find 
a smaller line segment that could be used as a "unit" or "ruler" with 
which to measure both lines, so that both would be a whole number of 
units. Then the ratio of their lengths would be the ratio of those two 
whole numbers. When it was discovered that some pairs of lines (such 
as the side and diagonal of a square) had a ratio that could NOT be 
written as such a fraction - that is, the ratio was irrational, and 
their lengths were not commensurable - it forced them to develop a new 
theory of ratios that did not depend on this assumption. This fiasco 
was one of the foundations of math as we know it: the recognition that 
everything has to be proved, and nothing assumed, because what seems 
obvious can be wrong.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/