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A trick to eliminate 2π (sometimes)

marci.gunyho.com

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Re: A trick to eliminate 2π (sometimes)

#151

Post by Misdicorl »

jacobolus wrote: post #149
Misdicorl wrote: post #148

Earlier quoted context omitted.

I understand your analogy, but I reject it's validity. Degrees/radians/turns map to meters/feet/angstrom. Decibels and octaves are true "number" multipliers. You could for instance talk about degrees in octaves or in dB if you like. It's just not particularly useful for the domain Edit: another example difference. I can't measure an octave or dB. I can measure a degree Edit2: we've reached reply limit but I concede y…

You absolutely can measure an octave or decibel. It's just a relative quantity, in just the same way any quantification of orientation is relative. (For example, you can measure an octave by marking out a particular fret on your guitar; it will make an octave change whichever particular note you start with.) You can feel free to "reject" whatever you want. You'll just be wrong/confused. ;-) (But you'll be in good com…

I think this will devolve at this point. But you should consider that you are perhaps far too confident in your position here. My last attempt is due you to consider what happens when you

A) multiply two degrees together

B) multiply two dBm values together

The output "units" in A change but do not in B. dB and angles are very different.

Edit: the units in B do change, but the dB part doesn't. Tired.

Re: A trick to eliminate 2π (sometimes)

#152

Post by NotYourLawyer »

adrian_b wrote: post #115
NotYourLawyer wrote: post #108

Earlier quoted context omitted.

Yeah, you’re right. And radians are what make the trig identities involving derivatives work out nicely.

The simpler derivation formula was important when such symbolic computation was done by hand. Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important. The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation. In real applications you almo…

If you’re using a computer for symbolic algebra or whatever, none of this matters anyway. The whole post is about simpler notation for the sake of making things less error-prone when working by hand.

Re: A trick to eliminate 2π (sometimes)

#153

Post by cycomanic »

twiss wrote: post #147
cycomanic wrote: post #60

The argument about why not to include the i in 2 pi i is incorrect. The problem is he says (e^x)^i2pi = e^i2pix does not work because ln(e^i2pi)=0. But he needs to use the complex logarithm. And for the complex logarithm ln(e^z) =z for z in C. If that wasn't the case calculation rules of logarithms and exponentials would depend on if arguments are complex or real, a lot of physics would become much more complicated s…

That's not his argument; he says defining Θ = e^2πi is not useful because e^2πi = 1, so Θ and Θ^x would also be 1. That's why he defined Θ = e^2π instead, so that Θ^x (or possibly Θ^ix) is a useful operation.

he writes:

> where ln(e^2πi)=ln(1)=0,

that is incorrect because the logarithm of a complex number is multi-valued. He even cites the correct source on wikipedia, but his argument is incorrect (and because the exponent is 2πi he actually would get a meaningful results I believe).

Re: A trick to eliminate 2π (sometimes)

#154

Post by scythe »

adrian_b wrote: post #111
scythe wrote: post #40

Earlier quoted context omitted.

You can always define angles in turns. But the problem is that it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is not so easily separated from the rest of mathematics.

There is no need for that definition. It is possible to completely remove the e^x function from mathematics without losing anything. It is possible to express everything using a pair of functions, the real function 2^x and the complex function 1^x. Then the cosinus and the sinus are the real and imaginary parts of 1^x (where x is measured in cycles a.k.a. turns). The only disadvantage of this approach is that symboli…

The function 1^x is constant everywhere. You can write (-1)^x to get the kind of effect you are looking for, but doing this with 1^x is madness.

>The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant.

There is more to it than that. The exponential function is not really e^x, it is lim_{N->∞} (1 + x/N)^N. The advantage of this definition is that, since it is valid everywhere on the complex plane, it allows the rigorous definition of real powers without recourse to inverse functions. We still teach students to prove things.

But also, isn't this a little contradictory? The goal of the original blog post was to simplify various differential equations. If you aren't simplifying symbolic differentiation, then what are you simplifying?

