Is there any way to use head of Real and Integer altogether?

Question

Is there any way to use the head of Integer and Real together like below? The below is not right, I know that, it is just for showing my thought.

f[x_(Integer||Real)] := x^2

The reason I ask this is because I need a function that receives an argument which is either integer or real.

Answer 1

To match your literal request you need Alternatives rather than Or.
Either x : (_Integer | _Real) or x_Integer | x_Real will work.

Following what Szabolcs and "Guess who it is" wrote you might define a realQ like so:

realQ = NumericQ[#] && Im[#] == 0 &;

f[x_?realQ] := x^2

f /@ {1, Pi, 1.3, 2/3, x^2, 7.1 - 2.8 I}

{1, π^2, 1.69, 4/9, f[x^2], f[7.1 - 2.8 I]}

Of note for those who are comfortable using undocumented functions:

Internal`RealValuedNumericQ /@ {1, Pi, 1.3, 2/3, x^2, 7.1 - 2.8 I}

{True, True, True, True, False, False}

There is also Internal`RealValuedNumberQ which passes only explicit numbers:

Internal`RealValuedNumberQ /@ {1, Pi, 1.3, 2/3, x^2, 7.1 - 2.8 I}

{True, False, True, True, False, False}

Answer 2

I frequently write functions that take one or more arguments which I limit to those quantities that mathematicians call real numbers. In Mathematica that means any quantity satisfying NumericQ excepting complex numbers. To facilitate writing such functions, I define a pattern

validNum = Except[_Complex, _?NumericQ];

This pattern is used like so:

f[x : validNum] := x^2

Update

As Guesswhoitis points out the above is not fool-proof. A more robust version is

validNum = Except[z_ /; Head[N[z]] === Complex, _?NumericQ];
f /@ {1, 1/2, .5, Pi, 1 + I/2, 1. + .5 I, Sqrt[-1], (-1)^(2/3), E + I Pi}

{1, 1/4, 0.25, Pi^2, f[1 + I/2], f[1. + 0.5*I], f[I], f[(-1)^(2/3)], f[E + I*Pi]}