Replacement of the mathematical constant Pi [modified]
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Apparently, Bob Palais of the University of Utah wants to prevent the use of the constant Pi in circular measure. He wants to substitute a value, symbolized by the Greek letter Tau, which has a value twice that of Pi; and to base all mathematical expressions on this Tau value rather than on Pi. The trigonometric double-angle formulas, sin(2A) = 2sin(A)cos(A) cos(2A) = cos2(A) - sin2(A) tan(2A) = 2tan(A)/(1 - tan2(A)) would then be replaced by sin(t) = 2sin(t/2)cos(t/2) cos(t) = cos2(t/2) - sin2(t/2) tan(t) = 2tan(t/2)/(1 - tan2(t/2)) and the trigonometric half-angle formulas, sin(A/2) = [+|-]SQRT((1-cos(A))/2) { + if A/2 is in quadrant I or II, - otherwise } cos(A/2) = [+|-]SQRT((1+cos(A))/2) { + if A/2 is in quadrant I or IV, - otherwise } tan(A/2) = [+|-]SQRT((1-cos(A))/(1-cos(A))) { + if A/2 is in quadrant I or III, - otherwise } would then be replaced by sin(t/4) = [+|-]SQRT((1-cos(t/2))/2) { + if t/4 is in quadrant I or II, - otherwise } cos(t/4) = [+|-]SQRT((1+cos(t/2))/2) { + if t/4 is in quadrant I or IV, - otherwise } tan(t/4) = [+|-]SQRT((1-cos(t/2))/(1-cos(t/2))) { + if t/4 is in quadrant I or III, - otherwise } The consequences of this alone don't look too intimidating. So what happens with the sum and product functions under this change? We have, currently, sin(A) + sin(B) = 2sin((1/2)(A+B))cos((1/2)(A-B)) sin(A) - sin(B) = 2cos((1/2)(A+B))sin((1/2)(A-B)) cos(A) + cos(B) = 2cos((1/2)(A+B))cos((1/2)(A-B)) cos(A) - cos(B) = 2sin((1/2)(A+B))sin((1/2)(B-A)) which would be replaced by sin(t/2) + sin(u/2) = 2sin((1/4)(t+u))cos((1/4)(t-u)) sin(t/2) - sin(u/2) = 2cos((1/4)(t+u))sin((1/4)(t-u)) cos(t/2) + cos(u/2) = 2cos((1/4)(t+u))cos((1/4)(t-u)) cos(t/2) - cos(u/2) = 2sin((1/4)(t+u))sin((1/4)(t-u)) and we have the current product formulas, sin(A)sin(B) = (1/2)(cos(A-B) - cos(A+B)) sin(A)cos(B) = (1/2)(sin(A-B) + sin(A+B)) cos(A)sin(B) = (1/2)(sin(A+B) - sin(A-B)) cos(A)cos(B) = (1/2)(cos(A-B) + cos(A+B)) which would be replaced by sin(t/2)sin(u/2) = (1/2)(cos((1/2)(t-u)) - cos((1/2)(t+u))) sin(t/2)cos(u/2) = (1/2)(sin((1/2)(t-u)) + sin((1/2)(t+u))) cos(t/2)sin(u/2) = (1/2)(sin((1/2)(t+u)) - sin((1/2)(t-u))) cos(t/2)cos(u/2) = (1/2)(cos((1/2)(t-u)) + cos((1/2)(t+u))) In these last four formulas, we have in particular to note that the quantities (1/2)(t[+|-]u) would be substituted for by the half-angle formulas noted above which involve the square roots. So
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Apparently, Bob Palais of the University of Utah wants to prevent the use of the constant Pi in circular measure. He wants to substitute a value, symbolized by the Greek letter Tau, which has a value twice that of Pi; and to base all mathematical expressions on this Tau value rather than on Pi. The trigonometric double-angle formulas, sin(2A) = 2sin(A)cos(A) cos(2A) = cos2(A) - sin2(A) tan(2A) = 2tan(A)/(1 - tan2(A)) would then be replaced by sin(t) = 2sin(t/2)cos(t/2) cos(t) = cos2(t/2) - sin2(t/2) tan(t) = 2tan(t/2)/(1 - tan2(t/2)) and the trigonometric half-angle formulas, sin(A/2) = [+|-]SQRT((1-cos(A))/2) { + if A/2 is in quadrant I or II, - otherwise } cos(A/2) = [+|-]SQRT((1+cos(A))/2) { + if A/2 is in quadrant I or IV, - otherwise } tan(A/2) = [+|-]SQRT((1-cos(A))/(1-cos(A))) { + if A/2 is in quadrant I or III, - otherwise } would then be replaced by sin(t/4) = [+|-]SQRT((1-cos(t/2))/2) { + if t/4 is in quadrant I or II, - otherwise } cos(t/4) = [+|-]SQRT((1+cos(t/2))/2) { + if t/4 is in quadrant I or IV, - otherwise } tan(t/4) = [+|-]SQRT((1-cos(t/2))/(1-cos(t/2))) { + if t/4 is in quadrant I or III, - otherwise } The consequences of this alone don't look too intimidating. So what happens with the sum and product functions under this change? We