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→Worst math article ever: please make more specific, it's hard to reply to a post when that post covers many different section's worth of discussion |
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:::::I would second what {{u|Deacon Vorbis}} says here. Specifically high school is the correct level for us to be aiming for, and the "{{math|''a'' + ''bi''}} definition" is better for our purposes here than the "(RxR, +, *) definition". And by the way I too was introduced to complex numbers in high school. [[User:Paul August|Paul August]] [[User_talk:Paul August|☎]] 16:05, 20 September 2020 (UTC)
:::::I also second this: High Schoolers do not learn Complex numbers as an extension of the Real numbers into the plane. They learn it first and foremost through the introduction of i as the square root of negative one, then are later shown that such a treatment gives rise to a convenient treatment of the plane. These two ideas are both important and should be given treatment, but extensive formaliation of these two pretty simple concepts right at the beginning of the article only serves to confuse the reader. '''<sub>[[User:IntegralPython| Integral Python]]</sub><sup>''[[User talk:IntegralPython| click here to argue with me]]''</sup>''' 17:49, 20 September 2020 (UTC)
:::::: Dear Deacon Vorbis, thanks for your reply.
:::::: Unfortunately, what you write shows that you do not understand the points that I am so desperately trying to make.
:::::: Obviously, writing {{math|(''a'', ''b'')}} is only superficially different from writing {{math|''a'' + ''bi''}} as strings. But this is not the reason for my proposal. The "(IR^2, +, *) definition" builds on well known objects for people with high school level knowledge (the plane, and +,* in IR). The statement
:::::: '''The complex numbers is the Euclidean plane together with a special multiplication for points in that plane.'''
:::::: is a very simple, complete, accurate description and certainly easily understandable for every layman. This is concrete and doesn't need any postulates. The imaginary unit is simply the point i=(0,1) in the plane. That is concrete and straightforward. The {{math|''a'' + ''bi''}} definition first introduces an indeterminate (means: not exactly known, established, or defined ''';-)''' ) variable/symbol i (in order to define polynomials) which then miraculously (by definition, though not exactly known) is a fixed number and solves i^2=-1. That's a lot of concepts in the package. Explaining why one can actually do that takes time. This is also considered 1700s math (see previous post/reply).
:::::: The multiplication formula can be motivated in a clear manner in about two sentences and 2 formulas (3 lines): ''Suppose we have a field which contains IR and a number i satisfying.... Then we would have the following multiplication rule....'' Now, we implement the latter insight as follows: ---- Note that this is a thought experiment for motivation and very different from postulating i. ---- Thus, any opaqueness can be removed easily.
:::::: '''Falsehood:''' It's the exact same definition, but with different notation. ---- One definition needs "new" postulates about i, the more modern one does not. Only the algebraic handling of real coefficients is the same.
:::::: '''Falsehood:''' ordered pair approach not used. ---- See that the ordered pair approach '''is''' used in practice in intro texts: https://people.math.gatech.edu/~cain/winter99/ch1.pdf [1]. This is the optimal approach to define and explain C. With the computation
:::::: (a, b) = (a, 0)+(0,b) = a (1,0)+b(0,1)= a*1+b*i=a+bi __(*)
:::::: a, b in the plane, i=(0,1) 2nd base vector, one then switches to the a+bi notation knowing that i=(0,1). Note that also the other reference which I posted uses the "(IR^2, +, *) definition". Modern books on analysis assume that the reader knows C and the above computation (*). And they proceed from there.
:::::: '''Falsehood:''' For a modern definition, the obvious choice is <math>\mathbb{R}[x]/(x^2 + 1).</math> ---- <math>\mathbb{R}[x]/(x^2 + 1)</math> is a beautiful math construction. However, it is '''NOT'''', '''ABSOLUTELY NOT''' an obvious choice. It can be a choice for an Algebra class, an exercise after introducing the quotient construction for rings. As outlined above in detail, <math>\mathbb{R}[x]/(x^2 + 1)</math> needs a substantial amount of concepts to be explained (3<= hours lecture time with all details), while the "(IR^2, +, *) definition" needs no new concepts beyond high school math (1.5> hours lecture time with all details). Observe that the above modern reference [1] uses the "(IR^2, +, *) definition". Note that also the other reference which I posted uses the "(IR^2, +, *) definition".
:::::: [[User:LMSchmitt|LMSchmitt]] 20:41, 20 September 2020 (UTC)
== Worst math article ever ==
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