Math brainlet here with another schizo revelation: Real numbers are just a shadow of Complex numbers.
It's pretty clear from the discourse here that it is very tricky to construct reals from rationals, the threads pop up constantly, and the discussion is always present. This got me thinking a bit about numbers and in particular about the intuitive understanding, which, I believe, is always reachable if you think hard enough.
In particular I started thinking about sqrt(2) and why this number is so different from the numbers it's constructed with. Why should it be that when you take a triangle with a side lengths of 1 and 1, simplest numbers possible, and try to calculate the length of hypotenuse, you get a number that is outside of the realm of rationals? What is it exactly that makes this new type of number creep in? This led me to a sudden realization: the sides of the triangle, while having the same length, are not identical: one of them, besides the length, encodes the information of ...rotation. If we assume that the segment (0,1) lies on a number line, then to construct a triangle we have to construct another segment that by definition does NOT lie on the same number line. In our case, to get the point we need, we simply rotate it by 90°, or in another words, multiply by i. This explains the apparent weirdness of getting sqrt(2) out of simple rationals: it is somehow glossed over, that you need 2 fucking dimensions and thus complex numbers in order to construct simple triangles and in fact all of the Euclidean geometry. I thus have a hunch that all the problems of constructing reals might potentially disappear if you introduce imaginary numbers first, although it's nothing more than a hunch based on intuition, which is probably incorrect, as i'm not a mathematician.
It's pretty clear from the discourse here that it is very tricky to construct reals from rationals, the threads pop up constantly, and the discussion is always present. This got me thinking a bit about numbers and in particular about the intuitive understanding, which, I believe, is always reachable if you think hard enough.
In particular I started thinking about sqrt(2) and why this number is so different from the numbers it's constructed with. Why should it be that when you take a triangle with a side lengths of 1 and 1, simplest numbers possible, and try to calculate the length of hypotenuse, you get a number that is outside of the realm of rationals? What is it exactly that makes this new type of number creep in? This led me to a sudden realization: the sides of the triangle, while having the same length, are not identical: one of them, besides the length, encodes the information of ...rotation. If we assume that the segment (0,1) lies on a number line, then to construct a triangle we have to construct another segment that by definition does NOT lie on the same number line. In our case, to get the point we need, we simply rotate it by 90°, or in another words, multiply by i. This explains the apparent weirdness of getting sqrt(2) out of simple rationals: it is somehow glossed over, that you need 2 fucking dimensions and thus complex numbers in order to construct simple triangles and in fact all of the Euclidean geometry. I thus have a hunch that all the problems of constructing reals might potentially disappear if you introduce imaginary numbers first, although it's nothing more than a hunch based on intuition, which is probably incorrect, as i'm not a mathematician.
