### Radix conversion in an unnormalized arithmetic system

N. Metropolis, R. L. Ashenhurst
1965 Mathematics of Computation
Introduction. The question of radix conversion of variable-precision binary numbers arises naturally in the context of unnormalized number representation , but may be of interest in other situations where it is desired to have number representations carry a reflection of significance. The present paper discusses a method for binary-decimal conversion of unnormalized numbers; this method differs in certain respects from one previously developed, and described elsewhere , for use with the
more » ... niac III computer. The question of decimal-binary conversion, taking into account explicit "uncertainty" in the decimal representation, is also investigated from the significance viewpoint. Binary-Decimal Conversion. First, consider the task of converting a number in unnormalized floating point binary form to a decimal in such a way that a discrepancy in the lowest-order binary digit corresponds to a discrepancy in the lowestorder decimal digit in the result (the standard maniac III routine for doing this has a "guard bit" provision for specifying a position other than the lowest-order one, but this feature can be incorporated by a preliminary transformation and so is neglected in the development here). The question of whether the decimal exponent of the result is represented explicitly or by the insertion of a decimal point character is irrelevant to the present discussion ; it will be assumed that the desired output is a string of decimal characters, representing an integer, and a second integer specifying an associated power of 10. Since 2~l = 10~ , equivalent precision in binary and decimal is given by numbers of digits in roughly the ratio 10 to 3; one could, of course, simply keep a count of decimal characters generated in a standard conversion procedure and stop at some approximately appropriate point. It seems not unreasonable, however, to ask for a conversion procedure which affords the user a more precise statement of the relation between the binary form and the decimal result. The observation that the exact conversion of integers gives also a true estimate of precision (i.e., the 10-for-3 criterion is naturally achieved), suggests that binary-decimal conversion can be accomplished by first transforming the floating point number to an integer expressed with the same significance, which differs from the original number only by a power of 10, and then taking the converted representation of this integer as the desired