- The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation un =anu„_,+bnun_2.
Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.
In the sexagesimal system the numerators of the successive fractions (the denominators of which were the successive powers of 60) were followed by', ", "', ", the denominator not being written.
Ã¯¿½; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators I and denominators continually increasing.
In refutation of Duchesne(Van der Eycke), he showed that the ratio was 3-, %-, and thence made the exceedingly lucky step of taking a mean between the two by the quite unjustifiable process of halving the sum of the two numerators for a new numerator and halving the sum of the two denominators for a new denominator, thus arriving at the now well-known approximation 3 6 3 - or ??