Talk:Decimal superbase

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[edit] Deleted the so-called "Bisand System"

In reality, there is no such like system.
See Google searches:  "Bisand system" or Bisand superbase.
Nothing, but original research!

This article needs still many works. John 2006 15:39, 1 August 2006 (UTC)

The Bisand system was constructed for pedagogical reasons. But, ok, maybe the Superbase million system suffices as a good example to explain the function of the prefix and the function of the suffix. --Najro 16:16, 1 August 2006 (UTC)

[edit] Quadrentum superbase 100 system

An artificial superbase hundred system might be constructed by using the suffix -entum derived from the roman word centum meaning hundred, and adding suitable prefixes. Counting in powers give; one, hundred, myriad, million, quadrentum, quintentum etc. Written as powers; 100⁰, 100¹, 100², 100³, 100⁴, 100⁵.

The table below compares some real and artificial 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	Artificial Quadrentum  Superbase Hundred system		Base Myriad Notation	Chinese Superbase Myriad system
100	1	ek	10(102) − 2 + 1.5	1	one	(102)0	1	yī	(104)0
101	10	das	(102) − 1 + 1.5	10	ten	10(102)0	10	shí	101(104)0
102	100	sau	10(102) − 1 + 1.5	1 00	hundred	(102)1	100	bǎi	102(104)0
103	1 000	sahastr	(102)0 + 1.5	10 00	ten hundred	10(102)1	1000	qiān	103(104)0
104	10 000		10(102)0 + 1.5	1 00 00	myriad	(102)2	1 0000	wàn	(104)1
105	1 00 000	lakh	(102)1 + 1.5	10 00 00	ten myriad	10(102)2	10 0000		101(104)1
106	10 00 000		10(102)1 + 1.5	1 00 00 00	million	(102)3	100 0000		102(104)1
107	1 00 00 000	crore	(102)2 + 1.5	10 00 00 00	ten million	10(102)3	1000 0000		103(104)1
108	10 00 00 000		10(102)2 + 1.5	1 00 00 00 00	quadrentum	(102)4	1 0000 0000	yì	(104)2
109	1 00 00 00 000	arab	(102)3 + 1.5	10 00 00 00 00	ten quadrentum	10(102)4	10 0000 0000		101(104)2
1010	10 00 00 00 000		10(102)3 + 1.5	1 00 00 00 00 00	quintentum	(102)5	100 0000 0000		102(104)2
1011	1 00 00 00 00 000	kharab	(102)4 + 1.5	10 00 00 00 00 00	ten quintentum	10(102)5	1000 0000 0000		103(104)2
1012	10 00 00 00 00 000		10(102)4 + 1.5	1 00 00 00 00 00 00	sextentum	(102)6	1 0000 0000 0000	zhào	(104)3

The columns with Quadrentum system was deleted by User:John 2006 because it was nowhere to be found on the Internet. No. But I felt there was a pedagogic need to show what a simple and large superbase 100 system might look like. If that can not be tolerated I will use the spoken English language instead as an example because it uses a tiny superbase 100 system. Therefore I have put back the superbase 100 columns but without using Quadrentum numerals. --Najro 08:47, 2 August 2006 (UTC)

Ok, Najro, without using personal "Quadrentum numerals" it's acceptable. Suggestions:

• In the "Superbase Myriad numeral systems" you can also put the Greek numeral system.
• The "Peletier system" – I.M.H.O. – is rather a "Superbase Million alike numeral system".

John 2006 09:21, 2 August 2006 (UTC)

It is not easy to find out how to count large numbers the modern Greek way. [1]. The numbers are grouped as superbase 1000 but on the other hand you have 1,000,000 ekatomirio and 100 ekato and mirio sounds like myriad 10000. And Greek number grouping?

And then there is ancient counting. Greek numerals#Higher numbers which I suppose is ancient is very shortly explained. Looking... --Najro 16:14, 2 August 2006 (UTC)

μύριοι	=	múrioi	=	myriad	=	1 0000
δισμύριοι	=	dimúrioi	=	bimyriad	=	2 0000
ένα εκατομμύριο	=	éna ekatommúrioi	=	one centmyriad	=	100 0000

This should be the traditional way of counting.