have, currently, sin(A) + sin(B) = 2sin((1/2)(A+B))cos((1/2)(A-B)) sin(A) - sin(B) = 2cos((1/2)(A+B))sin((1/2)(A-B)) cos(A) + cos(B) = 2cos((1/2)(A+B))cos((1/2)(A-B)) cos(A) - cos(B) = 2sin((1/2)(A+B))sin((1/2)(B-A)) which would be replaced by sin(t/2) + sin(u/2) = 2sin((1/4)(t+u))cos((1/4)(t-u)) sin(t/2) - sin(u/2) = 2cos((1/4)(t+u))sin((1/4)(t-u)) cos(t/2) + cos(u/2) = 2cos((1/4)(t+u))cos((1/4)(t-u)) cos(t/2) - cos(u/2) = 2sin((1/4)(t+u))sin((1/4)(t-u)) and we have the current product formulas, sin(A)sin(B) = (1/2)(cos(A-B) - cos(A+B)) sin(A)cos(B) = (1/2)(sin(A-B) + sin(A+B)) cos(A)sin(B) = (1/2)(sin(A+B) - sin(A-B)) cos(A)cos(B) = (1/2)(cos(A-B) + cos(A+B)) which would be replaced by sin(t/2)sin(u/2) = (1/2)(cos((1/2)(t-u)) - cos((1/2)(t+u))) sin(t/2)cos(u/2) = (1/2)(sin((1/2)(t-u)) + sin((1/2)(t+u))) cos(t/2)sin(u/2) = (1/2)(sin((1/2)(t+u)) - sin((1/2)(t-u))) cos(t/2)cos(u/2) = (1/2)(cos((1/2)(t-u)) + cos((1/2)(t+u))) In these last four formulas, we have in particular to note that the quantities (1/2)(t[+|-]u) would be substituted for by the half-angle formulas noted above which involve the square roots. So
Like rotating a three-dimensional puzzle: making something simpler from one perspective always makes it more complex from another perspective. What you think the simplest answer is depends on what perspective you care about the most. This is, of course, not the first some a person has gone on a lone crusade to change an accepted formal notation because he thinks that he has found something "better" in some way. My favorite example of this kind of crusade was Spencer-Brown's "Laws of Form". A book on formal logic, where Spencer-Brown actually builds an entire formal language from scratch and proves that it is formally equivalent to first-order predicate logic. Why? Because instead of the atomic concepts of his system being truth and falsity, the atomic concepts in his system are the act of making a "distinction" and making an "indication" (of one side or the other of a distinction boundary). Spencer-Brown expounds at length, in the appendix, about how there is no such thing as absolute truth, and how all of reality depends on the actions of a subjective observer: what an observer decides to distinguish from the background, and what he indicates as relevant. And wouldn't the whole world be more peaceful and open-minded and accepting of differences if we all just realized that there was no such thing as "truth" and "falsity"? And how can we become peaceful and open-minded if hegemonic notions such as "truth" are embedded in the roots of our very arithmetic?? Everybody wants to make things better. But alas! Like Dvorak Keyboards and Esperanto, these endeavors are destined to never be more than historical curiosities. "Simple" is always with respect to a viewer, after all, and based on the distinctions that that viewer makes and what that viewer think "matters". It's all very subjective. Isn't that right, Spencer-Brown? ;)
--Greg
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Apparently, Bob Palais of the University of Utah wants to prevent the use of the constant Pi in circular measure. He wants to substitute a value, symbolized by the Greek letter Tau, which has a value twice that of Pi; and to base all mathematical expressions on this Tau value rather than on Pi. The trigonometric double-angle formulas, sin(2A) = 2sin(A)cos(A) cos(2A) = cos2(A) - sin2(A) tan(2A) = 2tan(A)/(1 - tan2(A)) would then be replaced by sin(t) = 2sin(t/2)cos(t/2) cos(t) = cos2(t/2) - sin2(t/2) tan(t) = 2tan(t/2)/(1 - tan2(t/2)) and the trigonometric half-angle formulas, sin(A/2) = [+|-]SQRT((1-cos(A))/2) { + if A/2 is in quadrant I or II, - otherwise } cos(A/2) = [+|-]SQRT((1+cos(A))/2) { + if A/2 is in quadrant I or IV, - otherwise } tan(A/2) = [+|-]SQRT((1-cos(A))/(1-cos(A))) { + if