Swapping digits of pi and e
June 27, 2017 5:47 PM Subscribe
Over at Mathoverflow, user Erin Carmody asked the question "If I exchange infinitely many digits of π and e, are the two resulting numbers transcendental?" The answers given were great, and PBS Infinite Studios made this video about it.
I... I don't even understand the question...
posted by tivalasvegas at 6:36 PM on June 27, 2017 [6 favorites]
posted by tivalasvegas at 6:36 PM on June 27, 2017 [6 favorites]
Thanks for this. Just linked the video the the high school math team I coach (now they'll probably all want me to explain it when we get back from break.)
I'd never heard of normal numbers (used in one of the arguments) before either. Super neat! We know they're there (uncountably so) but can't seem to find (m)any examples.
posted by Wulfhere at 6:38 PM on June 27, 2017
I'd never heard of normal numbers (used in one of the arguments) before either. Super neat! We know they're there (uncountably so) but can't seem to find (m)any examples.
posted by Wulfhere at 6:38 PM on June 27, 2017
Check out the PBS video at about 3:00 for a concrete example of what they're thinking about with the exchanging thing.
Other terms they use: Irrational numbers have decimal expansions that don't repeat. A transcendental number is a special kind of irrational. There are transcendental-irrational and algebraic-irrational numbers. Algebraic-irrationals are solutions to polynomials (sqrt(2) is the solution to x2=2 ). Transcendental-irrationals are not solutions to a polynomial in that way.
posted by Wulfhere at 6:46 PM on June 27, 2017 [2 favorites]
Other terms they use: Irrational numbers have decimal expansions that don't repeat. A transcendental number is a special kind of irrational. There are transcendental-irrational and algebraic-irrational numbers. Algebraic-irrationals are solutions to polynomials (sqrt(2) is the solution to x2=2 ). Transcendental-irrationals are not solutions to a polynomial in that way.
posted by Wulfhere at 6:46 PM on June 27, 2017 [2 favorites]
The PBS video lost me when she said "Notice that if pi and e are only different at finitely numbered digits, then pi - e is rational." I don't see why that is the case.
posted by straight at 6:52 PM on June 27, 2017 [1 favorite]
posted by straight at 6:52 PM on June 27, 2017 [1 favorite]
straight: as a simple example, consider pi, which is irrational, and pi+1 which is also irrational. Their difference is rational (|1|) and they differ by a finite number of digits.
posted by idiopath at 6:57 PM on June 27, 2017 [5 favorites]
posted by idiopath at 6:57 PM on June 27, 2017 [5 favorites]
The PBS video lost me when she said "Notice that if pi and e are only different at finitely numbered digits, then pi - e is rational." I don't see why that is the case.
If they differ at finitely many digits, there exists some digit after which all of the digits are the same. That means that if you subtract one from the other, after that digit place you're subtracting identical digits from each other (resulting in a 0 at that place), and so your new number will terminate at the last differing digit.
posted by NMcCoy at 6:59 PM on June 27, 2017 [19 favorites]
If they differ at finitely many digits, there exists some digit after which all of the digits are the same. That means that if you subtract one from the other, after that digit place you're subtracting identical digits from each other (resulting in a 0 at that place), and so your new number will terminate at the last differing digit.
posted by NMcCoy at 6:59 PM on June 27, 2017 [19 favorites]
When I rule the world I shall call smart people before me daily to wax lyrical about their area of smartness. I will have no idea what they're talking about but damn watching smart people being smart and getting excited about being smart pushes all of my buttons.
posted by obiwanwasabi at 7:09 PM on June 27, 2017 [11 favorites]
posted by obiwanwasabi at 7:09 PM on June 27, 2017 [11 favorites]
Also I came to this after spending an hour plus watching the Hydraulic Press Channel. Surely there is some connection.
posted by Bringer Tom at 7:31 PM on June 27, 2017 [2 favorites]
posted by Bringer Tom at 7:31 PM on June 27, 2017 [2 favorites]
straight: Imagine they differ by only a finite number of digits; say the last such digit is at position n. Then pi - e must be rational because its last (nonzero) digit is at position n (which means we can express it as a fraction of two numbers of (n+1) digits).
posted by axiom at 8:42 PM on June 27, 2017
posted by axiom at 8:42 PM on June 27, 2017
The person who wrote the 2nd and 3rd answers to this question is Terence Tao, a true mathematical genius and a heck of a nice guy.
posted by Harvey Kilobit at 9:19 PM on June 27, 2017 [5 favorites]
posted by Harvey Kilobit at 9:19 PM on June 27, 2017 [5 favorites]
When I rule the world I shall call smart people before me daily to wax lyrical about their area of smartness. I will have no idea what they're talking about but damn watching smart people being smart and getting excited about being smart pushes all of my buttons.
For the time being, before that happens, you can always listen to In Our Time.
posted by Pyrogenesis at 1:12 AM on June 28, 2017 [5 favorites]
For the time being, before that happens, you can always listen to In Our Time.
posted by Pyrogenesis at 1:12 AM on June 28, 2017 [5 favorites]
When I rule the world I shall call smart people before me daily to wax lyrical about their area of smartness.
So basically TED talks, without the market-driven woo?
posted by rokusan at 1:16 AM on June 28, 2017 [1 favorite]
So basically TED talks, without the market-driven woo?
posted by rokusan at 1:16 AM on June 28, 2017 [1 favorite]
So basically TED talks, without the market-driven woo?
This is also basically In Our Time!
mathoverflow.net (as opposed to math.stackoverflow.com) is really interesting, because it's specifically intended for working researchers in mathematics. So you see a lot of recognizable names posting there regularly. CS Theory stack exchange is similar for that field. I'm not aware of anything like this in academic research previously, especially not with this much traction.
posted by vogon_poet at 8:12 AM on June 28, 2017
This is also basically In Our Time!
mathoverflow.net (as opposed to math.stackoverflow.com) is really interesting, because it's specifically intended for working researchers in mathematics. So you see a lot of recognizable names posting there regularly. CS Theory stack exchange is similar for that field. I'm not aware of anything like this in academic research previously, especially not with this much traction.
posted by vogon_poet at 8:12 AM on June 28, 2017
Well this is a wonderful reminder of why I didn't become a math major.
posted by ckape at 3:47 PM on June 28, 2017 [1 favorite]
posted by ckape at 3:47 PM on June 28, 2017 [1 favorite]
I bet Matt Hoverflow really regrets that choice of online persona given how often people mispronounce it
posted by DoctorFedora at 4:58 PM on June 28, 2017 [5 favorites]
posted by DoctorFedora at 4:58 PM on June 28, 2017 [5 favorites]
I don't understand this but can the math be used to calculate the volume or goodness of this drink?
posted by Wordshore at 5:32 AM on June 29, 2017
posted by Wordshore at 5:32 AM on June 29, 2017
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You might say it's giving me a math-owie. YEEEEAAAHHHH.
posted by fluffy battle kitten at 6:24 PM on June 27, 2017 [9 favorites]