However, like this document indicates: δέκα χιλιάδες = ten thousand, instead of myriad, seems now to be promoted.

What's about putting Peletier system to superbase Million? John 2006 17:49, 2 August 2006 (UTC)

No, I think Peletier is base 1000. When you go from one numeral to the next you multiply by a thousand. The multiplication factor is the superbase. If you ignore the -illiard numerals, then you are not analysing Peletier, then you are analysing Chuquet. But I agree with "Both Chuquet and Peletier naming conventions are referred to as long scale". The naming method of Peletier is base million (kind of), but the Peletier system is base thousand. And now I see the classification of Peletier as 10³ in the table is not quiet correct. It should be 10³ with "some appropriate description".

A person wanting to learn the Peletier system probably already know the meaning of million and milliard. To learn the rest it is best to first learn the Chuquet system and the apply the rule of inserting -illiards after the -illions. To learn the names you have to think base million in Peletier but the system itself is base thousand. --Najro 09:19, 3 August 2006 (UTC)

I don't agree with you, Najro.

See this table:

Comparison short scale	Base 10	Power	Chuquet	Peletier	SI Prefix	Comparison  Superbase 1000
unit	10  0	million 0	unit		[unit]	unit
thousand	10  3	million 0.5	mille		kilo	thousand
million	10  6	million 1	million		mega	"bi-thousand"
billion	10  9	million 1.5	thousand million	milliard	giga	"tri-thousand"
trillion	10 12	million 2	billion		tera	"quad-thousand"
quadrillion	10 15	million 2.5	thousand billion	billiard	peta	"quint-thousand"
quintillion	10 18	million 3	trillion		exa	"sext-thousand"
sextillion	10 21	million 3.5	thousand trillion	trilliard	zetta	"sept-thousand"
septillion	10 24	million 4	quadrillion		yotta	"oct-thousand"

It's self-evident that the full steps are the zillions!
The only difference between both systems is that Peletier replaces the half steps of thousand zillions by zilliards.
However, it − clearly! – stays a superbase million system, since the main steps are million, billion, trillion, etc.
John 2006 11:34, 3 August 2006 (UTC)

Well now I only can repeat my old argument that was: If you take the full step, you ignore the half step, that is you ignore Peletier. By taking the full step you are talking Chuquet.

My latest change on this article means:

Superbase million is (10⁶)⁰, (10⁶)¹, (10⁶)², (10⁶)³, (one, million, billion, trillion).

Scaled (by 1000=1000000^0.5) superbase million is (10⁶)^{0 + 0.5}, (10⁶)^{1 + 0.5}, (10⁶)^{2 + 0.5}, (10⁶)^{3 + 0.5}, (thousand, milliard, billiard, trilliard).

These two million base alike numeral systems are then interleaved to form a base thousand system. --Najro 12:33, 3 August 2006 (UTC)

I partially agree with your last sentence. Indead, it's a "interleaved system".

• The true SB thousand system is your theoretical "Zisand" system or my "zi-thousand" system. (Both, never without apostrophes!).
  
• And the Gillion System and the American Offset System, both are rightly ranged in the SB k system. Even if, both have the failing to use "-illon", by meaning "-mille".

Now, the two variants of the long scale system:

• Both use powers of million, both are principally SB million systems. This is attested by the prefixes used.
  • The older Chuquet System expresses hundreds, tens and units of thousands of the zillion.
  • Whilst the emerged Peletier System – by respecting its fundamental SB M character – conveniently replaced all thousand zillions by zilliards.
    It is the matter of half steps! This is attested by the used prefixes.
    So, we can say: billiard means million to the power of two and a half. Thus, it's clearly and principally a SB M system.
    However, admittedly, an interleaved Superbase Million System, with an own suffix for the thousands, in dependence of the fondamental system.
    Thus, it stays what it is: SB M.

John 2006 14:26, 3 August 2006 (UTC)

If I make a different example: Danish numerals. Some claim it is base 20, but its not, its base 10. (List of numbers in various languages) [[2]].

