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Keywords: Pi, approximation, rational, continued fraction, square root, spreadsheet, Excel
At Right Angles | Vol. 3, No. 1, March 2014 49 in the classroom
How to discover 22/7
and other rational
approximations to π
Gaurav Bhatnagar
Introduction
“Take ”. You and I, dear reader, know that
this cannot be right. We know perfectly well that is
an irrational number, and so cannot equal 22/7; for
22/7 is clearly a ratio o integers and therefore a
rational number. So the best we can say is ,
that is, is approximately 22/7.
rational approximations for and other irrational
numbers. The key idea here is to use a calculator (or a
continued fraction for an irrational number.
2
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50 At Right Angles | Vol. 3, No. 1, March 2014
The continued fraction for
It is easy to see that:
= 3 +
1
1/0.14159265 ...
We now compute 1/0.14159 ... in the denominator, using a calculator, and obtain:
1
7.06251331 ...
Repeating these steps, we obtain:
1
7 + 0.06251331 ... = 3 +
1
7 +
1
1/0.06251331 ...
= 3 +
1
7 +
1
15.99659440 ...
= 3 +
1
7 +
1
15 +
1
1/0.99659440 ...
= ⋯
= 3 +
1
7 +
1
15 +
1
1 +
1
292 + ⋯
.
continued fraction
representation of . To get approximations of , chop off the continued fraction suitably to get:
1
7 = 22
7 ;
≈ 3 +
1
7 +
1
15
= 333
106;
≈ 3 +
1
7 +
1
15 +
1
1
= 355
113.
The approximations 22/7 and 355/113 are quite popular, and have been known for thousands of years.
So now you know how to discover 22/7 and other rational approximations to . Let us try the same thing
with another familiar irrational number, namely √2.
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At Right Angles | Vol. 3, No. 1, March 2014 51
The square root of 2
Unlike , the continued fraction of √2 has a beautiful pattern. Here are the calculations:
√2 = 1.414213562 ... = 1 + 0.414213562 ...
= 1 +
1
1/0.414213562 ... = 1 +
1
2.414213562 ...
= 1 +
1
2 +
1
1/.414213562 ...
= 1 +
1
2 +
1
2.414213562 ...
= ⋯
Notice that the 2.414 ... has occurred earlier. So you would expect that
√2 = 1 +
1
2 +
1
2 +
1
2 + ⋯
.
√2, and is surely much nicer than its decimal
expansion 1.414213562 ... .
One can prove that the pattern repeats, and also avoid the use of a calculator, by noticing the following
equality which happens to be exact:
√2 = 1 +
1
1 + √2
.
Replace the √2 on the RHS by the expression
1 +
1
1 + √2
and see if you can tell why the pattern repeats!
√2. 3/2, 7/5, 17/12, 41/29 and 99/70.
√3 and √5 in the same way. The patterns
are every bit as nice as those in the continued fraction for √2.
rational approximations of a host o irrational numbers, such as , , . Why don't you try some
experiments of your own?
You should also try your hand at the simplest of all continued fractions:
1 +
1
1 +
1
1 +
1
1 +
1
1 + ⋯
.