Caption: Pi constant = π = 3.1415 9265 3589 7932 38462 6433 ... illustrated.
π is the ratio of the circumference of a circle to its diameter (which is 2 times its radius): π=C/D=C/(2R).
π is an irrational number---it's mad, bad, and dangerous to know---like Lord Byron (1788--1824).
Which means it is NOT a rational number: it cannot be expressed as a ratio of integers, and so is incommensurable with integers.
Famous irrational numbers include:
Commensurable and incommensurable explicated:
Naturally, incommensurable means that a ratio is an irrational number.
Say you have a set of experimental measurements of continuous quantities (i.e., quantities that form continuum). You can only measure them to within experimental uncertainty.
The quantities taken as exact values must be actually commensurable since each one is only known to a finite number of digits.
But there are two difficulties with treating the values as commensurable
First, if the number of digits is large for the values, then for practical purposes the values are incommensurable.
Sometimes a theory will tell you if the true values are or are NOT commensurable. That knowledge may or may NOT be useful depending on the case and on whether the theory is true.
The upshot is that in many cases you cannot use a set of experimental values as commensurable: they are effectively incommensurable.
Of course, they might be approximately commensurable for some purposes.
Numerical calculation on a computer can do accurate/precise eclipse prediction to the past and future for thousands of years, but probably NOT millions of years ???? due the aforesaid astronomical perturbations in a chaotic evolution.
Say event A happens every 2 time units, event B every 3 time units, and event C every 5 time units, and all the events happen simultaneously a time zero which is the first all-events event.
Because the time periods are commensurable, there will be an exact repeating cycle with the all-events event happening every repeat time of 2*3*5=30 time units which has each of the periods as divisor.
But if event B happens every π time units, then after the first all-events event there will NEVER be another exact all-events event.
To prove this, let's assume the contrary and show that there is a contradiction.
Say the all-events event repeats. Then it must happen at a time 2*f=π*g=5*h, where f, h and g are integers.
But this means that π=2*f/g which implies that π is a rational number.
But π is a known to be an irrational number.
So we have a contradiction. Thus, there is no repeat time for an all-events event.
The set of time periods with event B having time period π is incommensurable.