Number	Reading	Meaning	Meaning in a true base 20 system
60	tres*	3 times 20
61	enogtres*	1 and (3 times 20)
62	toogtres*	2 and (3 times 20)
63	treogtres*	3 and (3 times 20)
64	fireogtres*	4 and (3 times 20)
65	femogtres*	5 and (3 times 20)
66	seksogtres*	6 and (3 times 20)
67	syvogtres*	7 and (3 times 20)
68	otteogtres*	8 and (3 times 20)
69	niogtres*	9 and (3 times 20)
70	halvfjerds*	3½ times 20	10 and (3 times 20)
71	enoghalvfjerds*	1 and (3½ times 20)	11 and (3 times 20)
72	tooghalvfjerds*	2 and (3½ times 20)	12 and (3 times 20)
73	treoghalvfjerds*	3 and (3½ times 20)	13 and (3 times 20)
74	fireoghalvfjerds*	4 and (3½ times 20)	14 and (3 times 20)
75	femoghalvfjerds*	5 and (3½ times 20)	15 and (3 times 20)
76	seksoghalvfjerds*	6 and (3½ times 20)	16 and (3 times 20)
77	syvoghalvfjerds*	7 and (3½ times 20)	17 and (3 times 20)
78	otteoghalvfjerds*	8 and (3½ times 20)	18 and (3 times 20)
79	nioghalvfjerds*	9 and (3½ times 20)	19 and (3 times 20)
80	firs*	4 times 20
81	enogfirs*	1 and (4 times 20)
82	toogfirs*	2 and (4 times 20)
83	treogfirs*	3 and (4 times 20)
84	fireogfirs*	4 and (4 times 20)
85	femogfirs*	5 and (4 times 20)
86	seksogfirs*	6 and (4 times 20)
87	syvogfirs*	7 and (4 times 20)
88	otteogfirs*	8 and (4 times 20)
89	niogfirs*	9 and (4 times 20)

In a true base 20 system the numeral 70 does not exist. There aint no such thing as 3½ times 20, only integer factors are allowed. 3½ times 20 is just a name of the numeral 70 in a base 10 system. Maybe it was base 20 a long time ago until it transformed into base 10. For 75 the French say soixante-quinze (literally, "sixty [and] fifteen"). This is more like base 20 because they use 15 here, but they dont use 3*20 for 60.

Back to Peletier:

Yes we can say that billiard means million to the power of 2½. But that is no proof that the system is base million. In a true base million system there aint no such thing as a million to the power of 2½. If you for a moment forget about the number names, replace them by A,B,C,D... and just look at the structure of the numeral system, you might see that Peletier looks like base 1000. After the structure has been investigated, one can make some comments on how well the names of the numerals reflect that system. --Najro 16:46, 3 August 2006 (UTC)

• We can agree that in a "pure" vigesimal system e.g. 77₁₀ is not expressed neither as 3.5 x 20 + 7 nor as 7 + 3.5 x 20.

• However a true Base 20 system also not expresses 77₁₀ as 17 + 3 x 20  – unlike your assertion –  since in a true vigesimal system:  77₁₀   =   KH₂₀  (!)

• Let's covenant that
  • "Pure" bases (and "pure" superbases) are seldom in the cultural, traditional praxis, sub-bases or additional, auxiliary bases rather frequent.
  • Also it seems me useful to distinguish the oral number names and the writing use, the symbols, the digits.
    So, Romans for example used next to the symbol for the unit [I] and the symbol for the base [X], an auxiliary base [V] in writing, however never orally in the number names.
  • Whilst, for example, in the actual chinese decimal system there are no exceptions in numbering, exceptions are frequent in European number names: English, twelve, not ten-two; French, quatre-vingt-seize, not huitante-six; Danish etc. If in France and Danemark the current notation base is effectively ten, partially, the number names are Base 20.
  • Soixante-dix-sept: 6 x 10 + 10 + 7, I don't see a vigesimal system, rather a crossover tens, to prepare quatre-vingt-dix-sept.

Back to Peletier:
"If you for a moment forget about the number names, replace them by A,B,C,D... [...] [it] looks like base 1000."

Ha, that's true. If you replace: million by A, milliard by B, billion by C and billiard by D...  this would be a SB 1000 system!  But the Peletier system respects the Chuquet zillions.
Like another proposed at Talk:Long and short scales#Superbase_of_large_numbers the long scale billion must be written as 1, 000 000, 000 000 in a very logical manner.

