Arithmetic complement is a concept that links a number , written in any positional numbering system , with the unit of the next higher order; in certain applications, it facilitates the execution of operations.
Summary
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- 1 Definition
- 1 In decimal numbering
- 2 In ternary numbering
- 2 uses
- 3 Proposition
- 4 Corollaries
- 1 Idempotence
- 2 Capicua result
- 5 Sources
- 6 Linked topics
- 7 Notes and references
- 8 External links
Definition
If the number N has m digits in the base system k its arithmetic complement in the base k is the difference 10 m – N.
In decimal numbering
If the number abc, base 10, has three digits, its arithmetic complement is 10 3 – abc.
Calculation
In the case of abc it is 1000-abc; in practice c is subtracted from 10, then both b and a are subtracted from 9: (9-a) = e, (9-b) = f, (10-c) = g, then we juxtapose the differences, the complement is arithmetic = efg.
Example
Let’s find the complement of 472, the differences from left to right are 5 and 7 of 9, and 7 of 10, the CA is 527
In ternary numbering
The digits in this base are 0, 1 and 2, so to find the complement of xyst, we obtain the differences: u = 2-x, v = 2-y, w = 2-s finally z = 3-t, the complement ternary is uvwz.
Applications
In the subtraction of two numbers. the difference NP = N + Complement of P – 10 n , where n is the number of digits.
Proposition
- If the numeral N, in any numbering system, has pdigits, its complement at most has p
Corollaries
Idempotence
The complement of the complement of K is the same K, in any numbering system.
Capicua result
Given abc its complement is capicúa [1] if ac = 1, for example is K = 283, 3-2 = 1, then CA of 283 is 717, base 10.
If the number K has n digits N = ab … kl, its complement is capicua if la = 1 and the symmetric elements are equal: b = k, c = j, etc.
Sources
- “Arithmetic reasoned” by GM Bruño, edited by Editorial Bruño in Lima s / f
- “A walk through number theory” by GN Berman, publication of the USSR editorial, Moscow 2007
Linked Topics
- positional numbering system
- ternary numbering system
- Idempotence
- Capicúa number
