Constant base numbering

Let b be an integer greater than one. Writing an integer n in the base b means decomposing it into the powers of b, that is, determining the coefficients (also called figures) a_k such that:

• 0 ≤ a_k ≤ b-1
• n is the sum of the_kb^k, with k  0.

Constant base systems.

The most used foundation

It is well known that the system in force everywhere is the decimal; that is to say that the base ten is used: b = 10. The writing of any integer uses the powers of 10 like this:

1492 = 1000 + 400 + 90 + 2 = 1×1000 + 4×100 + 9×10 + 2×1 = 1x10³ + 4x10² + 9x10 ¹ + 2x10⁰.

To go from units to tens, and from tens to hundreds, multiply by the same number, here ten, which is why the system is said to be constant base numeration.

Other bases present in the history of humanity

Constant base numbering has been used, with bases other than ten, mainly bases five (the Aztecs) and twenty. Traces of the use of the base twenty remain in some Western languages, such as French (eighty is said quatre-vingts that is to say four twenty, since that the word huitante is used in Switzerland and Belgium), in Danish (for the numbers 50, 60 and 70), in English (score, a score, two-score, three-score, four scores for eighty), and in Latin (where 18 was not said 10 + 8 but 20 - 2).

Possible reason for use

It is believed that the choice of bases 5, 10 or 20 is due to biological causes, since man has always counted on his fingers (even his toes).

A foundation becomes relevant in the 20th century

By the middle of the 20th century, an enormous interest in base two or binary base, since it has the advantage of only needing two figures, 0 and 1. This is due to the development of electronic calculation and data processing. The system fits well with the two states of an electronic circuit: without current or with current. The pure binary system has the advantage of being simple, but its main drawback is that the expression of a number in base 2 is very extensive.

The technique adapts the binary code to shorter expressions

Regrouping the figures by four or by five gives the hexadecimal base (base sixteen) and the base thirty-two. When working on a base greater than ten, you have to invent new figures, to note the numbers that go from ten to b-1 (b is still the base). For example, when the base twelve is used, the alpha and beta digits for ten and eleven are added. For the hexadecimal base, the custom is to use the capital letters A, B,...F.

Variable base systems.

Historically, there have been variable base numerations, such as that of the Sumerians, who used a mixture of base 60 and base 10. They have bequeathed to the world today that one hour is divided into sixty minutes, and not in ten or one hundred, the week of seven days, the dozen and the one in which the circle is divided into 360 = 6×60 degrees = 12x30 degrees (in the Zodiac), and not in one hundred or four hundred (which also exists, but it is not so common).

The disadvantage of that system was that multiplying by 10 or 60 was not easy, since you cannot simply move the figures (to the left) and add an empty box (a zero, which had not been invented yet) to the right.

Disadvantages with larger databases

The larger the base, the more complicated it is to calculate on it, since you need to learn longer (multiplication) tables (with b² products).

The smaller the base, the longer the writing of the numbers becomes: for example, the writing of a number in binary base is ln 10/ ln 2 times longer than its writing in decimal base, that is 3, 3 times more, on average (length is proportional to the inverse of the logarithm of the base).

Base numbering system

The question of knowing which foundation is the most practical has no simple answer. However, it can be said that the decimal base is not exceptional, and that it is far surpassed by base six, which offers the advantage of having simple divisibility criteria to divide by 2, 3, 4,... up to eleven; while in decimal base, 7 does not have affordable criteria.

General information

Every real number can be written in base b, that is, decompose into the powers of b, the b^k, with k integer positive or negative . For example, in base ten:

42,58 = 4x10¹ + 2x10⁰ + 5x10^-1 + 8 10^-2.

If the decomposition requires an infinite number of figures, the number is said to be non-decimal. 1/3 = 0.333333333333... is not a decimal, but in base three, one third is 1/10 = 0.1 which is (we would have to invent a word like triemal to mean decimal in base three).
There is no uniqueness of writing a real in a base, as shown by the equality 1 = 0.999999999999...... but, if it is decided that the infinite sequence of digits b-1 in base b, it is shown that there is only one way to write a real in base b