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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 + ⋯

.