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Thank you. Got it, the ratio of algebraics to transcendentals is nothing. This has been bugging me lately... can you tell me why the transcendental e, signs it

by e3pi 14y ago

Thank you. Got it, the ratio of algebraics to transcendentals is nothing. This has been bugging me lately... can you tell me why the transcendental e, signs it's name in the first significant digits past the decimal point... ...where s3 is root(3), and gg is Euler's gamma/Masceroni's 0.5772... constant: e^(s3 times gg)/(e times s3 times gg) = 1.00000002718? Flirting? Fear of patent trolls, proprietary pride, just dumb luck, what? Note: HN edit coughed when I first used '*' for 'times'

ColinWright 14y ago
I'll look at that and come back to it tomorrow - I don't have time now to look into it. But I will reply with what I know (which may be nothing) Also, I think you you put spaces around the * sign you're OK. (Added in edit: Feels like a coincidence, but strange things do happen, as evidenced in my calculations for the birthday problem, where sqrt(ln(4)) surfaced.)
gjm11 14y ago
Just dumb luck, I think. In general, anything that depends essentially on working in base 10 tends to be just coincidence :-). If you compute some more places, you get 1.0000000271816967980437100574683393 so the digits of e don't continue for very long. The thing that's more surprising on the face of it is that the result is so close to 1. Let's rearrange it a little: exp(s3 gg) ~= exp(1) s3 gg which I prefer to write as exp(s3 gg - 1) ~= 1 + (s3 gg - 1) because the series for exp looks like exp(x) = 1 + x + ... -- so what gives you the value close to 1 is that s3 gg - 1 is rather close to 0, so that the first couple of terms of the series are close to the actual value of exp(); in other words, that gg is close to 1/s3. So far as I know, this too is just a coincidence. (There are lots of simple-ish constants between, say, 0.1 and 10; it's not really that surprising to find a pair that are approximately reciprocals.) The e-like digits that follow amount (after a little simple algebra) to the fact that (s3 gg - 1)^2/2 ~= e/10^8 or, if you prefer, 1 - s3 gg ~= sqrt(2e)/10^4. Which smells very, very coincidental to me. It's a basically random smallish number; why shouldn't it be close to sqrt(2e) / a power of 10?
jerf 14y ago
Somewhat more generally, the space of interesting numbers you can write compactly with traditional notation ("4", "sqrt(2)", "log(7)"), combined with the ways you can compactly write a relation between two numbers (+, /, Ackermann's function), combined with allowing two or three or maybe four numbers (even before they start showing up multiple times as you show there), combined with the number of ways in which a result may be "interesting", results in an inevitable population of combinations of numbers and operations that have an "interesting" result that aren't "really" interesting, or perhaps rather, are only interesting because of the selection function being used ("looks interesting to a human"). You may enjoy http://en.wikipedia.org/wiki/Mathematical_coincidence http://en.wikipedia.org/wiki/Mathematical_coincidence .
- e3pi 14y ago
  "........that have an "interesting" result that aren't "really" interesting, or perhaps rather, are only interesting because of the selection function being used ("looks interesting to a human"). No sir, you are mistaken. My linux netbook's perl regex's determined this is interesting. I myself, a human, merely scan a move-to-front list that suit's this week's curious fancy.

by e3pi 14y ago

ColinWright 14y ago

I'll look at that and come back to it tomorrow - I don't have time now to look into it. But I will reply with what I know (which may be nothing) Also, I think you you put spaces around the * sign you're OK. (Added in edit: Feels like a coincidence, but strange things do happen, as evidenced in my calculations for the birthday problem, where sqrt(ln(4)) surfaced.)

gjm11 14y ago

Just dumb luck, I think. In general, anything that depends essentially on working in base 10 tends to be just coincidence :-). If you compute some more places, you get 1.0000000271816967980437100574683393 so the digits of e don't continue for very long. The thing that's more surprising on the face of it is that the result is so close to 1. Let's rearrange it a little: exp(s3 gg) ~= exp(1) s3 gg which I prefer to write as exp(s3 gg - 1) ~= 1 + (s3 gg - 1) because the series for exp looks like exp(x) = 1 + x + ... -- so what gives you the value close to 1 is that s3 gg - 1 is rather close to 0, so that the first couple of terms of the series are close to the actual value of exp(); in other words, that gg is close to 1/s3. So far as I know, this too is just a coincidence. (There are lots of simple-ish constants between, say, 0.1 and 10; it's not really that surprising to find a pair that are approximately reciprocals.) The e-like digits that follow amount (after a little simple algebra) to the fact that (s3 gg - 1)^2/2 ~= e/10^8 or, if you prefer, 1 - s3 gg ~= sqrt(2e)/10^4. Which smells very, very coincidental to me. It's a basically random smallish number; why shouldn't it be close to sqrt(2e) / a power of 10?

jerf 14y ago

Somewhat more generally, the space of interesting numbers you can write compactly with traditional notation ("4", "sqrt(2)", "log(7)"), combined with the ways you can compactly write a relation between two numbers (+, /, Ackermann's function), combined with allowing two or three or maybe four numbers (even before they start showing up multiple times as you show there), combined with the number of ways in which a result may be "interesting", results in an inevitable population of combinations of numbers and operations that have an "interesting" result that aren't "really" interesting, or perhaps rather, are only interesting because of the selection function being used ("looks interesting to a human"). You may enjoy http://en.wikipedia.org/wiki/Mathematical_coincidence http://en.wikipedia.org/wiki/Mathematical_coincidence .

e3pi 14y ago
"........that have an "interesting" result that aren't "really" interesting, or perhaps rather, are only interesting because of the selection function being used ("looks interesting to a human"). No sir, you are mistaken. My linux netbook's perl regex's determined this is interesting. I myself, a human, merely scan a move-to-front list that suit's this week's curious fancy.