Wikipedia:WikiProject Mathematics/Mathematical intuition

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Many mathematical articles in Wikipedia (and elsewhere) fall short of giving readers a mathematical intuition for the subject. An example (which has been corrected) is symmetric matrices. In the context of transformation matrices, the symmetry of the matrix when you write it down may seem a seemingly-arbitrary property to a student and doesn't even hold for a symmetric tensor represented in terms of a non-orthogonal basis, but the orthogonality of the eigenvectors is an intuition that can be extended to other mathematical objects. This is an effort to give readers this sort of intuition.

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Complex conjugation – as far as I can tell the primary purpose of complex conjugation is to generalize the Pythagorean theorem to complex numbers so that $|x|=\sqrt{x^2}$ , $|\mathbf{x}|=\sqrt{\sum_i x_i^2}$ , and if $z = z + b i$ then $|z| = |a+bi| = \sqrt{z\overline{z}} =\sqrt{a^2+b^2}$ . To newbies, complex conjugation seems like an arbitrary operation (e.g., why not take the negative of the real part?). The article should say something like "yes it's arbitrary except that it's extremely useful for this reason" (or something else if I'm overlooking something).