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Simplify 1+Sqrt[2]I-complicated[1,Sqrt[2]] – Mathematica Stack Exchange retort

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Simplify 1+Sqrt[2]I-complicated[1,Sqrt[2]] – Mathematica Stack Exchange

Many what’s-going-on questions actually simply rotate out to breathe folks wanting a moor for his or her code. They simply do not win why it does not labor. Given the unostentatious code, perhaps a proof of how sophisticated works is definitely being sought.

There’s good-looking mighty the an identical drawback with 1/Sqrt[2] - Rational[1, Sqrt[2]] and 1/2 - Rational[1., 2]. sophisticated[x, y] just isn’t a precise expression except x and y are numbers, which is mirrored in how sophisticated[1, Sqrt[2]] is typeset within the Front stop. (Sqrt[2] just isn’t a quantity both. It’s a numeric expression. Compare Number[Sqrt[2]] and NumericQ[Sqrt[2]].) Generally, it is best to assemble an advanced quantity with I, not sophisticated, considerably affection your first two phrases. However, 1 + Sqrt[2] I can not breathe represented by a sophisticated quantity in Mathematica. It can solely breathe represented by a amalgam, complicated-numeric expression.

For sophisticated[x, y] to breathe precise, x and y can breathe any numbers, Integer, precise, Rational or aircraft sophisticated. However, sophisticated elements will breathe routinely simplified:

sophisticated[Complex[1, 2], 1]
(*  1 + 3 I  *)

Another undocumented quirk is that if one sever is MachinePrecision, the each elements will breathe made MachinePrecision (the docs array solely numbers with each elements entered with machine precision):

sophisticated[1., 2]
sophisticated[1., 2`500]
sophisticated[1.`6, 2]   (* might maintain mixed-precision non-MachinePrecision elements *)

(*
  1. + 2. I
  1. + 2. I
  1.00000 + 2 I
*)

Under the everything-is-an-expression philosophy, sophisticated[1, Sqrt[2]] is handled as an expression, however not one with internally outlined semantics apparently. So it’s not mixed with different numeric expression as if it represented an advanced quantity.

A precise quantity passes the NumberQ check (in addition to AtomQ, too):

sophisticated[1, Sqrt[2]] // NumberQ
sophisticated[1, 2] // NumberQ

(*
  faux
  correct
*)

P.S.
For Rational[x, y] to breathe precise, x and y every maintain to breathe an Integer with y nonzero, although Rational[1, 0] will charge to ComplexInfinity.

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