A Pi Question: Why do we use the awkward approximation 22/7 ?

[Agent Smith]

TL;DR Summary

pi = 314/100 = 157/50, much easier to compute with than 22/7

As a high school student we were told to use $\frac{22}{7}$ as a rational approximation for $\pi$.
However, to the same level of accuracy, $\frac{314}{100} = \frac{157}{50}$ is also $\pi$ and since there's a $100$ and a $5$ in the denominator many calculations would've been far easier than using $\frac{22}{7}$.

Why then are we asked to use the more awkward approximation $\frac{22}{7}$?

 

[Frabjous]

Check out continued fractions
https://en.wikipedia.org/wiki/Continued_fraction#

The successive approximations generated in finding the continued fraction representation of a number, that is, by truncating the continued fraction representation, are in a certain sense (described below) the "best possible".

 

[Steve4Physics]

TL;DR Summary: pi = 314/100 = 157/50, much easier to compute with than 22/7

As a high school student we were told to use $\frac{22}{7}$ as a rational approximation for $\pi$.
However, to the same level of accuracy, $\frac{314}{100} = \frac{157}{50}$ is also $\pi$ and since there's a $100$ and a $5$ in the denominator many calculations would've been far easier than using $\frac{22}{7}$.

Why then are we asked to use the more awkward approximation $\frac{22}{7}$?

Off the top of my head, here are a few reasons…

22/7 is short and easy to remember – easier (for me anyway) than 314/1000 (or 157/50).

22/7 is a little more accurate than 314/1000. Check it for yourself!

22/7 is accurate to about 0.04% which is more than good enough for most practical and teaching purposes.

For teaching purposes, the small numerator and denominator (22 and 7) facilitate elementary problems when a calculator is not allowed.

 

[fresh_42]

TL;DR Summary: pi = 314/100 = 157/50, much easier to compute with than 22/7

As a high school student we were told to use $\frac{22}{7}$ as a rational approximation for $\pi$.
However, to the same level of accuracy, $\frac{314}{100} = \frac{157}{50}$ is also $\pi$ and since there's a $100$ and a $5$ in the denominator many calculations would've been far easier than using $\frac{22}{7}$.

Why then are we asked to use the more awkward approximation $\frac{22}{7}$?

$\left|\pi -\dfrac{22}{7}\right| \approx 0.00126$ and $\left|\pi -\dfrac{157}{50}\right| \approx 0.00159\,.$

a) $22/7$ is a bit closer.
b) $22/7$ is easier to memorize.
c) $22/7$ is used in the book.
c) $22/7=154/49$ has smaller numbers than $157/50\,.$
d) $22/7$ practices manual divisions, dividing by $100$ not really.

All these arguments are guesses. "Why then are we asked ..." cannot be answered since we do not know who you asked, upon which ground they made this decision (curriculum, or one of the answers I speculated above), or even when you have been asked. The device I write this answer with uses $3,1415926535897932384626433832795$ so I don't need to think about approximation since the one the device uses is more than good enough for all real purposes. My thoughts kick back in when I think about why all these numbers must be approximations. Which one is better isn't important anymore.

 

[PeroK]

Also, there are 22 bones in the human cranium and 7 pillars of wisdom.

 

[Vanadium 50]

7 pillars of wisdom.

And seven deadly sins. I guess you can place them in one-to-one correspondence.

People know, or should know, 3.14 = 314/100 without being taught. 22/7? You have to teach them. (Like 355/113)

 

[fresh_42]

And seven deadly sins. I guess you can place them in one-to-one correspondence.

People know, or should know, 3.14 = 314/100 without being taught. 22/7? You have to teach them. (Like 355/113)

\begin{align*}
\pi \approx \dfrac{3}{\dfrac{57}{60}+\dfrac{36}{3600}}=3.125\quad&\text{Babylonian; for show-offs}\\[6pt]
\pi \approx 256/81 = 3,\overline{160493827}\quad&\text{China & Egyptian; for show-offs who were late to the history class}\\[6pt]
3+\dfrac{10}{71} < \pi < 3+\dfrac{10}{70} \quad&\text{Archimedes; for classical educated people}\\[6pt]
\pi \approx 22/7=3,\overline{142857}\quad&\text{Archimedes; for craftsmen (sic!) between 16th century and slide rules}\\[6pt]
\pi \approx 3.14 \quad&\text{modern times; for people who like engineering jokes}\\[6pt]
\pi \approx 3.1415926535897932384626433832795 \quad&\text{modern times; for lazy people who know calc.exe}
\end{align*}

 

[fresh_42]

Why then are we asked to use the more awkward approximation $\frac{22}{7}$?

