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

Question

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?

It's an arbitrary choice that's left us with multiplicative baggage. — Qiaochu Yuan, Mar 14 '11 at 13:58
Can someone give an example of an equation or a constant that would be simplified considerably? — ypercubeᵀᴹ, Mar 14 '11 at 14:51
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
Are there any arguments for π being "right"? If so, what are they? The τ manifesto was pretty convincing for me... — endolith, Mar 23 '11 at 19:56
Because then, "My turtle Pancho will, my love, pick up my new mover Ginger.", will be regarded as complete nonsense. You wouldn't want that, would you? — Mateen Ulhaq, Jun 05 '11 at 05:13
@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 06:02
I really think that this question should be closed. This is like asking whether 0 is a natural number. IOW it is a poll - not a question. — Jyrki Lahtonen, Jul 13 '11 at 06:43
@zulon: Thanks for the link, though those are some pretty weak arguments. If that's the best they can come up with, τ wins. — endolith, Sep 16 '11 at 14:36
More discussion and arguments back and forth here: http://spikedmath.com/forum/viewforum.php?f=30 — endolith, Sep 16 '11 at 15:50
Would you rather have $e^{\pi i}+1=0$ or $e^{\frac{\tau}{2}i}+1=0$? — E.O., Jun 15 '12 at 01:32
@E.O., I rather have the second. It explains itself. Half a turn in the imaginary plane plus 1 is zero. Also, $e^{\tau i}=1+0$ (and the zero was there from the $i\sin(\tau)$). — JMCF125, Jul 13 '13 at 15:04

score 46 · Accepted Answer · edited Jun 15 '12 at 00:29

46

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 MSNBC.com, 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.

edited Jun 15 '12 at 00:29

Théophile

24,627

answered Apr 01 '11 at 05:59

Carl Brannen

2,199

8

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 01 '11 at 08:40
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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 02 '11 at 00:01
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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 02 '11 at 02:39
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@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: http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus 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 02 '11 at 02:51
Same reference, also note that in computing the area of the circle, they take the measurement 9 to be the diameter, not the radius. The ancients used diameters because they were intensely practical people. – Carl Brannen Apr 02 '11 at 02:57
Geometry was studied independently in China and Japan. However, as far as I know, the conventions used by medieval Chinese mathematicians do not affect the notation we use today because they didn't pass their results or conventions to the West. The Ancient Greeks did pass on their results through Euclid, Archimedes, etc. So, where did the Ancient Greek mathematicians learn about $\pi$? Did they come up with it themselves, or did they learn of it through the writings of Egyptian tax collectors? Today, we try to put practical examples in elementary math texts, far more than in Euclid's Elements. – Douglas Zare Apr 02 '11 at 04:30
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By the way, that Wikipedia reference shows the approximation $\sqrt{\frac{\pi}{4}} \approx 8/9$ which doesn't indicate why we use $\pi$ instead of $\tau = 2\pi$ if our notation came from the Egyptians. Problem $48$ is also about areas, not the circumference, and it approximates the area of a circle inscribed in a square of side $9$. Again, that doesn't make it clear that they used $\pi$ instead of $\tau$. In fact, it looks like they would prefer a symbol for $\pi/4$ instead. – Douglas Zare Apr 02 '11 at 04:45
The definition of pi is the ratio of the circumference to the diameter. Presumably this is not described on the papyrus because the problem is too simple. The ratio for the circumference was surely known to the ancients long before the use of pi in the area of the circle. So what's important in the papyrus is "how the ancients described circles". Thus Problem 48 is described as a circle of diameter 9, not a circle of radius 9/2, and they were using diameters, not radius, and the natural definition is 3.14. – Carl Brannen Apr 02 '11 at 19:22
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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 02 '11 at 19:24
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-1 Please keep the tau manifesto and other religious wars outside of math forum. Ok. If you must: For the record, I am firmly in the $\pi$ camp. Can't turn back the clock, put the genie back into the bottle or such. Do observe that the people who started putting this idea forward are physicists. http://www.thepimanifesto.com/ – Jyrki Lahtonen Jul 13 '11 at 06:42
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For the record, I think the $\tau$ idea is a bad joke and that before it happens, spelling in English will be made phonetic. – Carl Brannen Jul 13 '11 at 10:44
Holy cow, this is a great answer. When I started reading, I thought there was nothing intelligent that could be said on the subject, and I was delighted to discover that I was mistaken. – MJD Aug 25 '12 at 02:14
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The mathematicians who appear responsible for popularizing the use of the letter $\pi$ (Jones and Euler) were much more prone to use radii and defined it using radii rather than diameter, introducing very explicitly a factor of 1/2. Euler therefore describes it as "half the circumference of a circle of radius one" in his Analysis of the Infinite, and Jones "half the periphery". – Marcel Besixdouze Nov 13 '13 at 08:37
@CarlBrannen 'before it happens' is a strange choice of words. We can use $\tau$ now if we want. In contrast, writing phonetic English is not currently practical. – May 27 '16 at 00:39

