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22/7 as an Approximation for Pi


Date: 04/01/98 at 09:25:46
From: Bob Faulkner
Subject: Pi  or fraction 22/7

If diameter is 10 ft., I multiply by 3.1412 for circumference. Or, I 
use 22/7 * diameter. Question: by what process was 22 over 7 found to 
be the same as 3.1412?


Date: 04/01/98 at 11:24:27
From: Doctor Rob
Subject: Re: Pi  or fraction 22/7

The usual figure used is 3.1416, not 3.1412. Note that:

   22/7 = 3.142857142857...

Neither of these is actually the correct number. The correct number is 
called Pi (a Greek letter, pronounced "pye" in English), and its value 
is approximately

   Pi =          
    3.141592653589793238462643383279502884197169399375105820974944...

You see that the two numbers 3.1416 and 22/7 are both close to Pi and 
to each other in value.

The actual circumference of a 10 foot diameter circle is 10*Pi feet, 
or about 31.4159265 feet, or 31 feet 4.991118 inches. If you use
10*3.1416 = 31.416 feet, you get 31 feet 4.992 inches. These answers
differ by only 0.000882 inches -- very close! If you use instead
10*22/7 = 31.4285714 feet, you get 31 feet 5.142857 inches. These 
answers differ by 0.1517387 inches, or about 5/32 inches. That isn't 
bad, either.

The result of this analysis is that for most purposes, 22/7 is a 
pretty good estimate of Pi (three significant figures), and 3.1416 is 
better (five significant figures), but if you need very high accuracy, 
you should use an even more accurate estimate of Pi. Decide what 
accuracy you need, and round the value of Pi to that many significant 
figures. Then use that times the diameter to get the circumference.

-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   


Date: 04/01/98 at 11:29:20
From: BFaulk1234
Subject: Re: Pi  or fraction 22/7

Thank you for responding.

My question was: how were the figures 22 and 7 arrived at? How did 
they go about finding a fraction 22/7?

Thanks.


Date: 04/01/98 at 12:27:12
From: Doctor Rob
Subject: Re: Pi  or fraction 22/7

Aha! That's a different story!

To find a fraction that is close to any real number x, the procedure 
is as follows.

Take the integer part of x, usually written [x], and subtract it from 
x, to get x - [x]; then take the reciprocal, 1/(x-[x]) = x1. Then:

   x = [x] + 1/x1

Repeat this with x1 to get x2 = 1/(x1 - [x1]). Then:

   x1 = [x1] + 1/x2
   x = [x] + 1/([x1] + 1/x2)

Repeat this with x2 to get x3 = 1/(x2 - [x2]). Then:

   x2 = [x2] + 1/x3
   x = [x] + 1/([x1] + 1/([x2] + 1/x3))

This process can be continued indefinitely if x is irrational (as Pi 
is). The expression you get for x is called a "simple continued 
fraction" expansion of x.

The fractions gotten by truncating the infinite simple continued 
fraction after some finite number of steps are called the 
"convergents" to x:

   [x]
   [x] + 1/[x1]
   [x] + 1/([x1] + 1/[x2])
   [x] + 1/([x1] + 1/([x2] + 1/[x3]))
    and so on. 
    
These fractions are close to x, and get closer the farther down the 
list you go.

In the case of Pi, we have:

   Pi = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/...))))

so the convergents are:

   3,
   3 + 1/7 = 22/7 = 3.142857142857...,
   3 + 1/(7 + 1/15) = 333/106 = 3.14150943396...,
   3 + 1/(7 + 1/(15 + 1/1)) = 355/113 = 3.14159292035...,
   3 + 1/(7 + 1/(15 + 1/(1 + 1/292))) = 103993/33102   
                                      = 3.1415926530119...,

and so on. They are alternately larger and smaller than Pi, and 
getting closer and closer. The two most commonly used convergents are 
22/7 and 355/113. The latter has a smaller denominator than

   3.1416 = 31416/10000 = 3927/1250

and yet is closer to Pi. The convergents all have this property: among 
all fractions whose denominators are less or equal, it is the closest.

As another example, let's take the cube root of 5 = 1.7099759466767...

   5^(1/3) = 
           1+1/(1+1/(2+1/(2+1/(4+1/(3+1/(3+1/(1+1/(5+1/(...))))))))),

so the convergents are:

   1/1 = 1.00000000000000...
   2/1 = 2.00000000000000...
   5/3 = 1.66666666666666...
   12/7 = 1.71428571428571...
   53/31 = 1.70967741935484...
   171/100 = 1.71000000000000...
   566/331 = 1.70996978851964...
   737/431 = 1.70997679814385...
   4251/2486 = 1.70997586484312...

which is correct to 8 significant figures.

-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   


Date: 04/01/98 at 13:22:11
From: BFaulk1234
Subject: Re: Pi or fraction 22/7

Thank you very much for the help. Gee, I should have remembered that 
formula from school ... except I have never heard of it.

Thanks again for the help.
    
Associated Topics:
High School Number Theory
High School Sequences, Series

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