Is $22/7$ an often used approximation for $\pi$?

Question

It is $\pi$-day and the internet is full of stories about $\pi$. One story mentions that "an approximation -- $22/7$ -- is used in many calculations."

I have never actually used $22/7$ as an approximation for $\pi$ in a calculation ... or at least I don't have a memory of doing that.

My questions are

1. Is it true that today $22/7$ is an approximation that is used in many calculations? Are there, for example, any computers that use this approximation? Is $22/7$ in general (in the USA) often used in handcalculations that involve $\pi$?

2. I could imagine that before the widespread use of calculators, people would actually use $22/7$ as an approximation. Are there any historical examples of this?

Answer 1

No, of course computers don't use $\frac{22}{7}$ as an approximation for $\pi$.

But people often do, when they need a quick back-of-the-envelope calculation of something, don't want to pull out a calculator/computer, and only need a few significant digits of precision anyway.

Answer 2

The two better known rational approximations of $\pi$ are $\dfrac{22}7$ ($3$ correct figures) and $\dfrac{355}{113}$ ($7$ correct figures).

Personally I don't use them as I consider that they bring no real benefit over the straight decimal representations $3.14$ and $3.141593$.

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These approximations did not arise by accident: they are the most efficient for a given magnitude of the denominator and can be retrieved from the theory continuous fractions.

Repeatedly that the inverse of the fractional part, and you get a sequence of integer parts

$$3,7,15,1,292\dots$$ so that

$$\pi=3+\frac1{7+\frac1{15+\frac1{1+\frac1{292+\cdots}}}}.$$

The first approximations can be drawn from this:

$$3,\frac{22}7,\frac{333}{106},\frac{355}{113},\frac{103993}{33102}\cdots$$

Answer 3

For the first part of your question about whether computers use this approximation, I think the answer is generally "no." Most modern languages contain some sensible numerical constant in a library that has $\pi$ to machine precision. The preference should be to use that. Before that, like old-style Fortran, the "correct" way was to make your own constant using $\pi = 4 \tan^{-1} 1$. (Of course that doesn't mean that no one ever hand-coded 3.14 or 22/7 into their program, but there were always better, more accepted solutions.)

For the second part, there's a list of historical approximations at Wikipedia, for example. It lists an old one going back to Archimedes:

In the 3rd century BCE, Archimedes proved the sharp inequalities 223⁄71 < π < 22⁄7, by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively).

https://en.wikipedia.org/wiki/Approximations_of_%CF%80#Early_history

If true that shows it used as a bound.

Answer 4

You're right that 22/7 isn't used in actual calculations today. It's still the first of an interesting sequence of remarkably good approximations, in the following sense.

To approximate $\pi$ using tenths, just look at the beginning of the decimal expansion $\pi \approx 3.1$. You can be sure that you are within $1/20$ of the correct value. With $\pi \approx 22/7$ you can be sure that you are within $1/14 = 0.071428\ldots$, but in fact you're much closer than that: $\pi- 22/7 = 0.003069\ldots$. So $7$ is a very good denominator to use when looking for rational fractions near $\pi$. The next really good approximations are $333/106$ and $355/113$.

These good approximations come from the continued fraction for $\pi$ - see http://mathworld.wolfram.com/PiContinuedFraction.html .

Answer 5

One mathematical use comes from the continued fraction expansion $$3 + \frac{1}{7 + \frac{1}{15 + \frac{1}{1 + \frac{1}{292 + \frac{1}{1 + \ldots}}}}} $$

The "convergents" of this expansion are the approximations that you get by truncating the continued fraction. The first convergent is $3$, the next is $3 + \frac{1}{7} = \frac{22}{7}$, the next is $3 + \frac{1}{7 + \frac{1}{15}} = \frac{333}{106}$, and so on.

The mathematical importance of the continued fraction convergents $\frac{m}{n}$ of any real number $t$ is that they are very good approximations in a very precise sense: they are the only fractions such that $$\bigl|t - \frac{m}{n}\bigr| < \frac{1}{c \, n^2} $$ where $c = \sqrt{5}$ (thanks to @Ian for the constant which I could not dig out of my brain at the right moment).