Making Big Numbers

The year was ~45^2, and I was in dire trouble. My large number skills were taking heavy hits, and I was in dire trouble when someone invoked Graham's Number.

Luckily, I had a Googological notation up my sleeve. One more powerful than Graham's Number. It's based upon hyper operators. Stay for...

Holo Notation
Let's start with the simple hyper operators:

x {4} y = x ^ x ... ^x with y x's

The observant of you will notice that this is tetration. We can continue this, with

x {5} y = x {4} x ... {4} x with y x's

This is also known as "pentation" (amusingly, Google spellcheck unhelpfully thought I wanted to type "penetration").

This can then be generalised to:

x {n} y = x {n-1} x ... {n-1} x with y x's

You could then, hypothetically, imagine a number such as x {y{y}y} x. This is simplified to x {{y}} x.

To simplify the writing of Graham's Number, I also defined a second type of Holo Notation, whereby

x {n} y = x <n-2> y

And, of course, we can extend that by

x <y<y>y> x = x <<y>> x

At this point, I was ready to beat back the Graham's Number. I defined 4 <<64>> 4. The Duseedu was born.

Old Kerbal Diaries: Mun Mission

Today, I went to the Mun in Kerbal space program, and took a lot of screenshots.

Here is my spacecraft on the launchpad. The design came to me when I realised that there was no reason that the lander engines should sit idle (using up precious payload mass) until the Mun is reached. This vehicle pumps fuel from the launch boosters into the lander, where it is used by the lander engine to provide additional launch power.

The vehicle also shows that it is capable of getting to the Mun, with this glorious encounter with a periapsis of 9.7 km. The only issue is that the periapsis is on the terminator, so the braking burn will have to wait until slightly after if I want to land in the light.

Bill Kerman decides that he wants to land on the other side of that huge canyon.

And that is exactly where he lands.

Interestingly enough, on the Mun the jump height of a kerbal is almost exactly the height of my lander. That leads to the interesting situation above, where Bill is able to leap up to the hatch without using any of his EVA fuel.

Reentry is safe, quick, and nothing bad happens (except for Bill spilling his drink. Who even drinks during high acceleration reentries?).

A safe and successful landing. Next stop: Minmus!

"Larger" Numbers: My Own System, Continied

In my last post, I stopped at 1000¹⁰⁰¹ with "Millillion", citing a lack of Latin numbers as my reason for stoppage. However, I've realised that there is an extended Latin system, already in existence! It starts at "bille" (=2000) and ends at "novemonaginongentille" (=999,000). This system is the extended -illions, highlighted in my previous post.

That means, of course, that we can continue counting up to the moderately-sized 1000⁹⁹⁹⁰⁰¹. Let's go, with a fun table:

1000x+1	Name
1000	Millillion
2000	Billillion
3000	Trillillion
4000	Quadrillillion
5000	Pentillillion
6000	Sextillillion
7000	Septillillion
8000	Octillillion
9000	Nonillillion
10000	Decillillion
20000	Vigintillillion
30000	Trigintillillion
40000	Quadragintillillion
50000	Quinquagintillillion
60000	Sexagintillillion
70000	Septuagintillillion
80000	Octogintillillion
90000	Nonagintillillion
100000	Centillillion
200000	Duocentillillion
300000	Trecentillillion
400000	Quadringentillillion
500000	Quingentillillion
600000	Sescentillillion
700000	Septingentillillion
800000	Octigentillillion
900000	Nongentillillion

That leaves one major issue though: how do I construct intermediate values? The previous method may not work - novemnonagintnongentillillion could be either 1000⁹⁹⁹⁰⁰¹ or 1000⁹⁹⁰⁰¹⁰, if the old method is used. The solution is simple - add in a separator to distinguish the "x times 1000" values (the "-illillion"s) from the additive values. "illi" serves this function well. Hence, novemnonagintnongentillillion is 1000⁹⁹⁹⁰⁰¹ and novemillinonagintnongentillillion is 1000⁹⁹⁰⁰¹⁰.

The highest number expressible thus far is 1000¹⁰⁰⁰², which is named the beautiful novemnonagintnongentillinovemnonagintnongentillillion.

But now what? We are out of -illions, and can only express tiny numbers (a novemnonagintnongentillinovemnonagintnongentillillion written out in full digital form would be a mere 14 km long, on the order of the length of the Strait of Gibraltar. This is too tiny!

