Continued Fraction Constants

All real numbers can be written as continued fractions independently of any base of positional notation. The Lévy constant arises from the co-efficients of most continued fractions and is explained on Wikipedia. It is: \( e^{\frac {\pi^2}{12 \ln{2}}  } \). As can be seen, it contains the number twelve.

Related to the Lévy constant are the Lochs constants, which also apply to most continued fractions but depend on the base of enumeration. For base \(B\), the Lochs constant is \( \frac{6 \ln{2} \ln{B}}{\pi^2} \). It can be interpreted as the ratio of the number of co-efficients of the continued fraction to the number of significant figures of positional notation of the same accuracy in the limit. For the decimal base ten, the Lochs constant is almost one, which means that the number of co-efficients in the continued fraction to achieve the same accuracy as a decimal number would be almost the same as but slightly fewer than the number of digits of the decimal number. Other bases of enumeration do not have this property. For base twelve, fewer significant figures are required than co-efficients of the continued fraction to achieve as much accuracy because the Lochs constant for base twelve is more than one. The whole number base where the Lochs constant is as close as possible to one such that the continued fraction and positional significant figure representations have the same number of terms is base eleven. The Lochs constants for the two bases ten and twelve on either side of eleven however remain close enough to one for the number of significant figures of a number to be a good indication of the number of co-efficients in its continued fraction to represent it as accurately. This appears to me to be a benefit of base twelve in mimicking this peculiar behaviour of decimal, such that if the base in society were to be changed from decimal to dozenal, this property would not be missed.

References/See also:

• https://en.wikipedia.org/wiki/L%C3%A9vy%27s_constant
  
• https://en.wikipedia.org/wiki/Lochs%27s_theorem
  

  https://en.wikipedia.org/wiki/Lochs%27s_theorem wrote:"The decimal system is the last positional system for which each digit carries less information than one continued fraction quotient"

  
  
• https://dozenal.forumotion.com/t24-dozenal-fifths-better-than-decimal-thirds#95
  In that post, I gave examples of fractional convergents resulting from the co-efficients of the continued fraction of the base \(e\) of the natural logarithms.