The commas give the main SB M and the non-breaking spaces give the auxilliary, intermediary base M ^N.5.  Orally:  as "thousand zillion" with Chuquet, as "zilliard" with Peletier.
It can't be true that a system what shares the half of all large number names is ranged in a class where it shares no number name, excepting the million itself. Or, in your logic, if someone decides to write the Chuquet "thousand billion" in one word as "thousandbillion" so it must be also ranged in SB k systems? Sure, now it has its own word. No, I don't agree.

In addition:  "Zilliard" is only a semantical variant of "thousand zillion", a synonym. That can't provoke – does not justify – a shift in another class.
Or, in other words, after restudying your classification, I assert that the Peletier system also possesses this Super-superbase million, just like the original Chuquet system.

Another topic:  Base Hundred Notation
Aryabhata invented a very interesting base hundred notation: one is क or ka, then 10², 10⁴, 10⁶, etc. are respectively ki, ku, kri, kli, ke, kai, ko and kau = 10¹⁶; 10¹⁸ = hau.

Perhaps you can use this names in Base Hundred. In any case, it seems me very important to develop the Aryabhata system at Wikipedia.

John 2006 09:17, 4 August 2006 (UTC)

PS.  I see, an article Katapayadi sankhya  – a later simplification (castration) of Aryabhata's ingenious system –  exists. Currently with a very justified cleanup-tag.

Vigesimal:

Yes a true Base 20 system does expresses 77₁₀ as 17 + 3 x 20.

77₁₀ = 3H₂₀ = (3 × 20¹ + 17 × 20⁰).

KH₂₀ = (19 × 20¹ + 17 × 20⁰) = 397₁₀.

(0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 = 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,J,K)

Let's covenant that:

I agree.

More comments in a few days. --Najro 16:12, 4 August 2006 (UTC)

My fault. You are right:  77₁₀  =  3H₂₀   (Excuse me, I seldom count in true vigesimal ;-)
Have a good week-end Najro,  John 2006 17:09, 4 August 2006 (UTC).

PS. However: (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 (20) = 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,I,J (K)  So:  KH₂₀ = (20 × 20¹ + 17 × 20⁰) = 417₁₀. (Perhaps!)
But the only important is, that – mutually – we understood what we meant.

Thanks. Vigesimal: I used the method that skips I (letter i) to avoid confusion with 1 (digit one). You dont use 20 in vigesimal, only 0-19, in the same way you use only 0-9, not 10, in decimal.

A,B,C,D: Yes, without looking at the names of the Peletier numerals, it is a SB 1000 system. And that is the point. I put the systems into different types according to their mathematical structure. Then you cannot skip the zilliards and keep them at the same time, as in the synonym argument. "But the Peletier system respects the Chuquet zillions". Yes of course, I repeat: The naming method of Peletier is base million (kind of), but the Peletier system is base thousand. So mathematics first, language second. But if you considered language first and mathematics second, then yes you might end up with a different classification.

'1, 000 000, 000 000': Yes, in the Chuquet system "," (comma) has been used as super-superbase (1000000) separator, " " (blank) used as superbase (1000) separator and "" (nothing) used as base (10) separator.

"thousandbillion": Yes it would be ranged a SB 1000 system, but only if you didn't notice that the meaning still was thousand billion. If you dropped several letters from this word it would be better disguised. As I understand this is what has happened to the Danish numerals. Wikipedia says: "Most Danes are unaware of the vigesimal roots of such numerals". But for the aware people the Danish numerals show their heritage from the base 20 system. And I mean the Peletier system shows its heritage from the base million system. The Danish numerals have now transformed into base 10 and Peletier transformed into base 1000.

Steps: In positional notation there are no fractions of the exponent like 1000000^1.5 = 1000000000. The base b is raised to the exponent k like b^k, and only integer values of k is allowed. But in a superbase system, ok, you could talk about steps by looking at a subbase of the superbase. E.g. in a pure superbase million system there are five possible steps (by the subbase = base 10) that is 10 = 1000000^{1 / 6},100 = 1000000^{2 / 6},1000 = 1000000^{3 / 6},10000 = 1000000^{4 / 6},100000 = 1000000^{5 / 6}. But it is not the superbase that is subdivided into steps, it is the exponent of the base that forms the superbase that is divided (10^{6 / 6}).