A/2 is in quadrant I or III, - otherwise } would then be replaced by sin(t/4) = [+|-]SQRT((1-cos(t/2))/2) { + if t/4 is in quadrant I or II, - otherwise } cos(t/4) = [+|-]SQRT((1+cos(t/2))/2) { + if t/4 is in quadrant I or IV, - otherwise } tan(t/4) = [+|-]SQRT((1-cos(t/2))/(1-cos(t/2))) { + if t/4 is in quadrant I or III, - otherwise } The consequences of this alone don't look too intimidating. So what happens with the sum and product functions under this change? We have, currently, sin(A) + sin(B) = 2sin((1/2)(A+B))cos((1/2)(A-B)) sin(A) - sin(B) = 2cos((1/2)(A+B))sin((1/2)(A-B)) cos(A) + cos(B) = 2cos((1/2)(A+B))cos((1/2)(A-B)) cos(A) - cos(B) = 2sin((1/2)(A+B))sin((1/2)(B-A)) which would be replaced by sin(t/2) + sin(u/2) = 2sin((1/4)(t+u))cos((1/4)(t-u)) sin(t/2) - sin(u/2) = 2cos((1/4)(t+u))sin((1/4)(t-u)) cos(t/2) + cos(u/2) = 2cos((1/4)(t+u))cos((1/4)(t-u)) cos(t/2) - cos(u/2) = 2sin((1/4)(t+u))sin((1/4)(t-u)) and we have the current product formulas, sin(A)sin(B) = (1/2)(cos(A-B) - cos(A+B)) sin(A)cos(B) = (1/2)(sin(A-B) + sin(A+B)) cos(A)sin(B) = (1/2)(sin(A+B) - sin(A-B)) cos(A)cos(B) = (1/2)(cos(A-B) + cos(A+B)) which would be replaced by sin(t/2)sin(u/2) = (1/2)(cos((1/2)(t-u)) - cos((1/2)(t+u))) sin(t/2)cos(u/2) = (1/2)(sin((1/2)(t-u)) + sin((1/2)(t+u))) cos(t/2)sin(u/2) = (1/2)(sin((1/2)(t+u)) - sin((1/2)(t-u))) cos(t/2)cos(u/2) = (1/2)(cos((1/2)(t-u)) + cos((1/2)(t+u))) In these last four formulas, we have in particular to note that the quantities (1/2)(t[+|-]u) would be substituted for by the half-angle formulas noted above which involve the square roots. So
I don't understand your objection - all of the half-angle formulae would stay exactly the same. Similarly, the definitions of sin, cos and tan would stay the same. The only formulae/identities that would change are ones in which π explicitly appears, rather than an arbitrary angle like A or B.
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I don't understand your objection - all of the half-angle formulae would stay exactly the same. Similarly, the definitions of sin, cos and tan would stay the same. The only formulae/identities that would change are ones in which π explicitly appears, rather than an arbitrary angle like A or B.
That's exactly right; and the reason I object is because those formulas do not and cannot change. It is at the point where substitutions are made that other dependencies arise; and because of this, I think it only leads to confusion to argue about a metric for the angular quantity. I don't see how complex arithmetic involving expressions of exponentials in terms of the circular functions is going to improve under his (Bob Palais') suggested improvements, either. If you can see this all as some isomorphism in which nobody is going to be able to, or care to, distinguish between present and future methods, show me some proof. I'm listening. But I don't think it's there to prove. How important is it, merely as a matter of heuristics?
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That's exactly right; and the reason I object is because those formulas do not and cannot change. It is at the point where substitutions are made that other dependencies arise; and because of this, I think it only leads to confusion to argue about a metric for the angular quantity. I don't see how complex arithmetic involving expressions of exponentials in terms of the circular functions is going to improve under his (Bob Palais') suggested improvements, either. If you can see this all as some isomorphism in which nobody is going to be able to, or care to, distinguish between present and future methods, show me some proof. I'm listening. But I don't think it's there to prove. How important is it, merely as a matter of heuristics?
I still don't get what you're talking about.
