Tangent of a Googol

There's an interesting math problem (with a fun backstory) in Feynman's autobiography "Surely You're Joking, Mr. Feynman!" The question is simply to calculate

tan(10¹⁰⁰).

I'm considering the problem in radians, but it's also challenging in degrees.
The number 10¹⁰⁰  can also be written as

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

This is a very big number that is also known as "a googol," which is pronounced the same as "Google."
Usual methods of calculating the tangent ("tan") function don't work well on very large numbers. If you Google "tan(googol)" or "tan(10^100)" you actually get different results (both wrong). You do get the correct result if you use Wolfram Alpha.
One way to get the right answer is to find a multiple of π that is very near a googol (call it nπ for some integer n). Then, because tangent is a π-periodic function,

tan(10¹⁰⁰)=tan(10¹⁰⁰-nπ),

and, since 10¹⁰⁰-nπ is a fairly small number, its tangent can be calculated accurately by normal methods.

Finding the number n and, say, 9 digits of 10¹⁰⁰-nπ requires knowing 109 digits of π, but this is easy to look up.

After some Googling, I couldn't find anyone that actually stated a decimal answer to this question, which is my main purpose here. It's about 0.401231962.

After the Ball in "My Fair Lady"

Before you go judging Professor Higgins and Colonel Pickering for ignoring Eliza after the ball: did you pay enough attention to her to notice that she said nothing (except a quick "Thank you" to Pickering) between the Ascot opening race and her moment alone after the ball? I have to wonder if the filmmakers did this deliberately.

Trump's Hot Mic Moment

This weekend, I've read about the 2005 recording of Trump's graphic, lecherous, swaggering conversation about adultery and sexual assault. People are reacting strongly, as though this recording has given us new information about Trump. We already knew he was a foul-mouthed, sexist braggart. It's not surprising that he says these things in private. It's only remarkable that we haven't caught him at it before.
Also, it took me a while to realize it, but Trump is insulting men as he defends himself. He has been trying to dismiss his behavior as "locker-room banter," as though men talk like this all the time. It's possible that I've heard talk like this in a locker-room, but I didn't pay much attention to middle-school delinquents. And to say that he has more respect for women than anyone? How degrading to the men he pretends to be above. Sorry Mr. Trump, but there are men who have more respect for women than you do—including all the decent ones.

Crushed

This blog is the closest thing I have to an influence on a culture and government that sometimes disgusts me. I write for my benefit. My mother might read this because she's subscribed, but I doubt anyone else will. If you've read this far, you know that I don't even write well, which compounds my problem. Like most people, many things about my life are dictated to me and I must accept them or be crushed.
I am at the mercy of forces too powerful for me to affect, defy, or ignore. Have you ever really pushed against a large boulder? It's heavy, hard and totally indifferent to the living consciousness seeking to influence it. If some other force were to set it in motion and you got in it's way, it would crush you. It would never know or care that a spark of life—which had thought itself important and unique—just became so much goo.
The immediate cause of these thoughts is that, due to a change of employer and not understanding that "open enrollment" means "the only time health insurance can be purchased," I don't currently have health insurance. This probably means I am stupid, but more importantly, I am young and healthy and likely to remain so. A couple of years ago, not buying insurance would have simply meant that I saved a few hundred dollars. Due to the Affordable Care Act, I am still uninsured but will be paying a fine. I am a student with a small income, and I won't get to keep that money.
I am experiencing a tiny sample of a life-crushing scenario faced by the truly poor. The "individual mandate" of the Affordable Care Act takes away the option to save money by not buying health insurance. Those who did not maintain insurance because they could not afford it are now worse off: they must now pay a fine with the money they risked their health to save. Some of the decisions made by my "representative" government offend me to my core.
I have a personal philosophy that I believe should rule my life. Instead, I will be ruled by other people's philosophies. Most of the people who will impose rules on my life will never hear me. They are a boulder.
I have a soul. How I feel about what happens to me should be regarded as an "end"—others should view it as a severe negative consequence that they have arranged my life circumstances to conflict with my values.  I believe in cooperation by common consent and noninterference by unaffected parties. I have the moral right to make a choice regarding the cash value of my health, and it is no one's concern but mine if I choose to risk death for the sake of another resource. I do it every day by crossing the street, which is also none of their business.
That's what I would say if I had a voice.

