Oh So Boring…

A while back it was announced that Bing would be using Wolfram|Alpha as their math engine.  Which is cool, because Bing is a pretty interesting engine, and W|A is a very cool one, indeed.

However, there is a bit of an issue with its integration, I believe. Namely: they give different answers!  I know this because I tried out an example in the first link up there, that of 2^2^2^2^2.  It turns out the answer to this is a bit vague, so differing solutions are not unexpected, but still…

Why? Exponentiation is not associative, unlike addition or multiplication (it might satisfy the Lie algebra equivalent of the Jacobi identity, I dunno), so the order matters. A lot.  Thus, while 2+(3+4) = (2+3)+4, and likewise for multiplication, 2^(3^4) !=  (2^3)^4. This being exponentiation, it really doesn’t. The latter is 4096, the former is 2417851639229258349412352. Juuuuuuuuuuuuuuuuuuuuust a bit different. So without parentheses, things are vague.

Now how about the example in the article? We’ll get to that. Let’s start with 3 2′s. (2^2)^2 = 16, 2^(2^2) = 16. Phew safe. So, 2^2^2 is fine.

2^2^2^2? Given that, Bing outputs the “bottom-up” answer: ((2^2)^2)^2 = 256. W|A outputs the “top-down”: 2^(2^(2^2)) = 65536. Ah ha.

The disparity is growing, but not bad. So adding one more ^2 shouldn’t be too horrible, right?

Wrong.

Exponentiation grows quickly so order reaaaaaallly matters.

2^2^2^2^2? As expected from above, Bing should do 256^2=65536 and it does. W|A? Well, it does 2^65536. This is a big number. How big? Well…

20035299304068464649790723515

602557504478254755697514192650

169737108940595563114530895061

308809333481010382343429072631

818229493821188126688695063647…

…8091458852699826141425030123391

That is a 19729 digit number. Big.

Therein lies the difference, but which is right? Hmm. Most mathematicians think of repeated exponentiation as “top-down”. So, my main question is why isn’t Bing using W|A as it says? Or is there some extra option I need to assemble?

Oh well. Just an odd thing I noticed.

So, this afternoon I was looking at a bottle of fuel additive and the instructions confused me. What it said was to only pour the additive into a nearly-empty tank and then fill it up.

My confusion is…why? Why couldn’t I just pour it into a half-filled tank and then fill the tank? It’s still one bottle of additive in one full tank.

My first thought is that maybe the additive isn’t very miscible, so it needs the physical action of the fueling to help mix? But, really, a fuel additive that isn’t miscible in fuel?

I suppose the real answer is that some yahoo had some overfilled gas tank and then added a bunch of flammable hydrocarbons to it and boom or something.

I dunno. Just irked me. Feel like they could just put “add to one full tank of gas”. But, well, we are the culture that came up with “Careful: Contents Are Hot” on a cup of coffee and all.

Saturday has been a busy day…well, busy by my lazy standards.  This morning I went to have my blood let and got a sweet button and pin proclaiming that. I then went to my local library‘s “Friends of” semi-annual booksale.

But the main event: cookie making! Specifically, a recipe my grandma used to make back in the day for Christmas. There is a Don Rockwell picnic tomorrow and this is my contribution. Now, to the picto-recipe after the cut!

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