Assumptions, Abs, etc.
in [Mathematica]
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From: S. B. Gray on 6 Jun 2010 06:41 Can anyone tell me why the following results appear? Clear[xc1, xc2] Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 < 0, xc2 < 0}] Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 < 0, xc2 > 0}] Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 > 0, xc2 < 0}] Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 > 0, xc2 > 0}] (xc1 - xc2)^2 (xc1 - xc2)^2 (xc1 - xc2)^2 (xc1 - xc2)^2 Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc2 < 0] Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc2 > 0] Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc1 > 0] Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc1 < 0] Abs[xc1 - xc2]^2 Abs[xc1 - xc2]^2 Abs[xc1 - xc2]^2 Abs[xc1 - xc2]^2 Abs[xc1 - xc2]^2 Simplify[Abs[xc1 - xc2]^2 - (xc1 - xc2)^2, Assumptions -> {xc1 -> 0}] Simplify[Abs[xc1 - xc2]^2 - (xc1 - xc2)^2, Assumptions -> {xc2 -> 0}] -xc2^2 + Abs[xc2]^2 -xc2^2 + Abs[xc2]^2 (* the next lines say that xc1, xc2 are Reals **) Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 \[Element] \!\(\* ButtonBox["Reals", BaseStyle->"Link", ButtonData->"paclet:ref/Reals"]\), xc2 \[Element] \!\(\* ButtonBox["Reals", BaseStyle->"Link", ButtonData->"paclet:ref/Reals"]\)}] Abs[xc1 - xc2]^2 I would expect all 9 answers to be (xc1 - xc2)^2. Wrong? Steve Gray
From: Sjoerd C. de Vries on 7 Jun 2010 08:05 As long as one of the variables can be Complex you cannot simplify Abs[xc1 - xc2]^2 any further. If for instance xc1 = 1+i and xc2 = 0 then Abs[xc1 - xc2]^2 = 2, but (xc1 - xc2)^2 = 2i. Cheers -- Sjoerd On Jun 6, 12:41 pm, "S. B. Gray" <stev...(a)ROADRUNNER.COM> wrote: > Can anyone tell me why the following results appear? > > Clear[xc1, xc2] > Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 < 0, xc2 < 0}] > Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 < 0, xc2 > 0}] > Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 > 0, xc2 < 0}] > Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 > 0, xc2 > 0}] > (xc1 - xc2)^2 > (xc1 - xc2)^2 > (xc1 - xc2)^2 > (xc1 - xc2)^2 > > Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc2 < 0] > Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc2 > 0] > Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc1 > 0] > Simplify[Abs[xc1 - xc2]^2, Assumptions -> xc1 < 0] > Abs[xc1 - xc2]^2 > Abs[xc1 - xc2]^2 > Abs[xc1 - xc2]^2 > Abs[xc1 - xc2]^2 > Abs[xc1 - xc2]^2 > > Simplify[Abs[xc1 - xc2]^2 - (xc1 - xc2)^2, Assumptions -> {xc1 -> 0}] > Simplify[Abs[xc1 - xc2]^2 - (xc1 - xc2)^2, Assumptions -> {xc2 -> 0}] > -xc2^2 + Abs[xc2]^2 > -xc2^2 + Abs[xc2]^2 > > (* the next lines say that xc1, xc2 are Reals **) > > Simplify[Abs[xc1 - xc2]^2, Assumptions -> {xc1 \[Element] \!\(\* > ButtonBox["Reals", > BaseStyle->"Link", > ButtonData->"paclet:ref/Reals"]\), xc2 \[Element] \!\(\* > ButtonBox["Reals", > BaseStyle->"Link", > ButtonData->"paclet:ref/Reals"]\)}] > Abs[xc1 - xc2]^2 > > I would expect all 9 answers to be (xc1 - xc2)^2. Wrong? > > Steve Gray
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