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swamp-thing submitted a new PF Insights post
Irrationality for Dummies
Continue reading the Original PF Insights Post.
Irrationality for Dummies
Continue reading the Original PF Insights Post.
If you take a proper rational ##q = frac{m}{n}## where ##m, n## have no common factors, then ##q^2 = frac{m^2}{n^2}## is clearly a proper rational. Where would the common factors of ##m^2, n^2## come from? (To be rigorous, appeal to the fundamental theorem of arithmetic and unique prime factorisations).
In any case, proper rationals square to proper rationals, never to whole numbers. Hence, only whole numbers and irrationals can square to whole numbers.
Isn't that it in a nutshell?
PeroK : proper rationals square to proper rationals
Thank you, JorisL.Haven't checked the maths thoroughly but I really like your style of writing.
10/10 would read your future insights.
Hi micromass,In your Insight you posted some kind of "walk" according to ##(1+ k/n)^2##. You said (and proved) that while doing this walk, you'll never land on an integer. Here's a question though: do you get arbitrarily close to an integer? For example, do you get closer than ##0.000001## to some integer?
If you take a proper rational ##q = \frac{m}{n}## where ##m, n## have no common factors, then ##q^2 = \frac{m^2}{n^2}## is clearly a proper rational. Where would the common factors of ##m^2, n^2## come from? (To be rigorous, appeal to the fundamental theorem of arithmetic and unique prime factorisations).
In any case, proper rationals square to proper rationals, never to whole numbers. Hence, only whole numbers and irrationals can square to whole numbers.
Isn't that it in a nutshell?
Ancient Greek mathematicians freaked out when they discovered that the square root of 2 is not rational. Like Swamp Thing, they were not dummies and realized that the existence of irrational numbers is a fact that is remarkable, deep, and a little scary.
Hippasus, according to Wikipedia, which says the story may be just legend.Was someone really murdered over it? Aristarchus? or a similar sounding name (apologies for the historical inaccuracy)