Gauss was the son of a bricklayer, and owed his education to the reigning duke. Ramanujan’s father was indeed accountant to a cloth merchant, but it may be doubted whether the position of an Indian accountant in the late nineteenth century was financially very much preferable to that of a German bricklayer in the late eighteenth. At any rate when Ramanujan, a married man of twenty-five, wrote his first letter to Professor Hardy enclosing statements of some of the results he had obtained in mathematics, he himself was employed as a clerk in the Accounts Department of the Port Trust Office of Madras and earning a salary of £20 a year. Ramanujan lived in a more democratic age than Gauss, so there was no reigning duke for him. Fortunate then that there were Indian and British mathematicians who were interested and impressed, and an able and sympathetic civil servant as chairman of the Madras Port authority.

In ordinary arithmetic a number ab … jk represents the sum of the place values of its digits a, b, …, j, k, these place values being k, 10j, 102i, etc. Also the representation is unique if the digits are restricted to being positive and less than 10. We are going to consider the representation of whole numbers by means of digits with place values, but with a different definition of place value. The digits a, b, …, j, k + 1 of a given number A are to be positive, and the number of them, n, is to be assigned, the units digit k + 1 (in the standard form) is to be greater than zero, and the place value of the digit a in the nth place counting from k + 1 is to be a. a + 1 … a + n - 1 divided by 1.2. …. n, or the number of homogeneous products of a letters taken n at a time; this number or place value is written (a)n.

It is perhaps somewhat remarkable that in such frequently recurring and comparatively simple processes as those of multiplication and division of decimals there should still be no final settling down by teachers of Arithmetic into one standardized method and arrangement of work, shown by experience to be the simplest, safest, least laborious, and most easily understood and assimilated by the pupil. Indeed there seems a tendency in some quarters, after trying in succession all the various modifications of method and set-out which have been suggested, each as an improvement on its predecessor, to come round full circle to the older methods in use, say, fifty years ago.