Aryabhata: Yes it would be nice to find a real superbase 100 system. One would like to find them in a table in English and then more or less just copy and paste it. But it is not that easy of course. The names may have been taken from the letters in some Indian alphabet? [[3]]

Katapayadi sankhya: This article was too difficult for me. --Najro 18:00, 8 August 2006 (UTC)

Thanks for your answer Najro.

Of course one doesn't need "digit 20" in base 20. By counting digit 17, I counted up to 20, then, instead of taking 3, I took this K of the full base, what we don't need as digit. However we can also write:  77₁₀  =  3H_K. Surely I hesitated to do so, when I erred.  ("...to avoid confusion with 1 and I." You see how useful it is to always make the up-bow of the digit 1 ;-)

"So mathematics first, language second. But if you considered language first and mathematics second, then yes you might end up with a different classification."
Your arguement is not consistent, since the only criterion to put Peletier in another class than Chuquet is a semantic argument, one word. Like you can see with many terms in different languages: What in one language is one word, in others, not seldom, needs two (or three) words. However, it's the same term.

Exactly in the same way, I consider "milliard" is practical contraction of the term "thousand million". Nothing more!
So in both systems the main base is million, in both systems "thousand zillion" is an auxiliary base, expressed – discretionary – as "thousand zillions" or "zilliards".
It's a semantic variant of the same system. The meaning of "billiard" is still "thousand billion". Everyone does notice this fact.

Peletier system (Peletier himself never used it, since he used milliard synonym for million) is not a transformation into base 1000, because the main base million still applies.

Steps & positional notation:  If there are some scholars who decree that "only integer values of k are allowed"  – in the article that's not mentioned –  then these scholars err.

[ PostScript:  If k means cardinal numbers, then k can't be used to discribe properly the Peletier system. Is not adequate. Basta! ]

Let's take Chuquet's own example number: 745 324, 804 300, 700 023, 654 321.  I don't see any other consistent way to discribe the systems than – respectively – like this:

Chuquet system
		(	(7×102 + 4×101 + 5×100)	× (103)	+	(3×102 + 2×101 + 4×100)	)	× (106) 3	+
	+	(	(8×102 + 0×101 + 4×100)	× (103)	+	(3×102 + 0×101 + 0×100)	)	× (106) 2	+
	+	(	(7×102 + 0×101 + 0×100)	× (103)	+	(0×102 + 2×101 + 3×100)	)	× (106) 1	+
	+	(	(6×102 + 5×101 + 4×100)	× (103)	+	(3×102 + 2×101 + 1×100)	)	× (106) 0
Peletier system
			(7×102 + 4×101 + 5×100)	× (106)3.5	+	(3×102 + 2×101 + 4×100)		× (106)3.0	+
	+		(8×102 + 0×101 + 4×100)	× (106)2.5	+	(3×102 + 0×101 + 0×100)		× (106)2.0	+
	+		(7×102 + 0×101 + 0×100)	× (106)1.5	+	(0×102 + 2×101 + 3×100)		× (106)1.0	+
	+		(6×102 + 5×101 + 4×100)	× (106)0.5	+	(3×102 + 2×101 + 1×100)		× (106)0.0

The Peletier system is only discribed accurately as a "half-step system". The good theory must allow .5 values in exponent.

[ PS2.  Or if this is not possible, Peletier may be discribed in the same way as you did for Chuquet: thousand billion = 10³((10³)²)².
Similary:  billiard must be interpreted as a synonym for thousand billion. Then, the both systems are intrinsically 1000^2 systems. ]

For the separators:  The inventor Chuquet, himself, didn't use, neither point nor comma separators but sup-script "apostrophes", like you can see here. He says "points", but obviously he used this word in a larger general marker sense. Many decimal separator "comma countries" traditionally used points as thousand separators, in the opposite, "point countries" traditionally used commas as thousand separators. Then, since SI was not able to obtain a consensus in the decimal separator form, therefore they recommend non-breaking spaces as thousand separators to avoid confusions.

I.M.H.O., in this topic, the modern way must be a renouncement of the "comma countries" to this possible, but losing graphem (used first by Snel in 1608) to come universally to the winning "decimal point", invented by Magini in 1592. [ PS3. or perhaps by others, cf. this link ]. This is the only, admittedly, courageous, but inevitable and progressive resolution.

