Is e Normal?

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by lisper • 7 years ago

29 22 7 years ago

empath75• 7 years ago

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.

infradig• 7 years ago

I would say the ratio of a circle circumference to its diameter is naturally occurring.

Someone• 7 years ago

Yes, that is π in Euclidean space, but that is rare (¿nonexistent?) according to our current understanding of nature.

coldtea3• 7 years ago

Are there circles in nature?

lidHanteyk• 7 years ago

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

roywiggins• 7 years ago

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).

SkyBelow2• 7 years ago

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.

occitan3• 7 years ago

The above comment is misinformed, but in no way obnoxious.

There was no reason to downvote it.

wglb3• 7 years ago

Downvotes are allowed to express disagreemnt.

ccvannorman• 7 years ago

It's fascinating how little we know about everyday numbers like pi and e, it's widely assumed that they are normal but there is no proof for this (and it's extremely difficult to do so). In fact we believe the fast majority of numbers are transcendental and normal but we only actually know of a single normal number, and it is a manufactured number.

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

Is e Normal?

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