Stupid Questions February 2015

Gondolinian

This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.

Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.

To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.

Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.

To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.

A lot of math and physics definitions feel like they have weird dross. Examples:

The Gamma function has this -1 I don't understand
The Riemann Zeta function ζ(s) negates s for reasons beyond me
cosine seems more primitive than sine
The gravitational constant looks like off by a factor of 4π
π seems like half the size it should be

After years of confusion, I was finally vindicated about π. That π is not 6.2831853071... is mostly a historical accident. Am I "right" about these other definitions being "wrong"? What are other mathematical entities are defined in ugly ways for historical reasons?

I used to think that the way mathematicians did things was forced to be the best way we could do because of the requirements to do things properly in order to advance in maths. But then the Tau Manifesto showed me I was wrong.

I think you're right about cosine. I think sine seemed simpler when it was named back in classical times, but then when complex numbers were discovered and their relationships to the trigonometric functions was discovered, cosine turned out to be simpler.

Here's one I come across as a programmer: which number is better for starting ind... (read more)

7Douglas_Knight11y

What is your complaint about Zeta? That it is the sum of n^-s, rather than the sum of n^s? It's the one that converges. Or are you bothered that zeta(-3) is rational, while zeta(3) is irrational? [...] Maybe useful for some purposes. Maybe that would be a good function to have when defining the Beta distribution, though there are other reasons for the normalization there. But in the context of the Riemann Zeta function (which is the context you have suggested), that is not at all the purpose of the Gamma function. Its role is as the Mellin transform of the exponential function. The Zeta function itself is a Mellin tranform and two interact well because of their common origin. Of course, that pushes back the question to why the Mellin tranform has a -1. What it really has is a dx/x. This measure is invariant under scaling, just as dx is invariant under translation. Indeed, the measures correspond under the exponential change of variables. (In fact, that is closely related to a justification for the normalization of the Beta distribution. B(0,0) is the measure invariant under logistic transformations; B(p,q) is the posterior after seeing p,q observations.)

9kpreid11y

I'd like to put in a word for sine: sin(0) = 0 sin(x) ≈ x These are highly useful properties in some contexts. That said, cos(0) being the unit prior to any rotation is also nice. (But the definition of a rotation in Cartesian coordinates contains exactly as many sines as cosines; and that generalizes to 3 dimensions where complex numbers do not.)

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Stupid Questions February 2015

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