Talk:Complex number: Difference between revisions

Revision as of 20:41, 20 September 2020

Revision as of 21:03, 20 September 2020

Line 373:

:::::: (a, b) = (a, 0)+(0,b) = a (1,0)+b(0,1)= a*1+b*i=a+bi __(*)

:::::: a, b in the planeIR, 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".