Since when did we use decimal numbers and are not using electronic devices?

Compare this with the time period we were using quotients. 22/7 was simply the approximation for many centuries and 3.14 was only even a question since the fifteenth, but realistically the end of the sixteenth century until about 1850, the year when production of slide rules in large numbers began that had π as a mark on them. This means at most three centuries of decimal numbers compared to eighteen centuries of ratios.

Now ask yourself why people should have replaced the approximation of π by 3.14 if they were already used to the better value of 3.1428... for centuries. 3.14 was simply never a true question.

Why then are we asked to use the more awkward approximation $\frac{22}{7}$?

seems to me to be the question of a school kid who has more trouble dividing by seven than constructors and carpenters had over eighteen, realistically twenty-one centuries without even using decimal numbers.

 

[Pikkugnome]

I doubt that woodworkers have used precise approximations for pi. A piece of wood longer than one meter is too warped. Has anyone ever seen a wood joint which doesn't grin, if not when build, but soon after.

 

[fresh_42]

I doubt that woodworkers have used precise approximations for pi. A piece of wood longer than one meter is too warped. Has anyone ever seen a wood joint which doesn't grin, if not when build, but soon after.

Sorry for not referencing my source.

In Western cultures, these calculations by Archimedes represented the status quo in terms of the accuracy of knowledge of π for a very long time - as in many other social and cultural areas. It was not until the 16th century that interest was revived. During this period - and until the advent of slide rules and pocket calculators - craftsmen used Archimedes' approximation.

https://de.wikipedia.org/wiki/Kreiszahl

Reliability of Wikipedia:

They sent 50 pairs of Wikipedia and Britannica articles on scientific topics to recognised experts and, without telling them which article came from which source, asked them to count the numbers of errors (mistakes, misleading statements or omissions). Among the 42 replies, Britannica content had an average of just under 3 errors per article whilst Wikipedia had an average of just under 4 errors — not as much difference, perhaps, as most people would expect.

https://blogs.nature.com/nascent/2005/12/comparing_wikipedia_and_britan_1.html (2007)
https://epub.uni-regensburg.de/15565/ (2010)

Reliability of those sources, see the links and references therein.

 

[Vanadium 50]

I'm trying to imagine a case where you need to know π in advance when building something and where 1/25 of a percent isn't good enough.

 

[Frabjous]

I'm trying to imagine a case where you need to know π in advance when building something

Knowing the magnitude would have helped here

 

[fresh_42]

I'm trying to imagine a case where you need to know π in advance when building something and where 1/25 of a percent isn't good enough.

I guess nobody has ever thought about anything else between $250$ BC and the $20$th century when someone was asked to implement $\pi$ in a calculator.

$22/7$ was taught and used. Why bother anything that is already known(!) within a margin of $2 \textperthousand$, I mean $2 \permil$, I mean $2$ per mille, namely $1/7 - 10/71=1/497=2/994$? (That **** MathJax ignores these LaTeX commands, and that **** spell checker wants me to write per mile.)


Such a question is academic, and senseless if the proposed replacement is a step backward. Those Roman aqueducts and medieval cathedrals at least don't seem to collapse anytime soon.

 

[Pikkugnome]

I do not really know about the topic, but I don't see an accurate approximation of pi being a thing. If you imagine making a wheel with a lining of metal strip along its perimeter. Is anyone going to measure the radius of the wheel and multiply with 44/7 to get an estimate what is the length of the strip. I know nothing about blacksmithing, but I would assume a standard length is known to a smith and then he hammers the ends to meet. I say the hammering is at odds with accuracy. I understand 22/7 was used for pi, but did it matter.

 

[Agent Smith]

@Frabjous Irrationals seem representable as continued fractions. Does that answer my question? I non intelligo.

22/7 is a little more accurate than 314/1000. Check it for yourself!

22/7 is accurate to about 0.04% which is more than good enough for most practical and teaching purposes.

As far as I can tell, the digits of $22/7$ and $\pi$ match in all $3$ places, doable with a denominator that's a power of $10$, no? Accuracy is a matter of finding the right numerator for a denominator that's a power of $10$.

