How do we assign rational value to numbers like pi and infinity, numbers that mathematically have no rational ending, if our brains cannot rationalize these numbers? The concept alone is as irrational as the numbers…
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“Irrational” in mathematics doesn’t mean “impossible to rationalize”. It means a number that cannot be written as a *ratio* of two whole numbers (*integers*). For instance, the number 0.5 is a rational number: it’s 1/2.
Pi is an irrational number because it cannot be written this way. There is no pair of numbers *a* and *b* so that a/b = pi. Some pairs get close. For instance, 22/7 = 3.14285…, whereas pi = 3.14159… But you can never get it exactly, and this has been proven (but there’s no ELI5 explanation of this proof).
However, pi as a concept is perfectly reasonable. It is the ratio between the circumference of a circle and its diameter. (Wait, a *ratio*? Doesn’t that mean it *is* rational? Well, no, because to be rational it has to be a ratio of *integers -* not just a ratio of any two numbers. Any circle with an integer diameter (e.g. 4) does not have an integer circumference, and a circle that has an integer circumference does not have an integer diameter.) This ratio is fixed, and its value is what we call “pi”. And we can even compute this value to a high degree of precision – just not with infinite precision.
There are certainly concepts in math that are hard to wrap your head around. Infinity is one of them. However, in addition to the point I made at the start about the definition of “irrational”, you have to accept that just because *you* cannot fathom something, that doesn’t mean other people – specifically mathematicians – are equally uncomfortable with it, or that it is fundamentally impossible to think about logically.
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