Complex Numbers are the last enhancement of the number system at High School
Some of the earliest mathematics dealt with the concerns of a large agricultural society. With areas, volumes and time, with what became known as Pythagoras Theorem and with quadratics, cubics and the early ideas of trigonometry. These are very much what Complex Numbers deal with, and unify. The timelines here will show when these ideas developed, perhaps by accident of their writing mechanism (on very durable clay) the Babylonians of 4000 years ago give us the earliest known serious mathematical work.
Mathematics is not something magic, fallen from the sky … we (people, mathematicians, scientists, philosophers seeking to understand the world around us and to communicate or teach aspects of that understanding) “just make it up”. As an individual we may have some abstract notion, perhaps a new way to think about and solve a difficult problem and if we do not have a way to describe it then it is very hard to retain it or to expand on it and almost impossible to share it with others. To work on that problem, to understand that aspect of the world, we need new language for the new idea. For some kinds of ideas mathematics is the language we develop and use for this. It is a very old debate indeed as to whether this process is one of invention or discovery … but either way this topic is one which offers an opportunity to help students see mathematics as constructing new structures, new approaches, new language in ways which extend the established mathematics in consistent and powerful ways. Caleb Everett in Numbers, the Making of Us (2017). writes of one of the very earliest inventions in mathematics, and about language and learning and what is innate, what is invented and taught.
For students to get a sense of mathematics as a way of thinking, as creative, rather than a dull set of routines that seem to have little point, a sense of the paths taken, and the people involved would surely help. For this they need to explore, and below are several links sites that provide stories of all sorts.
Complex Numbers
Complex Numbers are a difficult step, as we have argued elsewhere. The first steps were taken in the 1530s, at the very early stages of what might be called modern mathematics ... after the translations from arabic of much of the greek and eastern classics and the very substantial persian and arab contributions, and what must have been considerable eastern knowledge coming into europe via the silk road and into Italy.
Algebraic and geometric mathematics can seem quite distinct, when we teach trigonometry we have quite distinct definitions related to triangles and related to the cartesian plane. We teach about shapes and equations on the number plane, and the same shapes treated as geometrical objects with rules about angles and parallel lines and such. We try to relate these approaches to each other when possible. The 16th century was when these distinct threads came together. What might be called modern mathematics followed.
However it took a long time before Complex Numbers were developed enough to be the link, at first they offered some specific solutions to algebraic problems, much later they gained their geometric and trigonometric interpretations.
History sites
are quite variable, and sometimes alarmingly eurocentric ... so here I offer a selection I like:
The timeline page of www-history.mcs.st-and.ac.uk ... a very rich, old and constantly updated site.
Another good one, more narrative and friendly perhaps, is www.storyofmathematics.com ... with better details and less eurocentricity than many and a nice source list
www.counton.org has a simple graphical timeline with mathematicians starting from classical greek times to explore, with short biographies.
An overview, its 'other' column is a bit US focussed unfortunately: www.math.wichita.edu
More biographies from Thomas, this time calculus focussed (it isna calculis textbook)
a nice graphical presentation, not eurocentric, of calculation and devices from a finance site.