My hunch is that $22/7$ is of historical importance. Was it Archimedes who used it as the lower bound for $\pi$?

$\left|\pi -\dfrac{22}{7}\right| \approx 0.00126$ and $\left|\pi -\dfrac{157}{50}\right| \approx 0.00159\,.$

a) $22/7$ is a bit closer.
b) $22/7$ is easier to memorize.
c) $22/7$ is used in the book.
c) $22/7=154/49$ has smaller numbers than $157/50\,.$
d) $22/7$ practices manual divisions, dividing by $100$ not really.

All these arguments are guesses. "Why then are we asked ..." cannot be answered since we do not know who you asked, upon which ground they made this decision (curriculum, or one of the answers I speculated above), or even when you have been asked. The device I write this answer with uses $3,1415926535897932384626433832795$ so I don't need to think about approximation since the one the device uses is more than good enough for all real purposes. My thoughts kick back in when I think about why all these numbers must be approximations. Which one is better isn't important anymore.

Nice! Non cogito that computation by $22/7$ is easier than computation by $314/100$.
If you have a number that's not a multiple of $7$ had it! With $314/100$ we're guaranteed to get a nice number (it's just a matter of moving the decimal point for multiplication with $\pi$).
However, dividing by $314/100$ will be a problem (maybe that's why ...???)

@fresh_42 you mean to say $22/7$ is a hold over, a relic of past number systems? Computation with $7$ is difficult, no? Would you rather (say) divide by $7$ or $2$ or $5$?

I'm trying to imagine a case where you need to know π in advance when building something and where 1/25 of a percent isn't good enough.

Interesting.

I do not really know about the topic, but I don't see an accurate approximation of pi being a thing. If you imagine making a wheel with a lining of metal strip along its perimeter. Is anyone going to measure the radius of the wheel and multiply with 44/7 to get an estimate what is the length of the strip. I know nothing about blacksmithing, but I would assume a standard length is known to a smith and then he hammers the ends to meet. I say the hammering is at odds with accuracy. I understand 22/7 was used for pi, but did it matter.

Interesting.

 

[fresh_42]

@fresh_42 you mean to say $22/7$ is a hold over, a relic of past number systems?

Yes. It is an approximation that served well for around about two thousand years, plus it is extraordinarily accurate.

Computation with $7$ is difficult, no?

That depends on what you mean by computation. To find 1/5 or 1/7 by geometric means is comparably difficult, so why not use the better value? To find 1/5 or 1/7 on a pocket calculator is equally difficult. So what remains? Where and when did what people actually need to use an approximation? It is engineers and school kids. Engineers were used to 22/7 anyway, so there was no need to change for the worse. School kids are supposed to practice arithmetic. So why make life easy for them? They are supposed to learn long divisions, not making things easier and worse.

Your question simply didn't ever occur in the history of mankind. Only in classrooms, and roughly only since the seventeenth century, if at all. And until our modern times, no school kid would have ever complained that they had to divide by seven.

Would you rather (say) divide by $7$ or $2$ or $5$?

I don't care. Firstly, because I use calc.exe with $\pi$ implemented. Secondly, I would use 3.14159 if it wasn't implemented. Thirdly, dividing by seven doesn't take very much longer and I know particularly 1/7 by heart since it occurred so often during my school years. The rest of the formulas take much longer, e.g. calculating squares, cubes, or roots.

 

[Agent Smith]

@Halc

Gracias for explaining the importance of circularity and how $\pi$ is important to anything that can be reduced to "how many bricks" one might need for the circumference of a circle.

I wonder now about how objects that have circular components (spheres, cylinders, cones) are constructed at all and constructed as accurately as possible. You said $\pi$ isn't necessary to construct a circle (compasses don't use $\pi$). I can imagine using $\pi$ as some kind of a check for how circular an object is: measure circumference (c), measure diameter (d) and find the ratio (c/d). The closer c/d is to $\pi$ the more circular the object is, oui?

 

[Halc]

I've not been reading most of the posts, but I can comment on some recent ones.
You seem to be confusing numbers with their representations.

@FrabjousMy hunch is that $22/7$ is of historical importance. Was it Archimedes who used it as the lower bound for $\pi$?

