Cool Math Stuff

Anyway, in my Saturday morning isolation boredom, I learned (on my own) a couple of neat math oddities I hadn’t already known. I’ll start with a few similar ones I already knew, and end up with something I have proven to myself but that @Ether might have made a math quiz.

Known for years (I love φ):
x = (√5 - 1)/2 = 0.61803398874989484820458683436564…
φ = 1/x = 1.61803398874989484820458683436564…
φ² = 2.61803398874989484820458683436564…

x = √2 - 1 = 0.4142135623730950488016887242097‬…
1/x = 2.4142135623730950488016887242097‬…

Learned today (though it feeds into a tile pattern I used for a bathroom floor about a year before the guy who pulled me into FRC (@gixxy) was born!):
2 * atan(1/3) = atan(3/4)
2 * atan(1/2) = atan(4/3)
atan(1/3) + atan(1/2) = atan(1)

And also learned today. I have proven to myself, but I’d like someone else to explain it. I can’t give “credit” like the old CD, but I will like the first clear statement why the digits in the second case match those in the first.

x = 5 - √8 = 2.1715728752538099023966225515806‬…
x² = 4.715728752538099023966225515806‬…

yup, the same set of digits, one place down.

Also, I invite other cool/weird math stuff in this topic!

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x² = (5-√8)^2
x² = 25 - 10√8 + 8
x² = 33 - 10√8 --> Eqn 1


x = 5-√8 = 2.1715728…
10x = 21.715728
10x - 17 = 4.715728
10(5-√8) -17 = 4.715728
50 - 10√8 -17 = 4.715728
33 - 10√8 = 4.715728 --> Eqn 2


Combine Eqn 1 & Eqn 2
x² = 4.715728

/Edit: Formatting is awful, but I’m not in the mood to put it into LaTex

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That’s the key right there! x² = 10x - 17, so the digits repeat one place up.

It's * ok, you_{re} * a * wond \Sigma rful * human *anywa\gamma!

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OK, here’s another one. Write a program (pick your language) to efficiently* generate Pythagorean triples of the form:

a² + (a+1)² = c²

* Efficiently: the program should do more work rendering output in decimal than in calculating the values. My gawk script is significantly fewer than 200 non-comment characters, generates 21 properly ordered triples in well under 0.1 seconds on a 6 year old PC, and terminates when it recognizes it is beyond gawk’s resolution limits. It could go on all day with a while (0=0) if gawk supported arbitrary resolution. It uses two variables, four assignment statements, one while statement, and one printf statement. You don’t have to get that far down, but at least close. Oh, and explain why it works! (The explanation should be longer than the functional parts of the program!)

The first few triples are:
0² + 1² = 1² †
3² + 4² = 5²
20² + 21² = 29²
119² + 120² = 169²
696² + 697² = 985²
4059² + 4060² = 5741²

The last gawk can calculate (and the first one excel chokes on) is:
1235216565974040² + 1235216565974041² = 1746860020068409²

† OK, that one isn’t strictly Pythagorean.

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I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Less seriously, I hope this limerick will be okay for this thread:Math-Limerick
I have another one, but apparently I left it in my other pants pocket.

…and apparently the spoiler tags don’t work? And the details tag either?

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