In this study, for quantifying & comparing complexity of digits of irrational numbers, we used first one million digits to calculate fractal dimensions of digits of 10 irrational numbers with long sequence of known digits via box-counting algorithm. irrational numbers we studied are five algebraic numbers & five transcendental numbers (p, e, log(2), ?(3) & Champernowne's constant). For statistical analysis, we performed 2000 repeated calculations for each number with segment of digits with length 200, 300, 400 & 500. distributions of estimated fractal dimensions seem normal by histograms. Analysis of variance was used to test equality of means for fractal dimensions among numbers & within number for different digit lengths. pattern of complexity of digits in these numbers based on estimated fractal dimensions is similar except that of Champernowne's constant.
Many natural & social systems express complex, irregular patterns. One of commonly used techniques in quantifying complexity is based on estimating fractal dimensions  & . In time series setting, fractal dimensions can be expressed in various forms such as Korcak exponent & Hurst coefficient. Multi-dimensional scaling such as Korcak exponent was estimated on Japanese earthquake time series & Chinese earthquake time series & . Hurst effect was applied in monitoring clinical trials  & . Fractal analysis has been applied in many diversified fields .
In this paper, we report our estimation & statistical analysis of fractal dimension via box counting algorithm on complexity of digits of several widely used algebraic & trancendental irrational numbers  & . As we know real numbers can be classified into rational numbers & irrational numbers. Any rational number r can be expressed as the ratio of two integers p & q as r = p/q. rest of real numbers besides rational numbers are called irrational numbers. From the Lebesgue measure point of view, there are more irrational numbers than rational numbers. For example, in interval of [0,1], Lebesgue measure of rational numbers is 0, whereas it is 1 for irrational numbers. It is well know that digits of rational numbers will keep repeating their patterns as the finite number of digits. However, digits of irrational numbers show very complicated structures. Further, irrational numbers can be divided into algebraic & transcendental. THE transcendental number is any irrational number that is not the root of any polynomial with integer coefficients. An irrational number, which is the root of the polynomial with integer coefficients is called algebraic number. Almost all irrational numbers are transcendental, while all transcendental numbers are irrational.
number of decimal digits for any irrational number is infinite. It is the great challenge to mathematicians & computer scientists to find out the long sequence of accurate digits for irrational numbers. In our analysis of complexity of digit patterns of irrational numbers, we create “time series” from digits by treating position as “time” index & digit at position t as observed value of “time series” Z(t), where Z(t) representing digit at position t for the given irrational number. We studied five algebraic numbers & five transcendental numbers (p, e, ?(3), log(2) & Champernowne's constant). dynamics of digits of some irrational numbers have been extensively studied. For example, digits of well-known irrational number p ...