you lost me. any geometric ratio can be expressed exactly, mathematically. what's an example of a "ratio that cannot be expressed exactly, mathematically"?
“what’s an example of a ‘ratio that cannot be expressed exactly, mathematically’?”
I gave two examples. When expressed they are called “irrational” numbers, which really means a number which cannot be perfectly expressed as a ratio [it has nothing to do with rationality].
But I’ll give you one detailed example. It is Bertrand Russell’s recasting of Euclid’s explanation of incommensurables arising from the Pythagorean theorem.
“In a right-angled isosceles triangle, the square on the hypotenuse is double the square on either side. Suppose each side is an inch long; then how long is the hypotenuse? Let us suppose its length is m/n inches. Then m²/n²=2. If m and n have a common factor, divide it out, then either m or n must be odd. Now m²=2n², therefore m² is even, therefore m is even, therefore n is odd. Suppose m=2p. Then 4p²=2n², therefore n²=2p² and therefore n is even, contra hyp. Therefore no fraction m/n will measure the hypotenuse. The above proof is substantially that in Euclid, Book X.” (Bertrand Russell, History of Western Philosophy)
The reason I like this is because it illustrates that what are called irrationals and often expressed as incomplete decimals are not just a problem of precision. Irrationals are things which cannot technically be expressed mathematically—there is no number, positive or negative, that represents the ratio between the hypotenuse and either leg of an isosceles right triangle.
Hank