Rational numbers trace

Question

When I Trace a rational number, I get this quite complicated output:

1/2 // Trace

{{1/2,1/2},1/2,1/2}

which under the hood looks like

{{HoldForm[1/2], HoldForm[1/2]}, HoldForm[1/2], HoldForm[1/2]}

At the same time, 1/2 is represented as Rational:

1/2 // FullForm

Rational[1, 2]

But if we do:

Rational[1, 2] // Trace

we get very simple output, which would be expected:

{Rational[1,2],1/2}

with underlying output being

{HoldForm[Rational[1, 2]], HoldForm[1/2]}

So, the question is: why MMA does so many steps when evaluating a rational number entered in a "pure" form and doesn't do it when Rational is used explicitly?

Answer 1

Using FullForm should help illuminate things:

Hold[1/2]//FullForm

Hold[Times[1,Power[2,-1]]]

We see that 1/2 is a bit more complicated than you might have expected. To evaluate this, you need to first evaluate Power[2, -1], and then you need to multiply that output with 1. So, let's see what happens in the Trace:

1/2//Trace
% /. h_HoldForm -> FullForm[h]

{{1/2,1/2},1/2,1/2}

{{HoldForm[Power[2,-1]],HoldForm[Rational[1,2]]},HoldForm[Times[1,Rational[1,2]]],HoldForm[Rational[1,2]]}

The Trace shows that the evaluation proceeded exactly as expected based on the FullForm of the input.