Very interesting audition, but i miss a piece of paper in it, or a blackboard. First: reals. physicists don't need reals, they can leave only with rationals, and approach any physical magnitude with an interval of rationals. The lenght of the interval is the uncertainity.
Second, negative numbers. I don't agree with the discussion, I don't found difficulty expalining it. Are those numbers in the back, or at the bottom. Like celsius degrees or altitudes. More complicated is the zero number. Notice's readears saying "there are not temperature: zero degree!" are proverbial.
With complexes, here the program came more ... complex. Problem is with modern algebra books, that introduce complexes without discussions. In the oldest you can read how carefully are they treated. Firts, they are couple of sorted pairs of reals (a,b). Or, in more technical words: the elements of the cartesian product of R by R. Then old books go to define the operations with the new entities, saying always: "well, just because the complex (a,0) can be identity with the real a, let's try to conserve the most important properties of the old operations: +,-,*,/,exp,sin, etc". But always remember you that (a,0) isn't the same thing that a. (a,0) is a complex and a is a real! And remember you at a lot of times that a+b*i it's only a notation for the complex (a,b). Each definition have a full previous discussion about if it is "correct" in the sense of "usefull", but clarifiying that probably there are a lot of other correct definitions. So, introduce the operations as extensions of the similar real operations. New books looks like the formal operations are intrinsic in the new field, but they aren't.
Complexes have a very natural representation and implementation as points in the plane. Then, aren't that artificial as the history reveal. Old mathematician was impressed with the complexes properties just because they are not all translates to reals. But the plane representation must to be enough to say that they have a physical representation. So, using complexes to electromagnetics fields or other areas, must to be well accepted. For example, a book of complexes that have a very nice chapters interpreting plane fluid flows with complexes is Pollard (isn't? can't remember the author, and have not the book).
The more beatifull formula in math is, without a doubt, e^i + 1 = 0 which correlates the five more importants numbers in the history. It could be a very good starting point.
You will hear in the programme that one feature of these new numbers was that mathematicians had to ensure they were self consistent and didn't break existing rules, which is why I ended up having to go into so much detail for those that get rather perniketty. ! ! Tom G is actually very good at helping clarify these points.
Yes, I read some posts from Tom explaining what about angles and dimensions, and are very clear and are very true and real.
For most folk though, being able to use simple formula accurately (with real numbers!), will help a great deal. Trigonometric functions need angles.
Finally, I'm not sure which word you meant to use at the end of point 5 "I'm not sure if they are biunivoques." Could you suggest an alternative description (or a wikipedia article 😉 ?
http://es.wikipedia.org/wiki/Correspondencia_biun%C3%ADvoca
Sorry, can't find it in english. In the complex plane, problems about this are very ... again, complexes. You can transform apples and oranges into fruits, but can reverse the transformation: given 8 fruits, how many apples and oranges are it? When you define a complex frequencies, you do it from reals, ins't?; to ensure that you are doing the tings well, check that the correspondence is "biunívoca", mathematically equivalente to "iif" (if and only if). To have a magnitude also, a way is to prove that you can check when two are equals and adding any two. Tha'ts the case of a way to define the mass after having definitions for forces and accelerations.
Kind regards. Alvaro.
