I think it makes much more sense to make this statement for the rational numbers: It's the smallest field inside the real numbers that contains the naturals.
So every subset that allows you to do your daily calculations contains the rationals.
I think it makes much more sense to make this statement for the rational numbers: It's the smallest field inside the real numbers that contains the naturals.
So every subset that allows you to do your daily calculations contains the rationals.
They’re a field by construction, and yes, the initial field of characteristic zero, but otherwise don’t arise in any natural way. They’ll be there if you’re studying fields, but exact division by arbitrary integers doesn’t seem to be a very natural property outside the reals. Again, imagine doing any practical computations with rationals and see how far you get before resorting to decimal approximation.
I think teachers lie to children and say that decimals are just another way of representing rationals, rather than the approximation of real numbers that they are (and introduce somewhat silly things like repeating decimals to do it), which makes rationals feel central and natural. That’s certainly how it was for me until I started wondering why no programming languages come with rational number packages.