An irrational number is any real number that cannot be written as the ratio of two integers. How is this related to the definition of incommensurable magnitudes?
See solution.
Practice makes perfect
After doing some research, we can write the following definition of incommensurable magnitudes.
Two maginutes are incommensurables if these
magnitudes are of the same kind and its ratio
is an irrational number.
Recall that an irrational number is any real number that cannot be written as the ratio of two integers. The ratio of two incommensurable magnitudes is an irrational number. Therefore, it also cannot be written as a ratio of two integers.
Example
We can consider as an example a square of side length l. By the Pythagorean Theorem, we have that its diagonal has a length of sqrt(2)l.
The length of the sides and diagonal are magnitudes of the same kind and its ratio is an irrational number.
sqrt(2)l/l = sqrt(2)
Thus, the diagonal of a square and the side length are incommensurable magnitudes in the example.
