One of the tasks that ancient humans had to perform was counting. For this they came to the idea of having a base, that is, ten; to say 10 fingers a person, 10 people a family, 10 families a sub-clan, 10 sub-clans a clan, etc. Today it has been abstracted and we have simple units, tens, hundreds, thousands and so on. That is what we call the base ten numbering system. The anatomical structure of his hands suggested the numbering base . This is verified in pre-Columbian America, the Inca nation uses base ten. The Mayans used base 20; in Asia, the Sumerians base 60.
So we have 2 305 means
5 simple units
0 tens
3 hundreds
2 thousands, but instead of 10 as a base we can use a negative integer ≠ -1.
Summary
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- 1 Definition
- 1 Example
- 2 Affirmation
- 3 Examples
- 2 Source
Definition
We will say that abcd k = a × k 3 + b × k 2 + c × k + d, with the conditions
- k ≤ -2, the base k cannot be -1
- a, b, c, d, …, ≤ | k |, these represent the digits, positive integers, strictly less than the absolute value of the negative base.
Example
In levoquinary system, base -5: 113 (-5 = + 1 × (-5) 2 + 1 × (-5) + 3 = 25 – 5 + 3 = 23
In levosenario system, base -6: 53 (-6 = 5 × (-6) + 3 = -30 + 3 = -27
Affirmation
All integers, in a negative-based numbering system, can be written with non-negative integers. These negative-based systems provide a dialectical change in the notation of the integers: the sign – (minus) of the negatives disappears.
Examples
We are going to present several cases in the levocuaternary system; the basic digits are 0, 1, 2, 3.
- 5 = 131 (-4= 1 × 16 + 3 × (-4) +1; 6 = 132 (-4 ; 7 = 133 (-4 ; 8 = 120 (-4
- -5 = 23 (-4= 2 × (-4) +3
- 16 = 100 (-4= 1 × 16 + 0 × (-4) +0
- -16 = 1300 (-4= 1 × (-64) + 3 × (16) + 0 × 4 + 0
- -1 = 13 (-4; -2 = 12 (-4 ; -3 = 11 (-4 ; -4 = 10 (as is the normal writing of any base).