GAMerritt wrote:
the reason I object is because those formulas do not and cannot change.
You just said that they would!
GAMerritt wrote:
I think it only leads to confusion to argue about a metric for the angular quantity.
What metric? :confused: Pi is just a number, same as tau.
GAMerritt wrote:
I don't see how complex arithmetic involving expressions of exponentials in terms of the circular functions is going to improve under his (Bob Palais') suggested improvements, either.
It wouldn't change. It would just become easier and more natural to use tau than pi.
GAMerritt wrote:
If you can see this all as some isomorphism in which nobody is going to be able to, or care to, distinguish between present and future methods, show me some proof. I'm listening. But I don't think it's there to prove.
I keep rereading this, but it never makes any sense to me. :confused:
GAMerritt wrote:
How important is it, merely as a matter of heuristics?
Pi is a rather artificial and arbitrary choice, but tau isn't. Tau is extremely logical and natural, and I think it's worth switching over for that reason alone. Never mind the advantages of doing away with a whole lot of unseemly 2s.
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Apparently, Bob Palais of the University of Utah wants to prevent the use of the constant Pi in circular measure. He wants to substitute a value, symbolized by the Greek letter Tau, which has a value twice that of Pi; and to base all mathematical expressions on this Tau value rather than on Pi. The trigonometric double-angle formulas, sin(2A) = 2sin(A)cos(A) cos(2A) = cos2(A) - sin2(A) tan(2A) = 2tan(A)/(1 - tan2(A)) would then be replaced by sin(t) = 2sin(t/2)cos(t/2) cos(t) = cos2(t/2) - sin2(t/2) tan(t) = 2tan(t/2)/(1 - tan2(t/2)) and the trigonometric half-angle formulas, sin(A/2) = [+|-]SQRT((1-cos(A))/2) { + if A/2 is in quadrant I or II, - otherwise } cos(A/2) = [+|-]SQRT((1+cos(A))/2) { + if A/2 is in quadrant I or IV, - otherwise } tan(A/2) = [+|-]SQRT((1-cos(A))/(1-cos(A))) { + if A/2 is in quadrant I or III, - otherwise } would then be replaced by sin(t/4) = [+|-]SQRT((1-cos(t/2))/2) { + if t/4 is in quadrant I or II, - otherwise } cos(t/4) = [+|-]SQRT((1+cos(t/2))/2) { + if t/4 is in quadrant I or IV, - otherwise } tan(t/4) = [+|-]SQRT((1-cos(t/2))/(1-cos(t/2))) { + if t/4 is in quadrant I or III, - otherwise } The consequences of this alone don't look too intimidating. So what happens with the sum and product functions under this change? We have, currently, sin(A) + sin(B) = 2sin((1/2)(A+B))cos((1/2)(A-B)) sin(A) - sin(B) = 2cos((1/2)(A+B))sin((1/2)(A-B)) cos(A) + cos(B) = 2cos((1/2)(A+B))cos((1/2)(A-B)) cos(A) - cos(B) = 2sin((1/2)(A+B))sin((1/2)(B-A)) which would be replaced by sin(t/2) + sin(u/2) = 2sin((1/4)(t+u))cos((1/4)(t-u)) sin(t/2) - sin(u/2) = 2cos((1/4)(t+u))sin((1/4)(t-u)) cos(t/2) + cos(u/2) = 2cos((1/4)(t+u))cos((1/4)(t-u)) cos(t/2) - cos(u/2) = 2sin((1/4)(t+u))sin((1/4)(t-u)) and we have the current product formulas, sin(A)sin(B) = (1/2)(cos(A-B) - cos(A+B)) sin(A)cos(B) = (1/2)(sin(A-B) + sin(A+B)) cos(A)sin(B) = (1/2)(sin(A+B) - sin(A-B)) cos(A)cos(B) = (1/2)(cos(A-B) + cos(A+B)) which would be replaced by sin(t/2)sin(u/2) = (1/2)(cos((1/2)(t-u)) - cos((1/2)(t+u))) sin(t/2)cos(u/2) = (1/2)(sin((1/2)(t-u)) + sin((1/2)(t+u))) cos(t/2)sin(u/2) = (1/2)(sin((1/2)(t+u)) - sin((1/2)(t-u))) cos(t/2)cos(u/2) = (1/2)(cos((1/2)(t-u)) + cos((1/2)(t+u))) In these last four formulas, we have in particular to note that the quantities (1/2)(t[+|-]u) would be substituted for by the half-angle formulas noted above which involve the square roots. So
How can you replace Pi with a value twice PI, when Pi itself is a never ending number? you would still need Pi to reference Tau, e.g. T=2Pi So pretty poitnless if you ask me. You will be wanting to replace e[^] with something else next?
