Some Technical Tidbits

Gödel’s Lost Letter and P=NP 2018-03-12

Tid-bit: delicacy, dainty, snack, nibble, appetizer, hors d’oeuvre, goody, dipper, finger food

Adam Engst is the publisher of the site TidBITS. This is a site dedicated to technical insights about all aspects of Apple machines: from nano to servers.

Today I want to talk about mathematical tidbits not Apple devices, but look at the above site for information about Apple stuff of all kinds. By the way:

“tasty tidbits” is a small and particularly interesting item of gossip or information. Origin mid 17th century (as tyd bit, tid-bit ): from dialect tid «tender» (of unknown origin).

TidBITS has been running since 1990, when Apple made only a certain kind of bulky device that went by the name “personal computer.” The Internet did not exist yet, so the site was distributed via a platform called HyperCard, which combined publishing, presentation, and database functions.

Some Tidbits

${\bullet}$ A Math Result: An Interesting Pattern

$\displaystyle 111,111,111 \times 111,111,111 = 12,345,678,\mathbf{9}87,654,321.$

${\bullet}$ Another Math Result: A Bad Joke If you write out ${\pi}$ to two decimal places, backwards it spells “pie”:

${\bullet}$ The New York Times: Downgrades The Size of The Human Genome

In a recent article in the Times it was reported that a certain sequence effort found that a bacteria had a large genome. It was actually so large that it was a thousand times larger than our human genome. The article then added that the human genome is about 3.5 million bases. Wrong. It is actually about 3.5 billion bases. I reported this typo to the Times by the way.

${\bullet}$ Senator Jack Reed: On CNN Today

I happen to tune in CNN’s broadcast of the Senate hearing on security today. There Reed asked security experts whether or not the US has a coherent plan for quantum computing. They basically said that they could not say much in an open hearing. I do find it remarkable that quantum computing is mentioned in a Senate hearing.

Another Tidbit

${\bullet}$ Gifted: The Movie

This is a movie about a young child, Mary, who is a math savant, and whether or not she should be placed in a special school. I recently watched the movie with my wife Kathryn Farley. We enjoyed the movie although it had a pretty simple plot. Indeed see this:

Colin Covert of the Star Tribune gave the film 3/4 stars, saying, “Sure, it’s a simple, straightforward film, but sometimes that’s all you need as long as its heart is true.” Richard Roeper gave the film 4 out of 4 stars and said, “Gifted isn’t the best or most sophisticated or most original film of the year so far—but it just might be my favorite.”

Mary’s mother had been a promising mathematician, who worked on the famous Navier-Stokes problem, before taking her own life when Mary was six months old. Indeed the plot of the movie hinges on the mother actually solving the problem. Of course, in the movie we get no hint how she may have done this. One of the key parts of the movie concerns when Mary is being examined by a math professor—this is to determine if she really is gifted. Mary is asked to evaluate the definite integral that is written on a chalk board in a lecture hall. The integral is:

$\displaystyle \int_{-\infty}^{\infty} e^{x^{2}} dx.$

She cannot figure it out and leaves the exam without answering the question. But finally she explains to her grandmother that it is a trick question. The above integral of course is equal to infinity. She comes back to the lecture hall and explains that she was taught to never correct adults. Now she corrects the problem with a minus sign and solves it: $\displaystyle \int_{-\infty}^{\infty} e^{-x^{2}} dx = \sqrt{\pi}$

I must point out that when the problem came up I said to my wife that the first integral was incorrect. I thought perhaps the movie had an error in it. But no it was a trick question—very neat. No doubt this was based on advice from the math advisors—who include Terence Tao.

Open Problems

People have said that a mathematician is someone who thinks that

$\displaystyle \int_{-\infty}^{\infty} e^{-x^{2}} dx = \sqrt{\pi}$

is obvious. Well, if you multiply two copies of the equation, one with “ ${y}$ ” in place of “ ${x}$ ,” you get an integral with ${-(x^2 + y^2)}$ in the exponent on the left-hand side. Using ${r^2 = x^2 + y^2}$ converts it to polar coordinates, and then you can intuit why what you get equals ${\pi}$ . I wonder if there is a similar test for complexity theorists? Any suggestions?