# Why does “Turn! Turn! Turn!” equal 241217.524881?

If you search for “Turn! Turn! Turn!” on Google, then the second result is this YouTube video of The Byrds performing the Pete Seeger song of that name. But the first result is Google’s internal calculator displaying “241217.524881”. With a bit of experimentation, it appears that this number is a numerical approximation to $$Γ(2π+1)22π,\frac{\Gamma(2\pi+1)^2}{2 \pi},$$
where $$Γ\Gamma$$ represents the Euler gamma function.

I sort of understand why Google is interpreting “Turn” to mean $$2π2\pi$$, and the exclamation mark to mean $$x!:=Γ(x+1)x! := \Gamma(x+1)$$, as this is a relatively common (although not universal) choice of interpolation of the factorial function to the real numbers. But in that case, I would expect Google to interpret “Turn! Turn! Turn!” to represent $$Γ(2π+1)3≈18658774329\Gamma(2\pi+1)^3 \approx 18\, 658\, 774\, 329$$ instead of the expression above. Why isn’t it?

A possible partial solution: if you search “Turn! Turn” then you get the expected result $$7735.248≈Γ(2π+1)2π7735.248 \approx \Gamma(2\pi+1) 2\pi$$. But if you search “Turn! Turn!” then you do not get the expected result $$Γ(2π+1)2≈1515614\Gamma(2\pi+1)^2 \approx 1\, 515\, 614$$. Instead, you get 195.936, which appears to be the numerical approximation of $$Γ(2π+1)/(2π)\Gamma(2\pi+1)/(2\pi)$$. Moreover, Google reparses the input as “Turn ! (Turn !)”. To me, this suggests that it’s interpreting the second explamation mark as a factorial symbol, but the first exclamation mark to mean $$a!b:=b/aa ! b := b/a$$, i.e. division but with the usual order of arguments reversed. This explains the orginal result if Google is interpreting “Turn! Turn! Turn!” with the first exclamation mark representing reversed division (with a lower order-of-operations precedence than multiplication) but the second two exclamation marks representing factorial:
$$2\pi “!” (((2\pi)!)\ ((2\pi)!)) = \frac{\Gamma(2\pi+1)^2}{2\pi}.2\pi "!" (((2\pi)!)\ ((2\pi)!)) = \frac{\Gamma(2\pi+1)^2}{2\pi}.$$

Is this notation $$a!b := b/aa!b := b/a$$ standard? I’ve never seen it before. Can anyone explain how Google is parsing this string?

(This is one of those awkward questions where the (unknown) solution determines whether or not the question is on-topic for Math Stack Exchange. If the solution does indeed come down to unusual math notation, as I suspect, then the question is on-topic for Math SE. But if the resolution is just some black-box machine learning magic, then maybe the question isn’t on topic. I’m not quite sure what one does in this kind of situation.)

It appears that Google search uses the following syntax for conversions between different units or currencies: target_unit! <expression> (Though I can’t find any documentation for this). For example, searching for cm! 1m + 5 yards yields 557.2 centimeters. Interpreting a turn to be a dimensionless quantity equal to $$2\pi2\pi$$, we can try to make sense of the results:
• Turn ! (Turn !) means “calculate $$\text{Turn}!\text{Turn}!$$ and express it in the unit turn”, which comes out to $$\frac{\Gamma(2\pi +1)}{2\pi}\text{ Turn}\frac{\Gamma(2\pi +1)}{2\pi}\text{ Turn}$$, or approximately $$195.936116 \text{ Turn}195.936116 \text{ Turn}$$ (as observed by Mikael).
• Turn! Turn! Turn! is interpreted as (Turn ! (Turn !)) * (Turn !), where we make the calculation from the previous bullet point and are left with a product of something in the unit turn and a number, which Google evaluates by throwing away the unit information in the first term and calculating the product of the two numbers. Hence, we end up with $$\frac{\Gamma(2\pi+1)}{2\pi}\cdot \Gamma(2\pi+1)\frac{\Gamma(2\pi+1)}{2\pi}\cdot \Gamma(2\pi+1)$$ as the result.