How Awesome is Picard?

Captain Jean-Luc Picard of "Star Trek: the Next Generation" is pretty amazing. I've collected a few facts illustrating why I think he may be among the greatest men who will ever live.

• Captain Picard has his own holiday.
• Captain Picard met the devil and wasn't impressed. He sued her. And won.
• Captain Picard was mistaken for God. Twice.
• Captain Picard lived a lifetime in 25 minutes.
• Captain Picard can make stating the time sound impressive.
• Captain Picard regularly lectures a god on morality.
• Captain Picard threw a saddle at a pirate.
• Captain Picard went back in time in order to get stabbed through the heart. Again. And it still tickled.
• Captain Picard hired Whoopi Goldberg to tend his bar.
• Captain Picard mind-melded with a vulcan to help him with self-control.
• And he did it while overcoming a prim French name.

Chess in "Donald in Mathmagic Land"

In the Disney short "Donald in Mathmagic Land," Donald watches a game of chess. This is supposed to show how math is used in an everyday setting.
The game is shown without players, pieces moving on their own through stop motion. The game is played rapidly, lasting only about 24 seconds. In order to finish the game in such a short time, some moves begin to take place before the previous ones have finished.
The game begins as the Blackmar Gambit, but quickly devolves into a shallow novice game designed to remove pieces as quickly as possible rather than exhibit strong play. The simple discovered check in 16...Nc5+ seems to have come as a complete surprise to white. It is a flurry of movement to entertain and impress the uninitiated, but it is useless for serious study. I record the details here for the curious.

1. d4 d5
2. e4 dxe4
3. d5 c6
4. dxc6 Qxd1+
5. Kxd1 bxc6
6. Nc3 Nf6
7. f3 exf3
8. gxf3 e5
9. Bh3 Bxh3
10. Nxh3 Bb4
11. Bd2 Bxc3
12. Bxc3 e4
13. Bxf6 gxf6
14. fxe4 Nd7
15. Nf4 O-O-O
16. c3 Nc5+
17. Kc2 Nxe4
18. Rad1 Rxd1
19. Rxd1 Nf2
20. Rd6 Kc7
21. Rxf6 Rf8
22. Nd5+ cxd5
23. Rxf2 Re8
24. Rxf7+ Kc6
25. Rxa7 Re2+
26. Kb3 Rxh2
27. a4 Kb6
28. Rd7 Rd2
29. Rxh7 d4
30. cxd4 Rxd4
31. Rh6+ Ka5
32. Rc6 Rd3+
33. Ka2 Kxa4
34. Rc4+ Kb5
35. b3 Rd5
36. Ka3 Rd8
37. Rb4+ Kc5
38. Ra4 Kc6
39. b4 Kb5
40. Ra5+ Kc6
41. Ka4 Kb7
42. b5 Ra8
43. Rxa8 Kxa8
44. Ka5 Kb7
45. b6 Kb8
46. Ka6 Kc8
47. Ka7 Kd8
48. b7 Ke8
49. b8=Q+ Ke7
50. Qe5+ Kf8
51. Qe6 Kg7
52. Kb7 Kf8
53. Kc7 Kg7
54. Kd7 Kf8
55. Qe7+ Kg8
56. Qf6 Kh7
57. Ke7 Kg8
58. Qe6+ Kh8
59. Kf7 Kh7
60. Qg6+ Kh8
61. Qg7#
The white queen is shown capturing the red king.

Why Forward Must Be Up

When signs need to indicate a forward direction, they typically use an upward pointing arrow. "Forward" is actually perpendicular to the face of the sign, so an up- or down-arrow is equally wrong. However, roads leading away from you rise in your field of vision as they get farther away, so an upward vector is the more sensible choice.
I've wondered for years about the apparently arbitrary "up equals forward" convention, and I just today realized this satisfying explanation.