Is e Normal?(mathpages.com)
mathpages.com
Is e Normal?
https://www.mathpages.com/home/kmath519/kmath519.htm
23 comments
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> It’s widely believed that the two most naturally occurring transcendental numbers, p and e...
Not really germane to the larger topic of the essay, but I don’t think it’s true that those two numbers occur naturally very frequently except that they make it easier to calculate systems with cycles or exponential growth. It’s almost certainly exceedingly rare that someone would measure exactly pi or e in nature, and even in calculations they’re generally multiplied by some constant or other.
Not really germane to the larger topic of the essay, but I don’t think it’s true that those two numbers occur naturally very frequently except that they make it easier to calculate systems with cycles or exponential growth. It’s almost certainly exceedingly rare that someone would measure exactly pi or e in nature, and even in calculations they’re generally multiplied by some constant or other.
The above comment is misinformed, but in no way obnoxious.
There was no reason to downvote it.
There was no reason to downvote it.
Downvotes are allowed to express disagreemnt.
Actually there are no guidelines about voting whatsoever.
However arguably it's a bit rude to ding people simply for being (mildly) misinformed.
And the guidelines do ask us not to be rude.
However arguably it's a bit rude to ding people simply for being (mildly) misinformed.
And the guidelines do ask us not to be rude.
According to pg: https://news.ycombinator.com/item?id=392347
I would say the ratio of a circle circumference to its diameter is naturally occurring.
Yes, that is π in Euclidean space, but that is rare (¿nonexistent?) according to our current understanding of nature.
Are there circles in nature?
There are plenty of waves in nature, which use trigonometric functions to define them which share a relationship with e and π.
e and π themselves cannot exist in nature (per our current understanding) because our current reality isn't infinitely precise so any circle that exists won't be a perfect circle.
e and π themselves cannot exist in nature (per our current understanding) because our current reality isn't infinitely precise so any circle that exists won't be a perfect circle.
(Looks at sun, closes eyes.) Yes.
That's an ellipse.
FWIW, pi also appears in the formula for the area of an ellipse.
Really? You can tell that by looking at it with the naked eye and no scientific instruments? Are such instruments 'natural'? Wherever did we get the idea of 'circle' from then?
>You can tell that by looking at it with the naked eye and no scientific instruments?
How is that relevant?
We didn't ask if someone can consider or see with naked eye something that looks like a circle, but whether circles do exist in nature.
And we have the "scientific instruments" to check up on the sun (which has a very small, but existing, buldge in the equator, and which also has an uneven exterior with flares and all kinds of motion).
>Wherever did we get the idea of 'circle' from then?
From objects resembling circles, like cut tree trunks, the moon at night, and so on.
Same place we got the idea of line, while there are no lines in nature (for one, because everything we see line-like, is just a line segment: lines extend to infinity).
How is that relevant?
We didn't ask if someone can consider or see with naked eye something that looks like a circle, but whether circles do exist in nature.
And we have the "scientific instruments" to check up on the sun (which has a very small, but existing, buldge in the equator, and which also has an uneven exterior with flares and all kinds of motion).
>Wherever did we get the idea of 'circle' from then?
From objects resembling circles, like cut tree trunks, the moon at night, and so on.
Same place we got the idea of line, while there are no lines in nature (for one, because everything we see line-like, is just a line segment: lines extend to infinity).
So, and really just for the sake of argument, if we could look at the sun with one of its poles at the center, it would still be elliptical?
Or if we could somehow see cross-sections of atoms are elliptical and not circular too?
If so, how do you know that?
Or if we could somehow see cross-sections of atoms are elliptical and not circular too?
If so, how do you know that?
>So, and really just for the sake of argument, if we could look at the sun with one of its poles at the center, it would still be elliptical?
Missed the part where the sun's surface also isn't seamless, but an"uneven exterior with flares and all kinds of motion"?
>Or if we could somehow see cross-sections of atoms are elliptical and not circular too?
I seriously doubt it, as from what we know atoms don't look like little balls and aren't solid objects themselves -- they consist of electrons, protons, and neutrons, in orbits, and are "fuzzy". They're drawn as little balls for illustration purposes.
Missed the part where the sun's surface also isn't seamless, but an"uneven exterior with flares and all kinds of motion"?
>Or if we could somehow see cross-sections of atoms are elliptical and not circular too?
I seriously doubt it, as from what we know atoms don't look like little balls and aren't solid objects themselves -- they consist of electrons, protons, and neutrons, in orbits, and are "fuzzy". They're drawn as little balls for illustration purposes.
Transcendental numbers are usually characterized by a limiting iterative process. We don't say that a sunflower has exactly the Golden Ratio's worth of seeds, either; instead, we say that the Golden Ratio is inherent in the pattern of sunflower seeds. [0]
e is most famously in compound interest, or other continuous processes. An example in nature might be the radioactive decay of certain isotopes, often indicated by half-life. In particle physics, e manifests as the limiting process of septillions of particles operating independently with each other.
pi is more abstract. It might well only show up in idealized conic sections, which mean that all the natural conic sections we see, such as shadows cast by light sources, are imprecise. (This is because, extremely technically speaking, shadows are not physical!) There are still limiting processes for pi, like taking a large chunk of matter and chilling it into a constantly-curved smooth ball, which will generate an approximate sphere, but they are much harder to perform because gravity is fickle and atom boundaries are fuzzy. (That said, we can still make such artifacts when we want to, at great cost. [1])
In both cases, our understanding of the the number's construction is good enough that we no longer need any physical props to approximate their value, but instead can compute the values abstractly and then use them to refine our physical measurements.
[0] https://en.wikipedia.org/wiki/Spiral#In_nature
[1] https://www.youtube.com/watch?v=JKCBeDeVxkg
e is most famously in compound interest, or other continuous processes. An example in nature might be the radioactive decay of certain isotopes, often indicated by half-life. In particle physics, e manifests as the limiting process of septillions of particles operating independently with each other.
pi is more abstract. It might well only show up in idealized conic sections, which mean that all the natural conic sections we see, such as shadows cast by light sources, are imprecise. (This is because, extremely technically speaking, shadows are not physical!) There are still limiting processes for pi, like taking a large chunk of matter and chilling it into a constantly-curved smooth ball, which will generate an approximate sphere, but they are much harder to perform because gravity is fickle and atom boundaries are fuzzy. (That said, we can still make such artifacts when we want to, at great cost. [1])
In both cases, our understanding of the the number's construction is good enough that we no longer need any physical props to approximate their value, but instead can compute the values abstractly and then use them to refine our physical measurements.
[0] https://en.wikipedia.org/wiki/Spiral#In_nature
[1] https://www.youtube.com/watch?v=JKCBeDeVxkg
You can only measure anything to within a finite precision anyway. You won't measure exactly Pi inches, but you also won't ever measure exactly 1 inch. You are limited by your equipment's precision.
Electromagnetism is full of both e (exponential decay, etc) and Pi (charges rotate, polarization rotates, etc).
Electromagnetism is full of both e (exponential decay, etc) and Pi (charges rotate, polarization rotates, etc).
I wonder if there is a relationship to e^(πi) = -1. Are there any other pair of numbers (other than multiple of these e and π) that have this relationship? Perhaps these two numbers sharing that relationship are why they appear so commonly in our current mathematical equations.
Numberphile has an excellent video on the numbers we know about and the quest to find / define normal numbers: https://www.youtube.com/watch?v=5TkIe60y2GI