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Inspired by a paper (from 2001) entitled Pi is Wrong:

Why is $\pi$ = 3.14... instead of 6.28... ?

Setting $\pi$ = 6.28 would seem to simplify many equations and constants in math and physics.

Is there an intuitive reason we relate the circumference of a circle to its diameter instead of its radius, or was it an arbitrary choice that's left us with multiplicative baggage?

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It's an arbitrary choice that's left us with multiplicative baggage. – Qiaochu Yuan Mar 14 '11 at 13:58
8 – Brian Mar 14 '11 at 14:01
Setting pi to 6.28 would simplify things considerably. – Mitch Mar 14 '11 at 14:13
Would you rather have $2\pi r$ and $\pi r^2$ or $\pi r$ and $\frac\pi 2 r^2$? =) – Jens Mar 14 '11 at 14:55
@Jens: The latter makes apparent the fact that area is the integral of the circumference (as well as higher-dimensional versions of this fact), $r\to \frac{1}{2}r^2$. Now that I'm used to the 'multiplicative baggage', I'd have to say I like it more, aesthetically. – anon Jul 13 '11 at 6:02
up vote 41 down vote accepted

For mathematicians, $2\pi$ is a more natural number than $\pi$ because this is the circumference of the circle. The value $2\pi$ appears in things related to the circle such as Fourier transforms (as the complex units form a unit circle with circumference $2\pi$). Thus the symmetric, unitary formula for the Fourier transform in terms of angular frequency $\omega$, for a function $f(x)$ is:
$$ \hat{f}(\omega) = \frac{1}{\sqrt{2\pi}}\int f(x)\;e^{-i\omega x}\;dx$$ The subject has been surfacing recently, for instance see Science on, June 29, 2011: "Mathematicians want to say goodbye to pi."

The original use of $\pi$ had to do with the relationship between the circular measurement of circles (their circumferences) and the straight line measurement of them (their radius or diameter). If $\pi = 3.14...$ then it is the diameter that is related to the circumference. If $\pi = 6.28...$ then it is the radius that is related.

Relating the radius to the circumference may be more convenient for modern students, but $\pi$ was defined by carpenters and other artisans. It's easier and more accurate to measure the diameter than the radius. For example, if the object is a hoop, one always measures the diameter first and from this one obtains the radius.

Given a circle (perhaps on paper) one instinctively measures its diameter by maneuvering a ruler to obtain the largest difference between opposite sides. To measure the circle's radius an additional point is required, the center of the circle. This situation is fairly common in construction. For example, if one cuts a tree in two, the diameter is easily measured whereas the radius can be measured easily only if the tree has grown and been cut symmetrically. Otherwise the center of the circle must be found by construction and this process introduces measurement error and additional possibilities for mistakes.

In short, $\pi$ is defined as: $$\pi = \frac{\textrm{circumference}}{\textrm{diameter}}$$ because of the historical fact that $\pi$ was used for practical construction.

The oldest example of a calculation that a modern person would use $\pi$ in is the Rhind Mathematical Papyrus. The papyrus includes various questions. Unfortunately none requires the computation of a circumference of a circle. However, there is a problem where one computes the volume of a cylindrical granary. In that calculation, they use the diameter of the granary (as 9), rather than the radius of the granary (i.e. 4.5). Thus the oldest evidence we have for mathematical calculation verifies that the ancients were more inclined to measure diameters than radii. And consequently, $\pi$ was naturally defined by them as the ratio of the diameter to the circumference, rather than the ratio of the radius to the circumference.

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Do you have a reference for the statement that $\pi$ was defined by carpenters and artisans? I thought geometers and carpenters were in very different social classes in Ancient Greece. – Douglas Zare Apr 1 '11 at 8:40
Earliest textual evidence for pi is from the Egyptians; in 1900 BC they were already approximating pi by 256/81 = 3.16. And by "carpenters" I really mean those in the construction trades in general such as masons, architects, engineers, carpenters, metal workers, etc. Among those classes, I expect that there are people fully as smart as among the geometers. Even today, higher pay for engineers, business, and practical trades in general attracts people who would have made brilliant academics. So I see no reason to believe that pi was invented by academics. – Carl Brannen Apr 2 '11 at 0:01
In Ancient Greece, geometry was among the liberal arts, the education for a citizen, as opposed to a worker. Even if the carpenters invented $\pi$ or $\tau$, that does not explain how the geometers of Ancient Greece came to use $\pi$. Perhaps things were very different in Egypt. Perhaps the artisans were literate there, and somehow their writings were passed on to the Greeks and then to us. Do you have a reference, or are you just saying it could have happened? – Douglas Zare Apr 2 '11 at 2:39
@Douglas; Geometry was invented long before the Greeks. Pythagoras studied in Babylon and Egypt. Our oldest reference to pi has to do with the amount of wheat in a silo, see: This is practical calculation by practical people and is a function of the diameter and height: $V = (8d/9)^2h$, rather than the radius and height (which is actually simpler with the radius i.e. $V=\pi r^2h$, but the papyrus was apparently before the use of a letter to represent $\pi$). The diameter is easier to measure for someone who goes around taxing grain. – Carl Brannen Apr 2 '11 at 2:51
Similarly, the bible describes a bowl by its diameter and circumference famously assuming pi=3, rather than by its radius and circumference and having a ratio of 6. I have students who could easily measure the circumference and diameter of a hoop but who would be hard put to measure the radius since the hoop doesn't exist at the center point. – Carl Brannen Apr 2 '11 at 19:24

Considering that circular trigonometry is not the only one and that a class of curves can possibly be attributed an analogue of $\pi$, $2\pi$ does indeed seem to be more convenient in certain cases. Lemniscate constant plays the same role as $\pi$ for the lemniscate and since its definition is strictly analogous,i.e half length of the curve, the complimentary formula for lemniscate functions, for instance, looks like this: $$\operatorname{sinlemn} \phi = \operatorname{coslemn}\left(\frac{\tilde{\omega}}{2}-\phi'\right)$$ while it could happily do without $\frac{1}{2}$

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