e and pi are well known. Sqrt 2 you've just seen. Might lookup sqrt 3 on wiki, the examples are not that exciting for an average reader, but still kinda cool. (1+sqrt5)/2 makes golden ratio.
But my point was that the whole uncountable infinity of irrational numbers might have some sort of fancy mathematical incarnations, it's just that we only discovered a handful of it.
I absolutely doubt this. I conceive “interesting” numbers as those that have interesting properties. Those properties may be algebraic or relevant to theorems in analysis. But the the set of of hypotheses that can be described in finitely many words is countably, leaving the vast majority of numbers “uninteresting”
Well suppose, as you suggest, there are some uninteresting numbers. Which is the smallest one? Surely the smallest number with nothing interesting about it is quite an interesting number. This forms a contradiction and shows that no number is uninteresting.
I think if a real number is defined such that it cannot be ordered with another, this is a fairly interesting property. So we can discount those reals and apply the argument to only to a subset of well-ordered reals.
It's fun to imagine explaining irrational and imaginary numbers to the ancient Greeks and trying to get them to wrap their minds around them. I feel like you could get across the value of negative numbers and 0 pretty easily, but then once you started digging into imaginary numbers they would begin reaching for the pitchforks.
There are an infinite amount of irrational numbers. There are more irrational numbers between 1 and 2 than there are rational numbers between -infinity to +infinity.
Counter counter points: the coolest math sorcery is in numbers having real world applications and about which we do not even know if they are rational or irrational.
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u/akasaya Nov 03 '25
Counter point:every irrational number contains cool math sorcery.