Question

find the average of four number 6/4,5/3, 4/6 ,8/2.

Answer 1

A decimal expansion can be converted to a sequence of best rational approximations by an application of continued fractions.

The basic idea is to subtract off the integer part, and then reciprocate the fractional part. The result will have a nonzero integer part, and the process can be continued until the fractional part is zero (or forever, if the number is not rational).

Here we take the following steps, ignoring for the sake of illustration the "dot..dot..dot" portions of the number:

x0=4+0.8571428...x040.8571428...
x1=1/0.8571428=1+0.16666674444...x110.857142810.16666674444...
x2=1/0.16666674444=5+0.99999720016...x210.1666667444450.99999720016...
Notice that this x2x2 is very nearly 66, suggesting that if full precision had been used we would get exactly 66 and the process would halt there.

Putting the pieces together we have 4+11+16=46741116467.