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One correction. The root of rational and irrational is the word "ratio". Rational numbers are ratios of integers. Irrational numbers are not ratios of intege
by btilly 7y ago
One correction. The root of rational and irrational is the word "ratio". Rational numbers are ratios of integers. Irrational numbers are not ratios of integers.
It is a linguistic coincidence that we wound up with words that has another reasonable meaning.
- jonny_eh 7y agoIt's amazing that I went through high school and multiple calculus courses in college without ever learning that. Thanks!
- robocat 7y agoAlso a person that is rational, and a person that is rationalising, are opposite meanings: rationalisations are deeply irrational.
- cshimmin 7y agoThe author notes this in the article.
- smallnamespace 7y ago> linguistic coincidence It's not a coincidence. The root of both ratio and rational come from latin ratio, gerund of reri, 'to calculate, to reckon, to think'. To the ancients, just like us, computation, thinking, and reasoning are linked. Irrational numbers were literally numbers that 'could not be reckoned' in the normal sense. You can even go further with this analogy. The halting function is not computable because there is no finite procedure that tells whether any program halts. But this objection to infinity is also what the ancient Greeks had to continuity (Zeno's paradoxes) and irrational numbers, as there is no finite procedure that results in an exact answer. In fact, there is no bounded algorithm that can distinguish sqrt(2) on all possible numeric inputs.
- xamuel 7y agoNonsense, the digits of sqrt(2) most definitely are computable and there are bounded algorithms that can check, given inputs n and k, whether or not n is the kth digit of sqrt(2).
- smallnamespace 7y agoI said 'all inputs' for a reason. If I give you an infinite sequence that is sqrt(2) you cannot in finite time say it actually is. Now in practice this isn't a problem, but it reveals the problem with finitism.
- moefh 7y agoThat doesn't seem to be a fair argument, you can also do that with rational numbers: if I give you an infinite sequence that is exactly 1/3, can you in finite time say it actually is? If you choose to represent numbers as infinite streams of digits, then obviously you can't compare them for equality in a finite amount of time. The issue is: is it possible to use real numbers in a way that sidesteps this problem? In general, it's impossible -- most real numbers are uncomputable! -- but for some useful subsets (beyond the rational numbers) it's possible.
- smallnamespace 7y agoThere's a fair point there, which is that some computations may look hard simply because we picked a difficult representation, but I'm not sure that applies here. There's finite, exact representation for 1/3 in whole numbers, namely itself, but as far as I know (?) there isn't one for sqrt(2) unless, say, your choice of representation is the root of some polynomial. Is there a bounded procedure that shows whether any two polynomials represent the same set of roots? Getting back to the original etymological question, my point is that irrational in the sense of 'can't be reckoned, computed' is very close to how we would see it. The Greeks simply had a different conception of computation, one grounded in finitism and constructing things geometrically.
- messe 7y ago> Is there a bounded procedure that shows whether any two polynomials represent the same set of roots? Yes. Two polynomials share a root if their resultant vanishes.
- 7y ago
- yen223 7y agoI’ve always found it funny that Pi is an irrational number, when Pi is literally defined as a ratio of two numbers
- saagarjha 7y agoTwo numbers, at least one of which is irrational.
- quickthrower2 7y agoBut PI/PI is rational!
- saagarjha 7y agoThat’s not equal to pi.