Re: A trick to eliminate 2π (sometimes)

#155

Post by cycomanic »

cycomanic wrote: post #153
twiss wrote: post #147

Earlier quoted context omitted.

That's not his argument; he says defining Θ = e^2πi is not useful because e^2πi = 1, so Θ and Θ^x would also be 1. That's why he defined Θ = e^2π instead, so that Θ^x (or possibly Θ^ix) is a useful operation.

he writes: > where ln(e^2πi)=ln(1)=0, that is incorrect because the logarithm of a complex number is multi-valued. He even cites the correct source on wikipedia, but his argument is incorrect (and because the exponent is 2πi he actually would get a meaningful results I believe).

just to expand on this the issue is not that e^2πi = 1 but that (e^z)^x =/= e^(zx) when z is complex, because in general z^w . instead it would be e^(x ln(e^z)), where the logarithm of a complex number is a multivalued function.

We can compute the principle value of ln(e^2πi)=ln(1) + i(2π + 2πk) (where k is an integer) so therefore

(e^2πi)^x = e^(xln(e^2πi)) = e^(x2π(k+1)i)

Re: A trick to eliminate 2π (sometimes)

#156

Post by dTal »

bobbylarrybobby wrote: post #21
H8crilA wrote: post #13

Also, π is the wrong constant. The very definition is awkward: the ratio of two radiuses to the circumference. How about one radius? It is much more natural to work with 2π. Some people use the letter τ (Tau) to denote 2π, and it simplifies almost all naturally occurring expressions. For example, what is more elegant? e^(π*i) = -1 e^(τ*i) = 1

e^pi + 1 = 0, of course

e^pi + 1 = 24.14069263...

Re: A trick to eliminate 2π (sometimes)

#157

Post by dTal »

abecedarius wrote: post #140
DavidSJ wrote: post #57

To address the problem they discuss at the end with defining Θ = e^2πi, they could instead define Θ(x) = e^2πix, the circular analog to the exponential function exp (which is really more fundamental than exp(1) = e anyways).

Note there's an existing notation which I've mostly seen in lower-class settings like high-school textbooks: r theta, for the complex number r e^(i theta). Optionally leave out the r. So you have that "most beautiful formula in all of mathematics": tau = 1

And here it is in Unicode glory:

∠τ = 1

Re: A trick to eliminate 2π (sometimes)

#158

Post by jacobolus »

Misdicorl wrote: post #151
jacobolus wrote: post #149

Earlier quoted context omitted.

You absolutely can measure an octave or decibel. It's just a relative quantity, in just the same way any quantification of orientation is relative. (For example, you can measure an octave by marking out a particular fret on your guitar; it will make an octave change whichever particular note you start with.) You can feel free to "reject" whatever you want. You'll just be wrong/confused. ;-) (But you'll be in good com…

I think this will devolve at this point. But you should consider that you are perhaps far too confident in your position here. My last attempt is due you to consider what happens when you A) multiply two degrees together B) multiply two dBm values together The output "units" in A change but do not in B. dB and angles are very different. Edit: the units in B do change, but the dB part doesn't. Tired.

Multiplying degrees together is not a meaningful concept.

But you can compose rotations.

Multiplying decibels together is also not meaningful, but you can compose scaling.

Re: A trick to eliminate 2π (sometimes)

#160

Post by linuxdude314 »

failuser wrote: post #91

I’m surprised by the number of positive replies. Obviously the current notation is made up like all notations, but this is just a waste of time, 2п naturally arises in so many places. The h-bar for the Plank constant is just a product of not agreeing what constant to denote. sin(x)=x+o(x) is just too nice to give up. Switching units needs a way greater benefit than this.

Same.

This is amateurish math at best. Defining a new derivative operator that is mathematically equivalent to the normal derivative operator and then reformulating physics equations as some way to prevent human error is beyond absurd.

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