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I just want to ask all of the Pi/apathetic people-- how long did it take you to understand radians? For me, it was a week before I was comfortable naming any a

I just want to ask all of the Pi/apathetic people-- how long did it take you to understand radians? For me, it was a week before I was comfortable naming any angle in radians in a reasonable amount of time (this is after a week of drilling). This is just my point of view, but calculating radians was a significant roadblock into making quick trigonometric calculations. In fact, I'd have to say it was the biggest roadblock. This has nothing to do with how "clean" it looks or how I "feel" about how it's presented. That said, I don't think it's worth it to make the switch because of all the hassle. I'm just curious about all of the hostility towards Tauists. tldr; it has nothing to do with any mathematical formula looking "cleaner," but everything to do with teaching math effectively.

loup-vaillant 15y ago
The teaching argument (which by the way this article did not address) is for me the most powerful one. All the other arguments look secondary to me. > That said, I don't think it's worth it to make the switch because of all the hassle. It would be no hassle for pupils. I bet that it would be easier and faster to teach Tau first, then mention that Pi is half Tau. In my opinion, teaching should have the priority. I don't really care if the rest of the world use Pi, but I'll teach Tau first.
- flomincucci 15y ago
  I agree with teaching Tau first. Personally, it took me at least a week to begin to understand radians, and almost a month to understand it pretty well. Tau is more intuitive than Pi in this case.
amalcon 15y ago
It only took me about a day, but I suspect that's because I was already used to cycles at the time (1 cycle=2pi radians). Getting used to cycles took me about a week. To my mind, the problem is not with pi; the problem is with degrees. Everyone learns about degrees first, and then must "un-learn" these artificial numbers and begin thinking in fractions of a circle (with an extra constant thrown in there one way or the other, in the case of radians). If we began labeling globes, protractors, and the like in radians (or fractional cycles), this problem would go away.
- sesqu 15y ago
  I don't think in radians, I think in cycles (radians/tau). I used to think in degrees, and could never get the hang of radians. Unfortunately, I cannot imagine the mathematical world moving to cycles, ever. The most I can hope for is that teachers introduce others to a single intermediary constant (tau) rather than two (2,pi).
  - amalcon 15y ago
    The problem with using cycles as the primary unit of angle is that the wonderful trigonometric derivative symmetry only occurs when the functions are calibrated for radians. I've never understood the particular argument of "single intermediary constant (tau) rather than two (2,pi)". 2pi is one constant that contains multiple glyphs, just as 1/2 is one constant, just as tau/2 would be one constant. That it is derived logically from other constants does not make it two separate constants. Even multi-digit numbers are derived logically from their component glyphs (12 = 10+2).
    - sesqu 15y ago
      The two-glyph thing is a matter of description length and parametrization. 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. Thus, 10 carries more meaning than 12, even though both are constant - 10 is potentially the aforementioned parameter, but what is the 2? With sufficient study, 12 becomes a number of important constants as well (such as the number of inches in a foot, or the integral of a centered quadratic function), but not ones considered fundamental. Essentially, it's about the difference between a concept and a measurement, or the difference between (x+y)/x and 1+x/y. 2pi is an expression, pi is the concept.
      - amalcon 15y ago
        I've read this a few times, and I don't think I really understand. Perhaps you could explain further? 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. There is no notion of an "expression" as distinct from a "number" (or "function" if it involves a variable) in any branch of math apart from computer science[1]. In algebraic terms, (12) and (10+2) and (6x2) and (0xC) and (2^4-2) and "twelve" are all literally the same thing. Well, technically they are all equivalent notations for the same abstract concept. Thus, 10 carries more meaning than 12 Even if I accept this (which I'm not convinced I do), it's beside the point: 10 and 12 are not equal. Unlike with pi and a hypothetical tau, using one where the other is called for would be an error. [1]There is the notion of the limit, which is subtly different: limits do care how a function behaves at other points. One could make the case that this makes a limit into a sort of expression, but to be honest I think that only obscures the idea.

loup-vaillant 15y ago

The teaching argument (which by the way this article did not address) is for me the most powerful one. All the other arguments look secondary to me. > That said, I don't think it's worth it to make the switch because of all the hassle. It would be no hassle for pupils. I bet that it would be easier and faster to teach Tau first, then mention that Pi is half Tau. In my opinion, teaching should have the priority. I don't really care if the rest of the world use Pi, but I'll teach Tau first.

flomincucci 15y ago
I agree with teaching Tau first. Personally, it took me at least a week to begin to understand radians, and almost a month to understand it pretty well. Tau is more intuitive than Pi in this case.

amalcon 15y ago

It only took me about a day, but I suspect that's because I was already used to cycles at the time (1 cycle=2pi radians). Getting used to cycles took me about a week. To my mind, the problem is not with pi; the problem is with degrees. Everyone learns about degrees first, and then must "un-learn" these artificial numbers and begin thinking in fractions of a circle (with an extra constant thrown in there one way or the other, in the case of radians). If we began labeling globes, protractors, and the like in radians (or fractional cycles), this problem would go away.

sesqu 15y ago
I don't think in radians, I think in cycles (radians/tau). I used to think in degrees, and could never get the hang of radians. Unfortunately, I cannot imagine the mathematical world moving to cycles, ever. The most I can hope for is that teachers introduce others to a single intermediary constant (tau) rather than two (2,pi).
- amalcon 15y ago
  The problem with using cycles as the primary unit of angle is that the wonderful trigonometric derivative symmetry only occurs when the functions are calibrated for radians. I've never understood the particular argument of "single intermediary constant (tau) rather than two (2,pi)". 2pi is one constant that contains multiple glyphs, just as 1/2 is one constant, just as tau/2 would be one constant. That it is derived logically from other constants does not make it two separate constants. Even multi-digit numbers are derived logically from their component glyphs (12 = 10+2).
  - sesqu 15y ago
    The two-glyph thing is a matter of description length and parametrization. 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. Thus, 10 carries more meaning than 12, even though both are constant - 10 is potentially the aforementioned parameter, but what is the 2? With sufficient study, 12 becomes a number of important constants as well (such as the number of inches in a foot, or the integral of a centered quadratic function), but not ones considered fundamental. Essentially, it's about the difference between a concept and a measurement, or the difference between (x+y)/x and 1+x/y. 2pi is an expression, pi is the concept.
    - amalcon 15y ago
      I've read this a few times, and I don't think I really understand. Perhaps you could explain further? 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. There is no notion of an "expression" as distinct from a "number" (or "function" if it involves a variable) in any branch of math apart from computer science[1]. In algebraic terms, (12) and (10+2) and (6x2) and (0xC) and (2^4-2) and "twelve" are all literally the same thing. Well, technically they are all equivalent notations for the same abstract concept. Thus, 10 carries more meaning than 12 Even if I accept this (which I'm not convinced I do), it's beside the point: 10 and 12 are not equal. Unlike with pi and a hypothetical tau, using one where the other is called for would be an error. [1]There is the notion of the limit, which is subtly different: limits do care how a function behaves at other points. One could make the case that this makes a limit into a sort of expression, but to be honest I think that only obscures the idea.