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118 THE FASCINATION OF NUMBERS

The simplest irrational continued fraction of all is:

RO i SR S

I+I4T4+14+14 etc
or

I
141
141
141
I+1 etc.

and the convergents of this fraction are:

 

 

 

 

e o VBT SR S g
1> 2> 3 52 87 i 3 th.

Both the numerators and the dénominators are derived from
the series of Fibonacci numbers, each of which is equal to the
sum of the two preceding numbers:

L. 2 2 8.5 8,13 21 34 55 89 etc
The convergents of continued fractions are alternately
greater than and less than the true value, the error growing

smaller either way as further convergents are reached.

The use of continued fractions also gives a solution to any
equation of the form:

ax—by=1
where ¢ and 4 are relatively prime positive integers. To solve
the equation:
328 —51p=1

we set up the continued fraction for 32 and expand it as
follows:

 

I-+1
141
141,
241