22/7 is simple. The numbers are smaller and thus simpler than 314/100. The latter is only convenient if decimal arithmetic is used, and arguably not even then since I can multiply reasonable numbers by 22 in my head, and also divide by 7, but multiplying by 314 probably takes longer.
A non-human would probably not use decimal arithmetic, but then a non-human would probably not look for a simple approximation. A binary computer can divide by 2 with a simple shift operation, but when presented with x/y with y being 2, it will perform the op the hard way and not opt for the easy optimization.

22/7 is greater than pi, so I suppose it can express one upper bound of it, not a lower bound.

I wonder now about how objects that have circular components (spheres, cylinders, cones) are constructed at all and constructed as accurately as possible.

They're typically not constructed as accurately as possible, as evidenced by the total lack of a circle in the universe. They're made to specified tollerances, and it's an engineering problem. The LHC has very tight tollerances. A round building not nearly so much, and nowhere near as much effort needs to be expended on making it closer.

You said $\pi$ isn't necessary to construct a circle (compasses don't use $\pi$).

Another example: There are circles in nature. I have a spot nearby just below a small waterfall that gets a circular chunk of ice (perhaps 15 meters in diameter?) any winter when the conditions are right. It is very close to a circle, with the radius deviating probably less than a cm. There's a very large version of one about 500 km west of me.

https://en.wikipedia.org/wiki/Ice_circle

To my vast surprise, the 3rd picture down, long exposure, is the one by me. I can walk to that one. Ours is a little bit famous!

I can imagine using $\pi$ as some kind of a check for how circular an object is: measure circumference (c), measure diameter (d) and find the ratio (c/d). The closer c/d is to $\pi$ the more circular the object is, oui?

Not so sure. For one, if it isn't a circle, it doesn't have a diameter, by definition. But one can take a cross section of a near-circle. So I hand-draw a crude circle on a piece of paper. I use a wheel-pen that measured the length of the line I draw using a little trailing wheel, similar to the ones they put on cars for accurate road-length measurement. My 'circle' has a circumference of exactly 31.4 cm, but it's a botched job. Few can freehand a decent straight line, let alone a decent circle. Anyway, a cross section through the middle (a diameter) taken in the right place, will be 10 cm. Is this evidence that it is a good circle? Certainly not. The diameter measured at a different angle will be different.

Similarly, I have a perfect globe and I want to hand-draw an equator on it using a paint truck. But the truck goes around well to the north and its path deviates from side to side, leaving a bad circle again. But it's length is again exactly (by chance) $\pi$ times the radius of the globe, and this time every point is equidistant from that globe center.

Point is, your criteria is insufficient evidence of circularity.

 

[Vanadium 50]

Why then are we asked to use the more awkward approximation 22/7?

Who asked you to do this?

Is this any more than the student's complaint "when am I ever going to use this stuff?"

 

[fresh_42]

$1.265/1000$ versus $1.59/1000$ difference to the actual value of $\pi.$ Isn't this a quite academic discussion? Approximations are chosen according to a purpose. You need to know the purpose to decide which one is better or worse. $22/7$ is easy to memorize, easy to calculate with, and surprisingly accurate.

It is the upper bound that Archimedes has given us about 2,270 years ago! His lower bound has a distance of $0.75/1000$ from $\pi$ and is even more accurate. But nobody wants to calculate with $1/71$ so we have chosen to take his upper bound $22/7$ instead.

Decimal representations of numbers were foreign to him. The ancient Greeks considered numbers as ratios of lengths. It is astonishing, that already known for thousands of years, e.g. the Babylonians used $60$ as the base, positional notation didn't make its way into everyday use until recently. And we still distinguish between numbers that are ratios and those that are not. And the decimal system is in my opinion still artificial! Consider how long we are already debating about $3.14,$ or how often students mention $0.\overline{9}$ and want to discuss whether this equals one or not, or quite generally confuse how we write numbers by the numbers themselves. Whenever we want to write $\pi$ accurately up to all its infinitely many digits, then we can do that: we write $\pi.$

I haven't checked it, but Maya would have written $\pi \approx 3.2GCEG9GBHJ9D21HH2I4G.$

 

[Agent Smith]

@fresh_42 don't quite follow but an interesting equation is $\frac{24 - 2}{6 + 1} = \frac{21 + 1}{7} \approx \pi$

 

[Agent Smith]

Point is, your criteria is insufficient evidence of circularity.

