Tau (2π)

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En matematicas, tau (τ) es una constante propuesta por Bob Palais, Peter Harremoes, Hermann Laurent, Fred Hoyle, Michael Hartl, y otros, como reemplazo para la constante del circulo, π.^{[1] [2] [3] [4]} Su principal argumento es que los circulos son definidos mas naturalmente por su radio que por su diametroT.^{[note 1]} El valor de τ = 2π, o aproximadamente 6.28318,^[5] aparece mas frecuentemente en matematicas.

Varios simbolos han sido sugeridos, incluyendo 2π (Laurent), (Palais), (Harremoes), and τ (Hartl). El simbolo τ fue escogido en referencia a turn, vuelta en ingles, dado que en matemáticas τ-radianes son equivalentes a una vuelta completa.

Proposed advantages

Palais and Hartl claim a number of advantages of using τ instead of π.

• The so called "special angles", that need to be memorized when using π, simply become fractions of a whole circle, that is , , , and . It is easier to explain that one eighth of a circle corresponds to radians than to radians.^[6] Hartl describes the use of pi in this context as a "pedagogical disaster".
• The factor 2π, present in many formulae, such as normal distribution and Fourier transforms, can be eliminated, thus simplifying them.^[4]
• The periodicity of the cosine and sine functions is τ instead of 2π, which is simpler and arguably more intuitive.^[4]
• The formula for the circumference of a circle becomes simply τr, without introducing a factor 2.
• The formula for area of a circle and the formula for area of a circle sector have identical forms, so students have to memorize only one formula instead of two. (A whole circle is just a circle sector with θ = τ)
• The formula for the area of a circle falls in line with the power rule for integrals (e.g. kinetic energy ). Instead of A = πr², it becomes .^[4]
• Euler's identity is more straightforwardly expressed in τ than it is in π: e^iτ = 1 instead of e^iπ = − 1, or as it is usually expressed, e^iπ + 1 = 0.^[4]
• Related to the above, an n-th root of unity is expressed as e^{τi / n} rather than e^{2πi / n}.
• The reactance of an inductor is τfL instead of 2πfL. Similarly, the susceptance of a capacitor is τfC instead of 2πfC.
• Frequencies stand out much more clearly in the (most common time-periodic) functions sin ωt, cos ωt, and e^iωt. For example sin (876.89 τ t) is immediately recognizable as an 876.89 Hz sine wave while sin (1753.78 π t) is not.
• The sum of the exterior angles of a polygon is τ.

Possible disadvantages

• The area of a circle is expressed as rather than πr².
• Tau has many other unrelated mathematical meanings.
• Euler's identity is more precise in the simple phrasings above with π than with τ;^{[cita requerida]} e^iπ = − 1 is more specific than e^iτ = 1. That is, the first implies that i π is half of a period of the exponential function, plus a period, but not a period, and the latter implies that i τ is a period of the exponential function.
• The measures of the interior angles of a triangle in Euclidean space always add up to π. In general, the sum of the interior angles of a simple n-gon is (n − 2) π.
• The sum of the angles in a linear pair is π. ^{[cita requerida]}
• When a transversal intersects two parallel lines, the sum of the interior angles on the same side of the transversal is π. ^{[cita requerida]}

Historical notes

• Paul Laurent in Traité D'Algebra wrote equations using 2π as a single symbol.^[1]
• The famous Feynman point (six consecutive 9s early in the decimal digits of π) appears one digit earlier in τ, and is seven digits long instead of six (3.14...349999998... * 2 = 6.28...699999996...).^[7]
• Following the tradition of the pi day (March 14, or 3.14), "2pi day" has been celebrated^{[8] [9] [10]} on June 28 (6.28), and became more widely adopted (as "tau day") since the publication of Hartl's manifesto in 2010. It has been argued that this is a "perfect day" because 6 and 28 are the two first perfect numbers.^{[2] [11] [12]}

See also

• 2π theorem

Notes

1. ↑ For example: x = r cos(''t'') and y = r sin(''t''), or r² = x² + y²

References

Further reading

• Hartl, Michael (28 de junio de 2010). «The Tau Manifesto». Tau Day. Consultado el 03-07-2011.
• «The Pi Manifesto» (4 de julio de 2011). Consultado el 07-07-2011.
• «On National Tau Day, Pi Under Attack», NewsCore, Fox News Channel, 28 de junio de 2011. Consultado el 03-07-2011.
• Springmann, Alessondra. «Tau Day: An Even More Fundamental Holiday Than Pi Day», PC World, 28 de junio de 2011. Consultado el 03-07-2011.
• Palmer, Jason. «'Tau day' marked by opponents of maths constant pi», BBC News, 28 de junio de 2011. Consultado el 03-07-2011.

Enlaces externos

• Vi Hart (Author). Pi Is (still) Wrong (flv) [YouTube].
• Kevin Houston. Pi is wrong! Here comes Tau Day (flv) [YouTube].
• Robert Dixon (Author). Pi ain't all that (flv) [YouTube].