Decimal superbase

Many numeral systems with base 10 use a superimposed larger base of 100, 1000, 10000 or 1000000. It is a power of 10 and might be called a superbase or superradix of the numeral system. The superbase is mainly used in the spoken/written language but also apparent when writing large numbers with digits by grouping of digits, as a mental aid of measuring the number.

uperbases 1000 and 1000000

Counting geometrically in English goes like; one, ten, hundred, thousand, ten thousand, hundred thousand etc. Written as powers of 10 they look like; $10^\{0\}$, $10^\{1\}$, $10^\{2\}$, $10^\{3\}$, $10^\{4\}$, $10^\{5\}$ etc. There are unique names for the powers only up to a thousand - $10^\{3\}$ - demonstrating a superbase of 1000.

Now counting geometrically with common ratio 1000 in the constructed Gillion system goes like; one, thousand, million, gillion, tetrillion etc or written as powers; $1000^\{0\}$, $1000^\{1\}$, $1000^\{2\}$, $1000^\{3\}$, $1000^\{4\}$ etc. To better illustrate the relation of the base 10 and the superbase $10^\{3\}$ one could write; $(10^3)^\{0\}$, $(10^3)^\{1\}$, $(10^3)^\{2\}$, $(10^3)^\{3\}$, $(10^3)^\{4\}$ etc. Now, counting up to hundred thousand with common ratio 10 would give the sequence; $10^0(10^3)^\{0\}$, $10^1(10^3)^\{0\}$, $10^2(10^3)^\{0\}$, $10^0(10^3)^\{1\}$, $10^1(10^3)^\{1\}$, $10^2(10^3)^\{1\}$.

The table below compares some real and artificial numeral systems with superbases 1000 and 1000000.

Mathematical description

The spoken numeral system uses the ten "arithmetic numerals" zero, one, two , three, four, five, six, seven, eight and nine. It also uses the "geometric numerals" of the base, that is the names of the powers; one, ten, hundred, and it uses the geometric numerals of the superbase; one, thousand, million etc.

A number "a"_"n""a"_"n"-1..."a"₂"a"₁"a"₀ where a₀, "a"₁... "a"_"n" are all digits in base "10", the number can be represented as follows. $10^i$ is a "weight".

:$sum\_\{i=0\}^n\; a\_i\; imes\; 10^i$

(However we would rather count down from the highest weight to the lowest to make the formula look more like the number "a"_"n""a"_"n"-1..."a"₂"a"₁"a"₀).
The same number can be represented in superbase $10^\{3\}$ by:

:$sum\_\{j=0\}^\{lfloor\; n/3\; floor\}\; left(\; sum\_\{i=0\}^2\; a\_\{3j+i\}\; imes\; 10^i\; ight)\; imes\; 10^\{3j\}$

According to the formula $10^\{3\}$ looks like the base so 10 is rather a subbase of $10^\{3\}$ than $10^\{3\}$ is a superbase of 10. The example number 024 814 300 would expand to (after reversing everything):

:$left(0\; imes\; 10^2\; +\; 2\; imes\; 10^1\; +\; 4\; imes\; 10^0\; ight)\; imes\; 10^6\; +\; left(8\; imes\; 10^2\; +\; 0\; imes\; 10^1\; +\; 4\; imes\; 10^0\; ight)\; imes\; 10^3\; +\; left(3\; imes\; 10^2\; +\; 0\; imes\; 10^1\; +\; 0\; imes\; 10^0\; ight)\; imes\; 10^0$

Now substituting 1 - 9 by one - nine etc,
$10^\{1\}$ by ten, -teen or -ty,
$10^\{2\}$ by hundred,
$10^\{3\}$ by thousand,
$10^\{6\}$ by million
the number can be read out as:

(twenty four) "million" (eighthundred four) "thousand" (threehundred).

The European Chuquet superbase $10^\{3\}$ and super-superbase $10^\{6\}$ numeral system might be described as:

:$sum\_\{k=0\}^\{lfloor\; n/6\; floor\}\; left(sum\_\{j=0\}^1\; left(sum\_\{i=0\}^2\; a\_\{6k+3j+i\}\; imes\; 10^i\; ight)\; imes\; 10^\{3j\}\; ight)\; imes\; 10^\{6k\}$

The above table shows the base and superbases and their associated logarithms, which in the case of base 10 is the common logarithm $log\_\{10\}$. This logarithm converted to an integer is called the order of magnitude. The logarithm associated to superbase 1000000 is the million logarithm $log\_\{1000000\}$. It is not available on standard calculators but can be calculated as, using an example number 10¹², as $log\_\{1000000\}\; 10^\{12\}\; =\; log\_\{10\}\; 10^\{12\}\; /\; log\_\{10\}\; 1000000\; =\; 12\; /\; 6\; =\; 2$.

Some geometric numerals use the result of their logarithm applied to their weight, that is, their weights order of magnitude in their base, as prefix of their names. For example in superbase 1000000 the number 10¹² is called billion. The prefix bi- means 2, and $log\_\{1000000\}$ "billion" is also equal to 2. The million logarithm form another order of magnitude that is different from the common one.

The geometric numerals do "not" form a logarithmic scale, because of loss of continuity. For example in base 10 the logarithm of 100 is 2 and of 1000 is 3. In between of 100 and 1000 is 550 "five hundred fifty", but $log\_\{10\}\; 550\; =\; 2.74$ which has nothing to do with "five" etc. And in between of 2 and 3 is 2.5 but $10^\{2.5\}\; =\; 316$ which has also nothing to do with "five" etc.

ee also

*Decimal
*Radix
*Positional system