Busy Beavers

Thanks to Thomas for this exceptionally interesting essay entitled, “Who Can Name the Bigger Number?”. There is a lot of ground covered in the essay, so I definitely suggest sitting down with a cup o’ joe and reading the entire piece. My favorite part was the discussion surrounding Turing Machines, which, I hate to ruin it for you, have *everything* to do with naming the biggest number. Most people, albeit this is coming from someone with a degree in mathematics, can fathom the biggest number being an exponent raised to another exponent. So this would look like, 99^9^9. In Aaronson’s essay, he starts by describing these exponential numbers, something we can all grasp. Next he builds off this concept to introduce the Ackermann Sequence, ie: . At that point you are thinking, wow … those numbers are pretty huge. But wait … there is more! There are bigger numbers! Diving even further, he uses Turing Machines to describe the notation for the biggest numbers – Busy Beaver Numbers.

While reading the article, I felt the same way I did junior year in high school. I remember very clearly, sitting in an algebra class where my teacher, Mr. Carlton, introduced the imaginary number i. This imaginary number i was never mentioned until my 11th year of school?! I put the “?!” in there, because I remember being very surprised, and mildly annoyed, that there were numbers out there that I did not know about. Of course, I realized that this did not mean that these numbers did not exist. But, how could I have not heard about them before?

Funny, because as this feeling came over me and settled within, I got to the end of the essay where Aaronson suggests:

Could early intervention mitigate our big number phobia? What if second-grade math teachers took an hour-long hiatus from stultifying busywork to ask their students, “How do you name really, really big numbers?” And then told them about exponentials and stacked exponentials, tetration and the Ackermann sequence, maybe even Busy Beavers: a cornucopia of numbers vaster than any they’d ever conceived, and ideas stretching the bounds of their imaginations.

Certainly an interesting idea. Plant the seed early and let it grow. Maybe I wouldn’t feel so “cheated” if I had heard about the imaginary number i in the 2nd grade! It would definitely give people a different perspective on large, seemingly incomprehensible numbers. The earlier we grasp these concepts the more time we have to move beyond them, making the problems of today small ripples in the pond of tomorrow.

I’ll leave you with this, one of my favorite Dilbert cartoons:

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