Dave Find Me On: Web|Facebook|Twitter|LinkedIn
Folding Stats: Team CodeProject
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How can you replace Pi with a value twice PI, when Pi itself is a never ending number? you would still need Pi to reference Tau, e.g. T=2Pi So pretty poitnless if you ask me. You will be wanting to replace e[^] with something else next?
Dave Find Me On: Web|Facebook|Twitter|LinkedIn
Folding Stats: Team CodeProject
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Like rotating a three-dimensional puzzle: making something simpler from one perspective always makes it more complex from another perspective. What you think the simplest answer is depends on what perspective you care about the most. This is, of course, not the first some a person has gone on a lone crusade to change an accepted formal notation because he thinks that he has found something "better" in some way. My favorite example of this kind of crusade was Spencer-Brown's "Laws of Form". A book on formal logic, where Spencer-Brown actually builds an entire formal language from scratch and proves that it is formally equivalent to first-order predicate logic. Why? Because instead of the atomic concepts of his system being truth and falsity, the atomic concepts in his system are the act of making a "distinction" and making an "indication" (of one side or the other of a distinction boundary). Spencer-Brown expounds at length, in the appendix, about how there is no such thing as absolute truth, and how all of reality depends on the actions of a subjective observer: what an observer decides to distinguish from the background, and what he indicates as relevant. And wouldn't the whole world be more peaceful and open-minded and accepting of differences if we all just realized that there was no such thing as "truth" and "falsity"? And how can we become peaceful and open-minded if hegemonic notions such as "truth" are embedded in the roots of our very arithmetic?? Everybody wants to make things better. But alas! Like Dvorak Keyboards and Esperanto, these endeavors are destined to never be more than historical curiosities. "Simple" is always with respect to a viewer, after all, and based on the distinctions that that viewer makes and what that viewer think "matters". It's all very subjective. Isn't that right, Spencer-Brown? ;)
--Greg
Esperanto always reminds of Red Dwarf. In order to become a captain on a spaceship you were required to pass the Esperanto exam, even though everyone speaks English in space. The examples you give make in fact a lot more sense because at least they challenge the things we take for granted. Even if the Dvorak keyboard is useless, the research someone put into it might still be of use for other things. But changing Pi to Tau isn't even a very creative idea. If you take all the formula with pi and weigh their importance, then you find that most of them have 2*pi in them. That's the only argument repeated over and over again in different words in a multiple pages long manifest. It's like a bad manager randomly changing the names of the departments, because to make an impression but he doesn't know anything better to do. An empty idea.
Giraffes are not real.
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Esperanto always reminds of Red Dwarf. In order to become a captain on a spaceship you were required to pass the Esperanto exam, even though everyone speaks English in space. The examples you give make in fact a lot more sense because at least they challenge the things we take for granted. Even if the Dvorak keyboard is useless, the research someone put into it might still be of use for other things. But changing Pi to Tau isn't even a very creative idea. If you take all the formula with pi and weigh their importance, then you find that most of them have 2*pi in them. That's the only argument repeated over and over again in different words in a multiple pages long manifest. It's like a bad manager randomly changing the names of the departments, because to make an impression but he doesn't know anything better to do. An empty idea.
Giraffes are not real.
If you had the opportunity to change the direction of conventional current, would you?
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If you had the opportunity to change the direction of conventional current, would you?
Yes and No. I would make electrons positively charged, call them "positrons" and protons "negatrons". All the anti-matter crap will get a consistent anti-prefix. So positrons will be renamed to anti-positrons, etc... That way, the direction of current will be "correct", but it will still go from + to -. :laugh:
Giraffes are not real.
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Yes and No. I would make electrons positively charged, call them "positrons" and protons "negatrons". All the anti-matter crap will get a consistent anti-prefix. So positrons will be renamed to anti-positrons, etc... That way, the direction of current will be "correct", but it will still go from + to -. :laugh:
Giraffes are not real.
0bx wrote:
I would make electrons positively charged, call them "positrons" and protons "negatrons".
All the anti-matter crap will get a consistent anti-prefix. So positrons will be renamed to anti-positrons, etc...
That way, the direction of current will be "correct", but it will still go from + to -.:laugh: :) Pi is not so wrong that it needs replacing, in my revised opinion.
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If you had the opportunity to change the direction of conventional current, would you?
Only if we start calling it "anberic current" instead of "electric current". ;)
--Greg