I missed the part where complex numbers are convincingly “number” or convincingly “real”. Maybe you’re making a different argument than I thought. Just because

I missed the part where complex numbers are convincingly “number” or convincingly “real”. Maybe you’re making a different argument than I thought. Just because the full philosophical understanding and development of irrational numbers is recent doesn’t (to me) result in complex numbers being as natural as a scalar quantity. Vectors can be real numbers with a physical analog, and arguably a more natural algebra, and are more widely used than complex numbers. Does that have any bearing on what you were trying to express? > 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.