the idea that "wanting to do something new" sometimes led to "the need for a new kind of number" is clearly stated in both Morris Kline's book and in several videos in this playlist, both about history of maths.
Has anyone drawn a summarizing graph, to highlight, in maths history, how those "extensions" happened?
As far as I know, the graph should start from an empty node, labelled "no numbers", from which an arc labelled "I wanna do counting" leads to the node "natural numbers". Those are already close to "+" and "*", as indeed counting introduces those operations as "shortcuts".
From the "natural numbers" node, the arc labelled "I wanna 'close' the inverse of '+'" leads to the "integer numbers" node, while the arc labelled "I wanna 'close' the inverse of '*'" leads to the "rational numbers" node.
And so on, up to complex numbers, at least.
Questions:
1 - does such a thing already exist?
2 - would it be useful in teaching people maths?
PS: on question 2: see it as a sort of "you are here", on a tube map: quite helpful, right?