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A conjugate, with re, im and arg, as functions #2145
Description
Chapter III of Modenov's Problems in Geometry (#2131) writes every point as a complex number and every statement in z and z̄. Its lines are āz + az̄ = b, a point is on the unit circle when z̄ = 1/z, a slope is (z₂ − z₁)/(z̄₂ − z̄₁), and areas and similarity tests are determinants in z, z̄, 1. Chapter V states the distance, angle, area and collinearity in the same terms. About 150 of the book's rows need a conjugate. Today the parser refuses conjugate by name, so each was written out by hand.
Wanted, as #809's research planned it:
conjugate(z), withconjas an alias, rendered\overline{z}. It is an involution, distributes over+,*,/and integer powers but not across^(1/2), sendsito-iand fixes real numbers.re(z),im(z)andarg(z).
The rules above need no assumptions. What makes the book's method work is rewriting conj(a) as 1/a for a point on the unit circle, and as a for a real symbol. That needs facts attached to a symbol, which is a 3.0 sub-issue of #2131.
Until this lands, #2136 refuses these names, which today are read as products.
Activity
A codomain does not decide the argument
The complex conjugate and the principal argument are functions (#2181)
Rafael-SOWNet commentedon Oct 28, 2026
conjugate(z)(alsoconj) andarg(z)are functions since #2181. The conjugate is carried through the arithmetic and whole powers, not through roots or logarithms, and fixes what is known to be real.argis the principal argument in(-pi, pi], exact on numbers, with no value at0.Still open here:
re(z)andim(z). Making them functions changes howre(x)reads for a variable namedre, whichMissingFunctionNamesRefusedTest.TheTwoLetterNamesAreLeftAsProductskeeps as a product today. That is the next PR, with a breaking entry. Rewritingconj(a)as1/aon the unit circle needs facts on a symbol, which is #2148.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Happypig375 commentedon Oct 28, 2026
I feel that the R in
Reis capitalized, same forIm. Consult different sources for how this is written...Though it's also arguable that functions should be consistently lowercase. Do mathematicians upload this though?
Rafael-SOWNet commentedon Oct 28, 2026
You're right. The sources I checked:
Re zIm zRe[z]Im[z]Re(z)Im(z)re(z)im(z)real(z)imag(z)SymPy is the outlier, and the register in #2182 had followed it. So the next PR prints
Re(z)andIm(z)and draws them upright,\operatorname{Re}\left(z\right), as ISO writes them, rather than with the fraktur\Re. It also acceptsre,im,realandimagon input, since a SymPy or MATLAB user types those.ToSympystill emitssympy.re. The register's rows are updated.argstays lower case. ISO writesarg zfor the argument, SymPy'sargis also the principal value, and Mathematica'sArgis the one capitalised.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
The real and the imaginary part are functions, Re and Im (#2185)
Rafael-SOWNet commentedon Oct 28, 2026
Done:
conjugate/conjandargin #2181,ReandIm(alsore,im,real,imag) in #2185. Rewritingconj(a)as1/aon the unit circle needs facts on a symbol, which is #2148.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura