I was recently reminded that the following "functions" are settable, and I was surprised (even though I've seen this before). So I thought that it is valuable to share this information.

There are three critical properties of symbolic constants:

• The Constant attribute, which is used by Dt

• The ability to tell that it is a numeric expression (NumericQ[Pi] === True).

• The ability to compute the value to any precision.

We can implement all three by direct assignment. Let the symbol sqrt2 represent $\sqrt{2}$:

SetAttributes[sqrt2, Constant]
NumericQ[sqrt2] = True;
N[sqrt2, prec___] := N[Sqrt[2], prec]

Note that it wasn't necessary to unprotect any symbols.

Now sqrt2 doesn't evaluate:

sqrt2
(* sqrt2 *)

But it can be computed numerically:

N[sqrt2, 100]
(* 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573 *)

It is numeric:

NumericQ[sqrt2]
(* True *)

Which means that many functions work well with it without the need for additional definitions:

Positive[sqrt2]
(* True *)

Im[sqrt2]
(* 0 *)

Integrate[Exp[-sqrt2 x], {x, 0, Infinity}]
(* 1/sqrt2 *)

Integrate wouldn't give the same result with an arbitrary symbolic parameter:

Integrate[Exp[-s x], {x, 0, Infinity}]
(* ConditionalExpression[1/s, Re[s] > 0] *)

And of course it works with Dt

Dt[sqrt2 x, x]
(* sqrt2 *)

Did I miss anything? If so, let me know!

I was recently reminded that the following "functions" are settable, and I was surprised (even though I've seen this before). So I thought that it is valuable to share this information.

There are three critical properties of symbolic constants:

• The Constant attribute, which is used by Dt

• The ability to tell that it is a numeric expression (NumericQ[Pi] === True).

• The ability to compute the value to any precision.

We can implement all three by direct assignment. Let the symbol sqrt2 represent $\sqrt{2}$:

SetAttributes[sqrt2, Constant]
NumericQ[sqrt2] = True;
N[sqrt2, prec___] := N[Sqrt[2], prec]

Note that it wasn't necessary to unprotect any symbols.

Now sqrt2 doesn't evaluate:

sqrt2
(* sqrt2 *)

But it can be computed numerically:

N[sqrt2, 100]
(* 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573 *)

It is numeric:

NumericQ[sqrt2]
(* True *)

Which means that many functions work well with it without the need for additional definitions:

Positive[sqrt2]
(* True *)

Im[sqrt2]
(* 0 *)

Integrate[Exp[-sqrt2 x], {x, 0, Infinity}]
(* 1/sqrt2 *)

Integrate wouldn't give the same result with an arbitrary symbolic parameter:

Integrate[Exp[-s x], {x, 0, Infinity}]
(* ConditionalExpression[1/s, Re[s] > 0] *)

And of course it works with Dt

Dt[sqrt2 x, x]
(* sqrt2 *)

Update: How do we clear such a symbol? Like this:

ClearAll[sqrt2]
NumericQ[sqrt2] =.

I do not know where the NumericQ setting is stored. It doesn't seem to be associated either with NumericQ or sqrt2. Also, NumericQ is clearly set up to handle this usage and will not accept invalid values.

Did I miss anything? If so, let me know!