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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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