Real numbers very definitely were not known to humans until the 1700s, except in the vaguest sense that rational numbers seemed to have gaps in them that are the roots of certain polynomials. It was only Newton and Leibniz who really had to start thinking about what continuity and limits mean, and what they did manage was very hand-wavy, just enough to sort of convince themselves that analysis works.
Fully rigorous construction of the reals was not achieved until 1871, by Cantor, although Cauchy's work earlier in the 19th century, particularly the introduction of 𝛅–𝛆 reasoning, was certainly vital. It is important to note that by that time, complex numbers had already been introduced, and indeed were of great help in figuring out the reals, exactly because reals themselves were incomplete.
But really, I digress.
> Does it make real number sense to have 2+3i dollars or apples?
Does it make sense to have 𝛑 dollars or apples? Tangible quantities are integral or rational.
Every extension of number systems was inspired by the fact that some reasonable calculations did not seem to have an answer within the set of what was at the time understood as "numbers".
Zero was introduced to reason about the question of what remains if you have n sheep and give all of them away.
Negative numbers were introduced to reason about what happens if you want to buy a sheep that costs n coins and you only have n-1 coins, ie. debt.
Rational numbers were introduced to reason about dividing a deceased father's sheep and coins among several brothers, or a year or a day into sub-periods.
Real numbers were introduced to… Well, at this point it becomes more abstract, because the only justification for the reals was the philosophical desire to talk about the length of the hypothenuse of an ideal right isosceles triangle or the length of the circumference of an ideal circle.
Complex numbers were introduced because not all polynomials of degree n > 1 seemed to have real roots at all, and others only had some number < n. It was found that this, again, was only because reals were in some way incomplete, and the roots were all there if only we could have more numbers in some sense. Today we say that the field of complex numbers is algebraically closed [1] and reals are not. This is a tremendously important property and the very reason that ℂ is seen as mathematicians as the real deal and ℝ only interesting as a subfield of ℂ.
[1] https://en.wikipedia.org/wiki/Algebraically_closed_field
People invented mathematics to be able to explain things happening in nature, and refined their mathematical devices even further to even explain the nature of mathematics itself. Numbers did not exist before people invented them as sets for different tools to calculate stuff, as you have so eloquently put. To argue that "Real numbers very definitely were not known to humans until the 1700s" means nothing else than these numbers were not invented yet. Um, what is your point exactly?
> Zero was introduced to reason about the question of what remains if you have n sheep and give all of them away.
Hmmm, why do you think this? I have a doubt. We already know that language had ways to express having none of something long before zero appeared in math.
> Does it make sense to have 𝛑 dollars or apples? Tangible quantities are integral or rational.
We aren’t talking about what’s tangible, that wasn’t your original claim. Yes absolutely it makes complete sense to have pi apples, it’s a completely natural concept one can easily imagine, whether it’s tangible or not. It does not make any sense to have 2+3i apples. Not only is that not tangible, it also can’t be approximated and has no physical analog. I can easily obtain 3.14 apples or dollars, a rational approximation to pi. As you know, the rationals are a subset of the reals, so for the purposes of this discussion I’m even fine with talking about only finite reals or rationals. If you’re trying to make a point about an infinite irrational number or that I can’t have exactly pi apples, then I suggest that’s perhaps a straw man argument irrelevant to the question of whether 2+3i apples makes “number” sense. Pi apples is easy to imagine, easy to approximate, and easy to draw a picture of, while 2+3i apples is not.
Go on. Argue that please.