Textbooks in mathematics tend to contain very sloppy history. For example, one textbook on abstract algebra claims that mathematicians had disputes over negative and complex numbers because "the idea of a system of numeration which included negative numbers was far too abstract" and because "they couldn't find concrete examples or applications" (A Book of Abstract Algebra, 1990, p.2).
It is quite obvious that the disputes over negative and complex numbers were not due to lack of intelligence on the mathematicians' part nor to the lack of "concrete" (whatever that means) examples or applications. In fact, it was the same mathematicians who contributed most to the development of methods using negative and complex numbers, and these same thinkers had no problem finding "concrete" examples, chiefly that of debt and of squares.
Other books deal with the history of negative and complex numbers differently. In Negative Math: How Mathematical Rules Can Be Positively Bent, Martinez presents a much more nuanced history behind these disputes. Martinez shows, for example, that one chief concern with negative numbers is its lack of symmetry with positive numbers. While it is true that part of these problems were due to the types of intuitions which the mathematicians replied upon, it is perhaps more accurate to say that the chief problem was how to integrate and incorporate negative and complex numbers into the system of numbers as it was generally accepted at the time.
The problem, therefore, was not the lack of concrete applications, but rather the lack of a clear and concise way to relate negative and complex numbers to various other operations already accepted in mathematics.
This way of looking at the history of negative and complex numbers allows us to avoid certain prejudices which seem to color certain textbook account of the same history. According to the latter, mathematics prior to the 20th century was a jumble of confusion, and while it may hold some "historical" (meant in a derogative sense) interest for us, it does not present us with anything more interesting or worth grappling with. However, the fact of the matter is that this history is essential in understanding why negative and complex numbers are treated in the way they are today, at least at the university undergraduate algebra course level. As Martinez observes, many of those old problems disappear not because they are simply discarded, but because they are set aside unsolved and then are not taken up by others later on.
