What makes e natural? (2004)
komal.hu
What makes e natural? (2004)
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Re: What makes e natural? (2004)
#2e, 3, and pi be
Something of a trinity
(Wink from Diety)
Re: What makes e natural? (2004)
#3[deleted]
Re: What makes e natural? (2004)
#4Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful.
Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x.
In my opinion drilling down too much on conventions misses the point of math.
Re: What makes e natural? (2004)
#5Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful. Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x. In my opinion drilling down too much on conventions misses the point of math.
Without delving too far into the philosophy of math as concerns existence vs convention...
e pops up quite often when taking limits on a surprising number of varied phenomena. It is much more than a mere convention, unless you subscribe to the nihilistic, anti-epistemological notion that all of mathematics is merely convention. It seems to be the center of the conceptual space particularly around questions of relative and absolute scale.
It's true you can use any base for computing things but some are more natural than others in that specific parametrizations have natural interpretations especially when it comes to physics (timescales, information-theoretic optimality, etc.).
Re: What makes e natural? (2004)
#6e is natural because it's the smallest positive root of the equation e^(i pi) + 1 = 0.
Re: What makes e natural? (2004)
#7Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful. Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x. In my opinion drilling down too much on conventions misses the point of math.
No, we measure angles in radians so that d^4/dx^4 (sin x) = sin x.
Re: What makes e natural? (2004)
#8Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful. Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x. In my opinion drilling down too much on conventions misses the point of math.
No, we measure angles in radians so that d^4/dx^4 (sin x) = sin x.
No, we measure angles in radians so that e^(ix) = cos x + i sin x.
Re: What makes e natural? (2004)
#9Re: What makes e natural? (2004)
#10e is natural because it's the smallest positive root of the equation e^(i pi) + 1 = 0 .
In high school we were taught that it is because The slope of the function e^x is equal to e^x. Growth being proportional to magnitude is natural.