score 13 · Answer 2 · edited Nov 04 '17 at 05:26

For those stating that $\frac{\tau}{2}r^2$ is more cumbersome than $\pi r^2$ I would like to point out that the factor of $\frac{1}{2}$ is found naturally in many equations throughout physics and goes to show a fundamental relationship between the the area of a circle and the rest of physics that we miss when we cover it up with $\pi$. For example:

$\frac{1}{2}mv^2$ = kinetic energy, where $m$ is mass and $v$ is velocity.

$\frac{1}{2}kx^2$ = potential energy stored in a spring, where $k$ is the sprint constant and $x$ is displacement.

$\frac{1}{2}at^2$ = displacement, where $a$ is acceleration and $t$ is time.

That missing factor of $\frac{1}{2}$ shows an important relationship between the area of a circle and the rest of physics, something we've been missing due to our adherence to an old way of looking at circles. We don't make circles using their diameter, regardless of how easy diameter is to measure, we make circles using the radius. When you consider that all important factor of $\frac{1}{2}$ and you use $\tau \approx (6.28)$ instead of $\pi \approx (3.14)$ the area of a circle becomes $\frac{1}{2}\tau r^2$

score 6 · Answer 3 · edited Apr 13 '17 at 12:21

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 $\frac{1}{2}$

score 2 · Answer 4 · answered May 23 '17 at 08:02

No, $\pi$ is not wrong. But, if you prefer to use a constant that is equal to $2\pi$, you should look at the $\tau$ constant, which is exactly equal to $2\pi$. Here is a little video about it: Tau vs Pi - Numberphile

Both $\pi$ and $\tau$ denote a relation between the parameters of a circle though $\pi$ is the more researched and accepted one. The notion of $\tau$ came into existence because the definition and usage of $\pi$ seemed a little counter intuitive. Since a circle is defined as the set of points a fixed distance - the radius -from a given point, a more natural definition for the circle constant uses radius in place of diameter.

So, no $\pi$ is not wrong!!

score 1 · Answer 5 · answered May 23 '17 at 08:27

As others have said, several people have recently made the argument that the ratio of the circumference of a circle to its radius ($2\pi$, sometimes called $\tau$) is more natural than the ratio of the circumference of a circle to its diameter ($\pi$).

In my opinion, there is still a very good argument why $\pi$ is the better choice. If perimeter is what you're interested in, there is no need to restrict your thinking to circles. If you have any convex shape of constant diameter, the ratio of its perimeter to its diameter is $\pi$ (see this wiki article). But such shapes do not in general even have a radius.

Of course, if you are interested in area, not perimeter, the above is irrelevant. But in that case surely $\pi r^2$ is a more natural formula than $\tau r^2/2$.

score 0 · Answer 6 · answered Jun 27 '22 at 08:26

For some historical background:

When it because fashionable to use π in the 1700s, it was used for both 3.14 and 6.28. From Wikipedia:

Euler started using the single-letter form beginning with his 1727 Essay Explaining the Properties of Air, though he used π = 6.28..., the ratio of periphery to radius, in this and some later writing. Euler first used π = 3.14... in his 1736 work Mechanica, and continued in his widely-read 1748 work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1"). Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the Western world, though the definition still varied between 3.14... and 6.28... as late as 1761.

I've written up more about $2\pi=\tau$ here.

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

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