Decimal superbase

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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.

1 Superbases 1000 and 1000000
2 Superbases 100 and 10000
3 Super-superbase
4 No superbase
5 Classification of base 10 numeral systems
6 Mathematical description
7 See also

[edit] Superbases 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; $100$ , $101$ , $102$ , $103$ , $104$ , $105$ etc. There are unique names for the powers only up to a thousand - $103$ - 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; $10000$ , $10001$ , $10002$ , $10003$ , $10004$ etc. To better illustrate the relation of the base 10 and the superbase $103$ 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; $100 (10 3) 0$ , $101 (10 3) 0$ , $102 (10 3) 0$ , $100 (10 3) 1$ , $101 (10 3) 1$ , $102 (10 3) 1$ .

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

Superbase Thousand alike numeral systems							Superbase Million alike numeral systems
Number	Base Thousand Notation	Constructed Gillion Superbase Thousand system		European Peletier system	American Superbase Thousand with Offset system		Base Million Notation	European Chuquet Superbase Thousand² system		Artificial Superbase Million system
10⁰	1	one	$(10 3) 0$	one	one	$(10 3) 1 - 1$	1	one	$((10 3) 2) 0$	$(10 6) 0$
10³	1 000	thousand	$(10 3) 1$	thousand	thousand	$(10 3) 1 + 0$	1000	thousand	$103 ((10 3) 2) 0$	$103 (10 6) 0$
10⁶	1 000 000	million	$(10 3) 2$	million	million	$(10 3) 1 + 1$	1 000000	million	$((10 3) 2) 1$	$(10 6) 1$
10⁹	1 000 000 000	gillion	$(10 3) 3$	milliard	billion	$(10 3) 1 + 2$	1000 000000	thousand million	$103 ((10 3) 2) 1$	$103 (10 6) 1$
10¹²	1 000 000 000 000	tetrillion	$(10 3) 4$	billion	trillion	$(10 3) 1 + 3$	1 000000 000000	billion	$((10 3) 2) 2$	$(10 6) 2$
10¹⁵	1 000 000 000 000 000	pentillion	$(10 3) 5$	billiard	quadrillion	$(10 3) 1 + 4$	1000 000000 000000	thousand billion	$103 ((10 3) 2) 2$	$103 (10 6) 2$
10¹⁸	1 000 000 000 000 000 000	hexillion	$(10 3) 6$	trillion	quintillion	$(10 3) 1 + 5$	1 000000 000000 000000	trillion	$((10 3) 2) 3$	$(10 6) 3$

When writing a number one could insert spaces every third digit for improved readability and to further emphasize that the superbase is $103$ . In base thousand notation a million is written as 1 000 000. A base of 1000 needs 1000 symbols for each numeral, but since such are not available a group of three decimal numerals will do instead. In base million notation groups are made of six digits.

Gillion superbase thousand system makes sense when used with base thousand notation. At hexillion six full groups of digits are visible since the prefix hex- means six, and the superbase $103$ is raised to the power of 6 = log₁₀₀₀10¹⁸. However, the suffix -illion does not indicate that the superbase is thousand and group size is three. The suffix -ilia would, because it is derived from the Greek word chilia meaning thousand, creating gilia, tetrilia, pentilia, hexilia etc.
Superbase million system makes good sense when used with base million notation. At trillion three full groups of digits are visible since the prefix tri- means three, and the superbase $106$ is raised to the power of 3 = log_100000010¹⁸. The suffix -illion indicates that the superbase is million and group size is six.

The Gillion and Superbase Million systems have the advantage that the prefix of the numerals show how many full groups of digits there should be. It is then easy to remember the name of a numeral written with digits, and vice versa. The American system does not have this feature since the prefix is one less than the full digits groups count. It is like superbase thousand with a scale or offset. The American numerals have to be considered as just names of the powers, with no (strong) correspondence to the number notation. A similar comment might be made of the European Peletier system. By introducing the numerals milliard, billiard etc the system becomes superbase thousand. The -illiard numerals form a scaled superbase million system of its own that is interleaved with the standard superbase base million system, creating a superbase thousand system.

An artificial superbase million system might be constructed by starting with the European Chuquet system and use the name myriad for 10000 and lakh for 100000. Then there would be unique names for all the powers of 10 up to a million. Counting geometrically with common ratio 1000000 goes like; one, million, billion, trillion, quadrillion etc or written as powers; $10000000$ , $10000001$ , $10000002$ , $10000003$ , $10000004$ etc. The Chuquet system however is superbase thousand for numbers up to a million, but then introduces the base million that becomes a super-superbase. One could say that the Chuquet system is more like a superbase thousand squared system than a superbase million system. This is indicated in the table.

