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Some Inequalities Connected with a Method of Representing Positive Integers

Published online by Cambridge University Press:  03 November 2016

F. S. Macaulay
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Extract

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.

Type
Research Article
Information
The Mathematical Gazette , Volume 15 , Issue 207 , May 1930 , pp. 95 - 98
Copyright
Copyright © Mathematical Association 1930

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References

* The number of special numbers ≤ (a)n is (a)n-1 , and of non-special (a)n – (a)n-1 or (a—1)n ; and the ratio is that of n to a—1.