scale. Therefore, this is the distribution expected if the logarithms of the numbers (but not the numbers themselves) are uniformly and randomly distributed. For example, a number , constrained to lie between 1 and 10, starts with the digit 1 if , and starts with the digit 9 if . Therefore, starts with the digit 1 if , or starts with 9 if . The interval is much wider than the interval (0.30 and 0.05 respectively); therefore if log is uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9; the probabilities are proportional to the interval widths, giving the equation above (as well as the generalization to other bases besides decimal). Benford's law is sometimes stated in a stronger form, asserting that the fractional part of the logarithm of data is typically close to uniformly distributed between 0 and 1; from this, the main claim about the distribution of first digits can be derived. In other bases An extension of Benford's law predicts the distribution of first digits in other bases besides decimal; in fact, any base . The general form is For (the binary and unary) number systems, Benford's law is true but trivial: All binary and unary numbers (except for 0 or the empty set) start with the digit 1. (On the other hand, the generalization of Benford's law to second and later digits is not trivial, even for binary numbers.) Examples Examining a list of the heights of the 58 tallest structures in the world by category shows that 1 is by far the most common leading digit, irrespective of the unit of measurement (see "scale invariance" below): Another example is the leading digit of . The sequence of the first 96 leading digits (1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, ... ) exhibits closer adherence to Benford’s law than is expected for random sequences of the same length, because it is derived from a geometric sequence. History The discovery of Benford's law goes back to 1881, when the Canadian-American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages (that started with 1) were much more worn than the other pages. Newcomb's published result is the first known instance of this observation and includes