[edit] Superbases 100 and 10000

Counting arithmetically in English with common difference 100 goes; ... ,twelve hundred, thirteen hundred, fourteen hundred etc. This is a superbase 100 system. However the series is not complete as it usually ends at nineteen hundred and superbase 1000 being used further on.

The Indian numeral system is a superbase 100 system but it has a scale of $100 1.5 = 1000$ that unaligns it with the other systems. The modern Chinese numeral system labeled 2 in the article is a superbase 10000 system. Counting the powers; yī, wàn, yì, zhào etc and with numbers; $100000$ , $100001$ , $100002$ , $100003$ etc.

The table below compares some numeral systems with superbases 100 and 10000.

Superbase Hundred alike numeral systems							Superbase Myriad numeral systems
Number	Indian System Notation	Indian Superbase Hundred with Scale system		Base Hundred Notation	Spoken English Superbase Hundred system		Base Myriad Notation	Chinese Superbase Myriad system
10⁰	1	ek	$10(10 2) - 2 + 1.5$	1	one	$(10 2) 0$	1	yī	$(10 4) 0$
10¹	10	das	$(10 2) - 1 + 1.5$	10	ten	$10(10 2) 0$	10	shí	$101 (10 4) 0$
10²	100	sau	$10(10 2) - 1 + 1.5$	1 00	hundred	$(10 2) 1$	100	bǎi	$102 (10 4) 0$
10³	1 000	sahastr	$(10 2) 0 + 1.5$	10 00	ten hundred	$10(10 2) 1$	1000	qiān	$103 (10 4) 0$
10⁴	10 000		$10(10 2) 0 + 1.5$	1 00 00		$(10 2) 2$	1 0000	wàn	$(10 4) 1$
10⁵	1 00 000	lakh	$(10 2) 1 + 1.5$	10 00 00		$10(10 2) 2$	10 0000		$101 (10 4) 1$
10⁶	10 00 000		$10(10 2) 1 + 1.5$	1 00 00 00		$(10 2) 3$	100 0000		$102 (10 4) 1$
10⁷	1 00 00 000	crore	$(10 2) 2 + 1.5$	10 00 00 00		$10(10 2) 3$	1000 0000		$103 (10 4) 1$
10⁸	10 00 00 000		$10(10 2) 2 + 1.5$	1 00 00 00 00		$(10 2) 4$	1 0000 0000	yì	$(10 4) 2$
10⁹	1 00 00 00 000	arab	$(10 2) 3 + 1.5$	10 00 00 00 00		$10(10 2) 4$	10 0000 0000		$101 (10 4) 2$
10¹⁰	10 00 00 00 000		$10(10 2) 3 + 1.5$	1 00 00 00 00 00		$(10 2) 5$	100 0000 0000		$102 (10 4) 2$
10¹¹	1 00 00 00 00 000	kharab	$(10 2) 4 + 1.5$	10 00 00 00 00 00		$10(10 2) 5$	1000 0000 0000		$103 (10 4) 2$
10¹²	10 00 00 00 00 000		$10(10 2) 4 + 1.5$	1 00 00 00 00 00 00		$(10 2) 6$	1 0000 0000 0000	zhào	$(10 4) 3$

[edit] Super-superbase

The European Chuquet system uses superbase 1000 and super-superbase 1000000. Base = 10, superbase = base³ and super-superbase = superbase². This could be written in compact form as $(10 3) 2$ . This is less consistent than other systems described here because the superbase is not raised to the same power as the base.

The Knuth -yllion system uses superbase 100 and super-superbase 10000. It then continues and introduces super-super-superbase 100000000 and so on. This could be written in compact form as $(((10 2) 2) 2) ...$ . This is really another kind of numeral system where the weights increase as a power of a power rather than geometrically. The weights have been powers in the numeral systems described so far. The Knuth -yllion base, superbase, super-superbase, super-super-superbase, super-super-super-superbase , super-super-super-super-superbase etc are; ten, hundred, myriad, myllion, byllion, tryllion etc or as numbers; $101$ , $102$ , $104$ , $108$ , $1016$ , $1032$ etc. One n-yllion is $10^{2^{n+2}}$ so the term +2 in the formula makes the name of the weight not (strongly) connected to its size. The ancient Chinese system labeled 4 in the article is similar but starts with superbase 10000.

[edit] No superbase

The ancient Chinese system labeled 1 in the article is a system that does not have any superbase. It is a pure base 10 system.