There's a beacon of hope though. Remember that table I put at the end of my last post? I'll show it:

1000x+1	-illion Name	1/x	SI Prefix
10	decillion	10	deci-
100	centillion	100	centi-
1000	millillion	1000	milli-
106	???	106	micro-

It just so happens to be that our notation ends at 1000^10002+1. How convinient that this table offers the solution to our conundrum. MEET THE MICRILLION!

1000x+1	Name
10002	Micrillion

As with the millillion, this can be modified to give numbers up to 1000¹⁰⁰⁰³. I'll give some examples to demonstrate how:

1000x+1	Name
E6	Micrillion
2E6	Bicrillion
5E6	Penticrillion
E7	Decicrillion
E8	Centicrillion
999E6	Novemnonagintnongenticrillion

Fun, easy, and surprisingly effective. Intermediates are constructed by putting whatever you're adding the the power of a thousand before it. An example would be best, showing the largest number possible thus far. This number is a novemnonagintnongentillinovemnonagintnongentillillinovemnonagintnongenticrillion. Glorious. It's equal to a small 1000¹⁰⁰⁰³ (14 Mm when written as digits, on the order of a terrestrial planet's diameter). Next time, we will define yet larger numbers, on my quest to name ever larger "-illion"s.

--Thomas

"Large" Numbers: My Own System

An experienced googologist knows that there are many systems for naming -illions. I am going to add my own to the list, for no real reason.

Small -illions are the same as they are in the canonical system. They will be listed here, for completeness.

1000x+1	Name
0	Thousand
1	Million
2	Billion
3	Trillion
4	Quadrillion
5	Pentillion
6	Sextillion
7	Septillion
8	Octillion
9	Nonillion
10	Decillion
11	Undecillion
12	Duodecillion
13	Tredecillion
14	Quattuordecillion
15	Quindecillion
16	Sexdecillion
17	Septendecillion
18	Octodecillion
19	Novemdecillion
20	Vigintillion

Does this go up to a usefully high amount? Of course not! A vigintillion, wrote out in decimal form, is a mere 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (10⁶³, E63, or <10,63> depending on which notation you prefer). It doesn't even approach the number of m³ in a Mpc³, let alone larger numbers.

Following the trend set by "decillion", it is trivial to extend the numbers up to 10⁹⁰.

1000x+1	Name
21	Unvigintillion
22	Duovigintillion
23	Trevigintillion
24	Quattuorvigintillion
25	Quinvigintillion
26	Sexvigintillion
27	Septenvigintillion
28	Octoviginitillion
29	Novemvigintillion

Are these so high they are useless? Not at all! Looking at Wikipedia's page on orders of magnitude, it looks like we still need to beat decimal32 floating point numbers (the canonical prefixes beat 32-bit binary numbers, with a mere three hundred and fourty undecillion, two hundred and eighty two decillion and three hundred and fourty nonillion the highest number expressible). Anyway, we seem to have reached a roadblock, given that there is no word for 1000³¹ in the canonical prefixes. However, this is easily fixed, by looking at etymology.

The word "vigintillion" is formed from the Latin word "vīgintī", meaning twenty. To find the next "-illion", we simply find the Latin word for 30, which happens to be "trīgintā". Application is simple:

1000x+1	Name
30	Trigintillion

With -illions number 31 through 39 being constructed in the same way as 21 through 29 and 11 through 19. But why stop at 30? The romans had numbers up to 1,000 (M), and these can easily be used as prefixes for numbers up to 1000¹⁰⁰¹. Given that many people can't count in Latin, I'll include the names of the -illions from 40 to 100 here. Intermediates can be generated as normal.

1000x+1	Name
40	Quadragintillion
50	Quinquagintillion
60	Sexagintillion
70	Septuagintillion
80	Octogintillion
90	Nonagintillion
100	Centillion
101	Uncentillion
111	Undecentillion (Un-dec-centillion)
200	Duocentillion
300	Trecentillion
400	Quadringentillion
500	Quingentillion
600	Sescentillion
700	Septingentillion
800	Octigentillion
900	Nongentillion
999	Novemonaginongentillion (Novem-nonagint-nongentillion)
1000	Millillion

Now, we reach a second roadblock. The Romans had no reason to use numbers over 1000 (pfft, ameutures) and hence haven't named them with single words(they still have notation for it though, which increases at a rate of 1000^x, and they can also express larger numbers by saying them directly, like "eight thousand"). In my next post, I will describe how roman numbers can be extended to stretch beyond 1000 (mīlle), and up to the low -illions.

One last thought for this post:

1000x+1	-illion Name	1/x	SI Prefix
10	decillion	10	deci-
100	centillion	100	centi-
1000	millillion	1000	milli-
106	???	106	micro-

--Thomas