You're of course stating the error that is inherent in measuring system, oui?

If I find an circular object and measure its circumference and its diameter (both accurately) and the find the ratio between the two is $\pi$ then it could be something other than a circle???

 

[fresh_42]

If I find an circular object ...

This does not exist. A circle is a philosophical construct or (equivalently) the solution of an equation. They do not exist in real life.

... and measure its circumference and its diameter (both accurately) ...

... which is again impossible ...

... and the find the ratio between the two is $\pi$ ...

You won't. You will measure an approximation of $\pi.$ Always.

You can derive by mathematical means that the ratio between the circumference and the diameter of the set of points that are equally far from a distinguished point is always the same number. We call the distinguished point center, the set of solution points a circle, and the resulting ratio $\pi.$ All these are theoretical terms, or if you like that better, ideas in Plato's heaven. They do not exist in our discrete and imperfect reality.

... then it could be something other than a circle???

This is a circular argument. You basically asked: If there is a circumference and a diameter, is it a circle? You began already by assuming the existence of a circle and then ask at the end whether it is a circle. That doesn't make sense. $\pi$ itself is just a number like any other number, too. Ok., it is a transcendental number and plays an astonishing role in our description of nature, but so is $e.$ However, you have asked about approximations and in this respect it isn't qualitatively any different from, say $2^\sqrt{2}.$

 

[Agent Smith]

@fresh_42

We use a compass, draw an object such that all of its points are equidistant from a given point. Measure its perimeter (circumference) and its diameter (2 opposite points and the center). If this turns out to be (close to) $\pi$ is the object not a circle?

 

[Frabjous]

We use a compass, draw an object such that all of its points are equidistant from a given point. Measure its perimeter (circumference) and its diameter (2 opposite points and the center). If this turns out to be (close to) $\pi$ is the object not a circle?

Do you know a good way to measure the perimeter?

 

[Agent Smith]

Do you know a good way to measure the perimeter?

A piece of string? Pain the rim of a wheel and roll it on the ground? Je ne sais pas.

 

[fresh_42]

@fresh_42

We use a compass, draw an object such that all of its points are equidistant from a given point. Measure its perimeter (circumference) and its diameter (2 opposite points and the center). If this turns out to be (close to) $\pi$ is the object not a circle?

Yes, close to. How close depends on the microscope you have at hand.

 

[Frabjous]

A piece of string? Pain the rim of a wheel and roll it on the ground? Je ne sais pas.

So you will need to take measurement accuracy into account if you want to distinguish between a “good” circle and a “great” circle.

 

[Mark44]

@fresh_42 don't quite follow but an interesting equation is $\frac{24 - 2}{6 + 1} = \frac{21 + 1}{7} \approx \pi$

How is this interesting? It's nothing more than a slightly disguised $\frac{22}7$, the well-known and long-used approximation for $\pi$.

 

[Agent Smith]

How is this interesting? It's nothing more than a slightly disguised $\frac{22}7$, the well-known and long-used approximation for $\pi$.

Cogito it should be $\frac{21 + 1}{6 + 1}$

 

[Agent Smith]

So you will need to take measurement accuracy into account if you want to distinguish between a “good” circle and a “great” circle.

Si and I also say that the closer c/d is to $\pi$ the more circular it is.

Is $\pi$ a measure of a circle's curvature?

 

[DrJohn]

Also, consider the speed of calculation without a calculator compared to using a three or five figure decimal value.
And the final result is often close enough for the majority of non-scientific requirements.

 

[fresh_42]

Is $\pi$ a measure of a circle's curvature?

The usual curvature of a circle depends on its radius. Imagine a circle that is astronomically large. Wouldn't you agree that such a circle is far less curved than one you can draw in the sand? Therefore, $1/r$ is the usual measure for the curvature. The larger the radius the less curved is the circle.

 

[fresh_42]

Cogito it should be $\frac{21 + 1}{6 + 1}$

The Babyloninas thought it should be $\dfrac{3}{\dfrac{57}{60}+\dfrac{36}{3600}}$ and Archimedes thought it should be $\dfrac{223}{71}.$

 

[Frabjous]

Si and I also say that the closer c/d is to $\pi$ the more circular it is.

Is $\pi$ a measure of a circle's curvature?

The curvature of a circle is one over its radius.