Dick Moores wrote: > I need to figure out how to compute pi to base 12, to as many digits as > possible. I found this reference, <http://mathworld.wolfram.com/Base.html>, > but I really don't understand it well enough.
How many stars are in "*************************"? You probably answered "25". This means that, for convenience, you've broken down the row of stars into ********** + ********** + *****, that is, 2 tens with 5 left over, which the base-10 numeral system denotes as "25". But there's no reason other than tradition why you should arrange them into groups of 10. For example, you could write it as ******** + ******** + ******** + *, or 3 eights plus 1. In octal (base-8) notation, this is written as "31"; the "tens" place in octal represents eights. In general, in the base-r numeral system, the nth digit to the left of the ones digit represents r^n. For example, in the binary number 11001, the place values for each digit are, right to left, 1, 2, 4, 8, and 16, so the number as a whole represents 1×16+1×8+0×4+0×2+1×1=16+8+1=25. This analogous to 25=2×10+5 in base-10. It's also possible to write it as 3×8+0×4+0×2+1×1 = 3001 base 2, but by convention, base-r only uses the digits in range(r). This ensures a unique represenation for each number. This makes "11001" the unique binary representation for decimal 25. Note that for bases larger than 10, the digits will be numbers that are not single digits in base 10. By convention, letters are used for larger digits: A=10, B=11, C=12, ... Z=35. For example, the number (dec) 2005 = 1×12³+1×12²+11×12+1×1 is represented in base-12 by "11B1". Fractions are handled in a similar manner. The nth place to the right of the radix point (i.e., the "decimal point", but that term is inaccurate for bases other than ten) represents the value radix**(-n). For example, in binary, 0.1 = 1/2 = dec. 0.5 0.01 = 1/4 = dec. 0.25 0.11 = 1/2 + 1/4 = 3/4 = dec. 0.75 0.001 = 1/8 = dec. 0.125 0.01010101... = 1/4 + 1/16 + 1/64 + ... = 1/3 0.0001100110011... = 1/10 = dec. 0.1 The last row explains why Python gives: >>> 0.1 0.10000000000000001 Most computers store floating-point numbers in binary, which doesn't have a finite representation for one-tenth. The above result is rounded to 53 signficant bits (1.100110011001100110011001100110011001100110011010×2^-4), which is exactly equivalent to decimal 0.1000000000000000055511151231257827021181583404541015625, but gets rounded to 17 significant digits for display. Similarly, in base-12: 0.1 = 1/12 0.14 = 1/12 + 4/144 = 1/9 0.16 = 1/12 + 6/144 = 1/8 0.2 = 2/12 = 1/6 0.3 = 3/12 = 1/4 0.4 = 4/12 = 1/3 0.6 = 6/12 = 1/2 0.8 = 8/12 = 2/3 0.9 = 9/12 = 3/4 0.A = 10/12 = 5/6 Notice that several common fractions have simpler representations in base-12 than in base-10. For this reason, there are people who believe that base-12 is superior to base-10. (http://www.dozenalsociety.org.uk) > Could someone show me how to do what I need? You'll need 3 things: (1) An algorithm for computing approximations of pi. The simplest one is 4*(1-1/3+1/5-1/7+1/9-1/11+...), which is based on the Taylor series expansion of 4*arctan(1). There are other, faster ways. Search Google for them. (2) An unlimited-precision numeric representation. The standard "float" isn't good enough: It has only 53 bits of precision, which is only enough for 14 base-12 digits. The "decimal" module will probably work, although of course its base-10 internal representation will introduce slight inaccuracies. (3) A function for converting numbers to their base-12 representation. For integers, this can be done with: DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ" def itoa(num, radix=10): is_negative = False if num < 0: is_negative = True num = -num digits = [] while num >= radix: num, last_digit = divmod(num, radix) digits.append(DIGITS[last_digit]) digits.append(DIGITS[num]) if is_negative: digits.append("-") digits.reverse() return ''.join(digits) For a floating-point number x, the representation with d "decimal" places count be found by taking the representation of int(round(x * radix ** d)) and inserting a "." d places from the right. -- http://mail.python.org/mailman/listinfo/python-list