[edit] Classification of base 10 numeral systems

The classification is done by examining what mathematical structure the unique number names of the system create. Then the semantics of these names is compared to the structure.

Classification	Superbase		Super-superbase		Super-super-superbase		Example systems
$10$	-	-	-	-	-	-	Ancient Chinese labeled 1 in the article
$102$	100	$102$	-	-	-	-	Spoken English
$102$ with scale $100 1.5$	100	$102$	-	-	-	-	Indian
$103$	1000	$103$	-	-	-	-	Constructed Gillion
$103$ by interleaving two $106$ alike	1000	$103$	-	-	-	-	European Peletier
$103$ with offset 1	1000	$103$	-	-	-	-	American
$104$	10000	$104$	-	-	-	-	Modern Chinese labeled 2 in the article
$106$	1000000	$106$	-	-	-	-	Superbase million
$(10 3) 2$	1000	$103$	1000000	$106$	-	-	European Chuquet.
$(10 4) 2$	10000	$104$	100000000	$108$	-	-	Ancient Chinese labeled 3 in the article. Since there does not seem to be any names of the powers $105$ , $106$ , $107$ within the labeled 3 system it is not superbase $108$ .
$(((10 2) 2) 2) ...$ with exponential offset 2	100	$102$	10000	$104$	100000000	$108$	Knuth -yllion. There is also a super-super-super-superbase and so on.
$(((10 4) 2) 2) ...$	10000	$104$	100000000	$108$	10000000000000000	$1016$	Ancient Chinese labeled 4 in the article. There is also a super-super-super-superbase and so on.

[edit] 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_na_n-1...a₂a₁a₀ where a₀, a₁... a_n are all digits in base 10, the number can be represented as follows. $10 i$ is a weight.

$\sum_{i=0}^n a_i\times 10^i$

(However we would rather count down from the highest weight to the lowest to make the formula look more like the number a_na_n-1...a₂a₁a₀).
The same number can be represented in superbase $103$ by:

$\sum_{j=0}^{\lfloor n/3\rfloor} \left( \sum_{i=0}^2 a_{3j+i}\times 10^i \right)\times 10^{3j}$

According to the formula $103$ looks like the base so 10 is rather a subbase of $103$ than $103$ is a superbase of 10. The example number 024 814 300 would expand to (after reversing everything):

$\left(0\times 10^2 + 2\times 10^1 + 4\times 10^0\right)\times 10^6 \ \ \ +\ \ \ \left(8\times 10^2 + 0\times 10^1 + 4\times 10^0\right)\times 10^3 \ \ \ +\ \ \ \left(3\times 10^2 + 0\times 10^1 + 0\times 10^0\right)\times 10^0$

Now substituting 1 - 9 by one - nine etc,
$101$ by ten, -teen or -ty,
$102$ by hundred,
$103$ by thousand,
$106$ by million
the number can be read out as:

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

The European Chuquet superbase $103$ and super-superbase $106$ numeral system might be described as:

$\sum_{k=0}^{\lfloor n/6\rfloor} \left(\sum_{j=0}^1 \left(\sum_{i=0}^2 a_{6k+3j+i}\times 10^i \right)\times 10^{3j} \right)\times 10^{6k}$

Base 10	$log 10$	Superbase 1000	$log 1000$	Superbase 1000000	$log 1000000$	Superbase 100	$log 100$	Superbase 10000	$log 10000$
10⁰	0	$(10 3) 0$	0	$(10 6) 0$	0	$(10 2) 0$	0	$(10 4) 0$	0
10¹	1
10²	2					$(10 2) 1$	1
10³	3	$(10 3) 1$	1
10⁴	4					$(10 2) 2$	2	$(10 4) 1$	1
10⁵	5
10⁶	6	$(10 3) 2$	2	$(10 6) 1$	1	$(10 2) 3$	3
10⁷	7
10⁸	8					$(10 2) 4$	4	$(10 4) 2$	2
10⁹	9	$(10 3) 3$	3
10¹⁰	10					$(10 2) 5$	5
10¹¹	11
10¹²	12	$(10 3) 4$	4	$(10 6) 2$	2	$(10 2) 6$	6	$(10 4) 3$	3

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 $log 10000001012 = log 101012 / log 10 1000000 = 12 / 6 = 2$ .

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.

[edit] See also

Categories: Numeration

Decimal superbase

From Wikipedia, the free encyclopedia

Contents

[edit] Superbases 1000 and 1000000

[edit] Superbases 100 and 10000

[edit] Super-superbase

[edit] No superbase

[edit] Classification of base 10 numeral systems

[edit] Mathematical description

[edit] See also

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