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Pi is Wrong: http://www.math.utah.edu/~palais/pi.pdf http://www.math.utah.edu/~palais/pi.pdf It should be e^(pi i) = 1, but pi was unfortunately defined at hal

by GavinB 17y ago

Pi is Wrong: http://www.math.utah.edu/~palais/pi.pdf http://www.math.utah.edu/~palais/pi.pdf It should be e^(pi i) = 1, but pi was unfortunately defined at half the appropriate value in the 17th century.

amalcon 17y ago
Pi is defined as the circumference of a circle of unit diameter. This isn't somehow less correct than the alternative of making that circle one of unit radius. It causes some annoying extra factors (as the PDF points out), but the alternative would cause different complications. The area of a circle, for example, would be r^2*pi/2.
- sofal 17y ago
  I'm not convinced that redefining Pi as 2*Pi would cause an equal amount of complications (besides confusing everyone with the change). The radius is more fundamental to a circle. The diameter is just derived from the radius. The idea of a circle with a unit radius is so widely used that it is simply called a "unit circle" with the obvious implication that the radius is what the "unit" refers to. Every time I've done math with radians, the fact that Pi is only half way around has added a subtle but noticeable disconnect. I have to force my intuition to fit the definition.
  - amalcon 17y ago
    The radius is only more fundamental to the notion of a circle because of the modern definition of a circle (set of all points exactly r away from the given point). One could also use a definition like: "Continuous set of points such that each is distance d from exactly one other, and further from none." The modern definition is convenient because it's a special case of a convenient definition of ellipse, but there's nothing terribly fundamental about it. And yes, every time you've done math with radians, that problem has haunted you. That's because we decided to use radians instead of "diameter-ans", to keep with the above definition.
- abecedarius 17y ago
  Yes, the alternative would cause different complications, but fewer of them. The pi-is-wrong pdf goes into it. It was programming a lot that brought me to realize this. I got in the habit of always questioning the primitives I'm given, in the hope of simplifying the definitions built on them. It was a little boggling to notice that pi, something I'd grown up with, deserved the same kind of questioning.
- mhartl 17y ago
  The area of a circle, for example, would be r^2 pi/2. This is actually my favorite example of why pi is wrong—it's the "exception" that proves the rule. To see why, set τ = C/r = 2 pi, and then consider the following chart of common quadratic forms: integral of u 1/2 u^2 kinetic energy 1/2 m v^2 distance fallen 1/2 g t^2 spring energy 1/2 k x^2 triangular area 1/2 b h circular area 1/2 τ r^2 We see that, far from causing "different complications", using the right circle constant brings the area of a circle into a more natural form. The 1/2 in the formula for circular area is actually a missing factor; using tau in place of pi restores it. N.B. I made a previous comment along these lines, but put it in the wrong place. If you're still a "pi is wrong" skeptic, considering the improvement in radian angle measure may yet convince you: The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc. In these terms, Euler's formula would be recast as e^(i τ) = 1 That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.
  - christopherolah 17y ago
    Hm, is using tau a standard convention? I've thought pi is horrible... and adopted using a loopy pi as 2pi. I loop the first vertical line over the horizontal, right, and then make it the horizontal, and so on. If that makes any sense. I'm thinking of making a LaTeX package containing it... (and also some other personal conventions) Also, to strengthen your point, 2pi makes the radian system a lot more natural.
    - mhartl 17y ago
      I made up the usage of τ myself, partially because of its typographic similarity to π, partially because it leads naturally to the usage "τ = turn". It's too bad that π has two legs while τ has only one; it would be poetic if π were 2τ, but it wasn't to be. N.B. I have a secret master plan to spread the use of τ, but this comment is too small to contain it. ;-)

by GavinB 17y ago

amalcon 17y ago

Pi is defined as the circumference of a circle of unit diameter. This isn't somehow less correct than the alternative of making that circle one of unit radius. It causes some annoying extra factors (as the PDF points out), but the alternative would cause different complications. The area of a circle, for example, would be r^2*pi/2.

sofal 17y ago
I'm not convinced that redefining Pi as 2*Pi would cause an equal amount of complications (besides confusing everyone with the change). The radius is more fundamental to a circle. The diameter is just derived from the radius. The idea of a circle with a unit radius is so widely used that it is simply called a "unit circle" with the obvious implication that the radius is what the "unit" refers to. Every time I've done math with radians, the fact that Pi is only half way around has added a subtle but noticeable disconnect. I have to force my intuition to fit the definition.
- amalcon 17y ago
  The radius is only more fundamental to the notion of a circle because of the modern definition of a circle (set of all points exactly r away from the given point). One could also use a definition like: "Continuous set of points such that each is distance d from exactly one other, and further from none." The modern definition is convenient because it's a special case of a convenient definition of ellipse, but there's nothing terribly fundamental about it. And yes, every time you've done math with radians, that problem has haunted you. That's because we decided to use radians instead of "diameter-ans", to keep with the above definition.
abecedarius 17y ago
Yes, the alternative would cause different complications, but fewer of them. The pi-is-wrong pdf goes into it. It was programming a lot that brought me to realize this. I got in the habit of always questioning the primitives I'm given, in the hope of simplifying the definitions built on them. It was a little boggling to notice that pi, something I'd grown up with, deserved the same kind of questioning.
mhartl 17y ago
The area of a circle, for example, would be r^2 pi/2. This is actually my favorite example of why pi is wrong—it's the "exception" that proves the rule. To see why, set τ = C/r = 2 pi, and then consider the following chart of common quadratic forms: integral of u 1/2 u^2 kinetic energy 1/2 m v^2 distance fallen 1/2 g t^2 spring energy 1/2 k x^2 triangular area 1/2 b h circular area 1/2 τ r^2 We see that, far from causing "different complications", using the right circle constant brings the area of a circle into a more natural form. The 1/2 in the formula for circular area is actually a missing factor; using tau in place of pi restores it. N.B. I made a previous comment along these lines, but put it in the wrong place. If you're still a "pi is wrong" skeptic, considering the improvement in radian angle measure may yet convince you: The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc. In these terms, Euler's formula would be recast as e^(i τ) = 1 That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.
- christopherolah 17y ago
  Hm, is using tau a standard convention? I've thought pi is horrible... and adopted using a loopy pi as 2pi. I loop the first vertical line over the horizontal, right, and then make it the horizontal, and so on. If that makes any sense. I'm thinking of making a LaTeX package containing it... (and also some other personal conventions) Also, to strengthen your point, 2pi makes the radian system a lot more natural.
  - mhartl 17y ago
    I made up the usage of τ myself, partially because of its typographic similarity to π, partially because it leads naturally to the usage "τ = turn". It's too bad that π has two legs while τ has only one; it would be poetic if π were 2τ, but it wasn't to be. N.B. I have a secret master plan to spread the use of τ, but this comment is too small to contain it. ;-)