Oh So Boring…

Our long national nightmare is over. I have new glasses!  Presenting…

I hope everyone likes them…because they are staying! (Read, pricey.)

If you are anything like me…well, no one is like me. I’m unique. But, if you want to be like me, here is an activity that you can do if you are stuck inside during a blizzard like I am now or want to pretend to be. That activity? Looking at old menus!

I know, I know, not that exciting, but, man, is it fascinating! The object of my focus this afternoon is the LA Public Library’s online Menu Collection. Just give it a whirl and I dare you not to be sucked in.  Marvel at what ten bucks would buy you 30 years ago compared to today.

I’m not the first to find this site, but I’m damn sure not going to be the last!

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.