While waiting that this judiciousness has arrived the last "loiterer", I consider, at least in the "point countries", yet we can use the consistent form 745 324, 804 300, 700 023, 654 321. As well for Chuquet, as well for Peletier variant. The inconsistent short scale – at all hazards! – will perish. Joky, that nonsense can be noted as 745, 324, 804, 300, 700, 023, 654 321.

Āryabhaṭa:  Yeah, this genial mathematician not even used our positional decimal system, since he was coeval to the its development. Perhaps in his old days he was acquainted with this system, probably not in 510 when he invented his own system, which attests a perfect awareness of the principle of positions. However, without zero-sign! Hence, it is not really a positional system. Notwithstanding, he named the numbers in a very economical way.

You see: Sometimes the fortuity is favorable ;-)  Right now a new article Sanskrit numerals was created. But it is not finished.

Despite of his great renown till our days his numeration was never really used by anyone excepting by himself. Several reasons: The main reason was the objective superiority of our positional decimal system, invented in India during Āryabhaṭa's life-time (perhaps some precursors before, but that's not sure). The second important reason was: It's true the system is relativly complicated. The are also some superfluous redundancies, since 24 for example can be expressed either in only one syllable "bha" and one written ligature consonant + vowel or it can be expressed in two syllables "gha-na" (What's more logical!) and thus written with two ligatures. Nevertheless, his system was ingenious.

So, Āryabhaṭa was an "exotic". Never somebody before him invented a similar system and never afterwards. At least this was true already about 16 years ago.

No, I don't think, that it would be nowadays sensefull to note digits with a combination of consonants and vowels. This is contrary to the progressive principle: One digit, one symbol.

However as number names, this is a good way. If you have time and interest, therefor you can study these two links [1] and [2]. If this is the case, your mathematical analysis of this proposed system would be interesting.

John 2006 10:51, 9 August 2006 (UTC)

PS. The Katapayadi sankhya article is, first of all, badly edited. So no wonder one don't understand. The musical part, me neither, I didn't kept, so badly explained it is. However, the mathematical part is easy. Many followers of Āryabhaṭa in later centuries, took his numerals, but attribuated values (generally) equal to the rest of a division per ten. So they obtained several variants of each-one of the ten digits, including divers zero digits. But then, I.M.H.O., one can take directly the common decimal digits. It's true, it simplifies the original system. It becomes fully positional, but without real advantage w.r.t. the standard decimal system. However, that's Katapayadi sankhya.

Hello,

I dont see how "milliard" could be a contraction or abbreviation of the "thousand million". Some "linguistic" explanation would be needed. And that explanation would still only be valid in French and Latin languages. To other languages milliard and million seems like two different words (but somewhat related).

If you still think Peletier is SB 1000000 you could write something in the article that it can be disputed that Peletier should be classified SB 1000. Your main argument seems to be that the zilliards are not real numerals, they are only intermediate numerals. Etc. My opinion for the moment is that Peletier is SB 1000.

The k of b^k in the summation formula is a cardinal number as I understand it. "one usually uses letters from the middle of the alphabet (i through q) to denote integers". But the positional notation formula is maybe more descriptive than being a "law".

I think the Hexadecimal metric system is a base 16 and superbase 1048576 system, classified as 16⁵. Very compact numerals indeed. I interpret mir- as 1048576 to the power of .... Then there is also the difficult mi, me, ma, ma, mu series. If m = 1048576 then mi = 1048576 * 16⁰ = 1048576. But me = 1048576 * 16¹ is not equal to 1048576². If we interpret the suffix -i, -e, -a... as 1048576 to the power of 0,1,2... if prefixed by m then the series seems to be superbase 16⁵ with offset 1 because without offset mi should mean 1048576⁰ = 1 but it means 1048576¹ = 1048576. Najro 15:40, 14 August 2006 (UTC)

[edit] The word "Apparently" is not encyclopedic

In the phrase "so the superbase is apparently 1000." in the first paragraph of the first section, the word apparently is used. This is not encyclopedic, per Wikipedia's guidelines. I have rephrased it.