Transparent Equations

I am currently spending my free time between family visits reading The Man who Loves only Numbers. by Paul Haffman.  This is a biography about Paul Erdos whom I will probably discuss later, but I want to mention a mathematician named Ramanujan whom Haffman discusses near page 83.  I previously heard the story of this mathematician as an undergraduate.  This is a man who stumbled upon a book of mathematical equations designed for exam preparation.  Akin to an SAT or GRE prep book.  It contained many equations but no theory behind why those equations existed.

From this dense book, Ramanujan re-derived those equations and found further equations without any concept of a proof or pictorial representation.  This is completely unfathomable for me.  I usually find myself convinced by an image and then have no need to draft an equation.  I think Ramanujan is more lucky than I.  He found the equations to be obvious.  The only downside was due to his lack of math education; he would discover “new” equations that had already been proved when he simply hadn’t seen them before.  I think the ability to see straight to the heart of the symbols is quite beautiful.

Perhaps this man truly understood what an elegant proof was.  The equations are insightful and complete yet are very brief.  How charming!

Which Calculator to use when?

Counting devices, like the ever famous abacus, have been around since 2400BC -ish. But, what you may not know is that these beauties were first seen in their ’clockwork’ form in 1623. Is that earlier than you expected too? And the pocket calculator was readily available by the mid-1970’s. Now we have several hierarchies of calculators; Basic, scientific, graphing, and the full-on computers. (But I’ll do more computer stuff next post) Most of us can get by with the most basic of calculators, we don‘t need anything fancy. Which type do you use?

My current hypothesis says the type of calculator you use is whatever calculator you were comfortable with at the end of High School. By that point the lines are set, our fingers know the keys, and we can’t imagine using anything else. Case in point: I never liked graphing calculators. So, in High School, learned the methods of how to solve the problem using just my scientific calculator. In my mind it was checking my arithmetic, but I was doing “the real math“. Even now, unless I need a complex computer program to complete my work, I turn to my scientific. To be frank, it’s actually the same calculator the school made me buy in 8th grade when I started pre-algebra. But I don’t think that the choice in calculator is only based on the fate of our high school past.

I think elegance is a big factor in an individual’s choice in calculator. First off, you can big a calculator that has larger or smaller buttons, with any color you could want or have your calculator integrated into other things like your office phone or your computer’s accessories file. These are all to do with social elegance and the use of daily items. I want to talk about the choice between the four kinds of calculators more than the aesthetics of a particular one.

I like the use the simplest calculator that is appropriate for the job. I don’t like graphing calculators because I don’t know how to use half the buttons and there is a lot of functionality on a graphing calculator that I will just never use. (like the entire graphing feature) However, I know how to use every button and function on my scientific and I use them all on a regular basis. But an average guy who just needs to check the math on his monthly bill, or determine a restaurant tip will use a basic calculator. Chances are he will reach for his cell phone. (And isn’t that just strange?– you couldn‘t do that 10 years ago!)

I think that despite the level of complication one prefers, we all still appreciate the little machine that took centuries to perfect and to shove into your Blackberry for those just in case, on the fly, calculations.

Sophistication not required?

An interesting idea occurred to me while I was thinking about some feedback I heard from the last elegance post. When we prove something we can use various levels of mathematical sophistication to do so. So my current question is, does the level of math required to prove the theorem effect the proof’s elegance somehow?

For example, if I need to use calculus to solve a theorem that is taught in high school, then a teacher can not teach the theorem to the high school students. They are relegated to saying something akin to, “you’ll just have to trust me.” However, when you reach that proof it’s quite lovely. I think there may exists proofs that are not lovely as well, but for now lets just consider the ones that are pretty when you get there. And if you are not a advanced math person and you are not sure that you believe these lovely proofs exists; then I guess you’ll just have to trust me.

Look at Pythagoras’s proofs. There are billions of them, and some are really basic and some require really interesting and complex mathematics. (I am using the term ‘billions’ loosely.) I would argue that the basic geometric proofs are more appealing to me because when Pythagoras first worked on this proof, he didn’t have calculus so I think the simple proofs that require almost no calculation are the coolest. These simple proofs give a great example of how the Pythagoreans may have thought about it. However, I can also appreciate the beauty of the more complex proofs. So I would say that this idea is important but doesn’t really change the beauty of the proof, but it is definitely an aspect of proofs that people may have opinions on.

Do you like proofs that are of an equivalent math knowledge to their own or can do appreciate simpler proofs? Does it seem logical for someone without mathematic sophistication to be able to answer the previous question? I’m not sure. That’s why I asked you! Tell me what you think.