22/7 as an Approximation for PiDate: 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. |
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