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I have been learning at a relatively slow pace since last time. I decided to hold off on Calc II until I finish trigonometry in school since Calc II requires a lot more trig than Calc I, and trig requires a lot of memorization, where the drills of school are actually useful. School work has been piling up, I’m trying to catch up on my reading, and I have an ever nagging desire to learn Russian.

 

I like to make excuses, but I haven’t stopped. I’m about half way through this book:

img_20160226_235035

I think it is a good book by a good author, and I plan to read some of his other work. In fact I talked someone who had him as a professor on a forum the other day, it was a surprising coincidence. It is a general math book for everyday audiences but it goes into more detail than books of a similar type. I like it because it teaches some fun math history and fills in many little gaps in my knowledge.

Another thing I wanted to talk about is hyper-operations, something that I found very interesting. It comes from the basic idea that multiplication is repeated addition, and exponentiation is repeated multiplication etc. But what about what comes after exponentiation? Before addition? Well, the answers, respectively, are tetration and succession. They’re very simple questions that have been studied since Euler in the 18th century,(1) and I wish it was brought to my attention sooner. But beyond the concept it gets a little more difficult. Indeed, neither tetration nor its successive operations are very well studied. There are many open problems, and this is due to the fact that they require so much computing power. It is not known if \pi ^{\pi ^{\pi ^{\pi }}} is an integer or not. It seems obvious that it would be irrational, but we just can’t know because the number is just so huge that it’s impossible to look at.(2) Hyper-operations are very exciting because it’s likely they’ll be an important area of new study in the future.

My resources: Tetration.org, A neat forum document about notation, since there isn’t really any standards yet, Wiki articles for a quick overview: one and two

Another nice thing I found is this lady’s YouTube channel. She’s going through one of the oldest and most important books in mathematics, Euclid’s Elements, piece-by-piece in nice little videos. I watched quite a few of them and while Geometry is not one of my favorite areas of math it’s still very important and interesting. Constructions can be very fun.

regular_pentagon_using_carlyle_circle
How to construct a regular pentagon By Aldoaldoz (Own work) [CC BY-SA 3.o (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)%5D, via Wikimedia Commons
 And finally, I’m going to try to study category theory after I publish this post, the theory is sometimes called “a sound basis for mathematics,” but I think everyone knows math is way too weird to have a basis.

(1)Euler, L. “De serie Lambertina Plurimisque eius insignibus proprietatibus.” Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921.

(2)Charles (http://math.stackexchange.com/users/1778/charles), Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?, URL (version: 2016-02-11): http://math.stackexchange.com/q/853463

 

 

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