Large numbers; short scale and long scale

[Humberto]

I want to share some information about large numbers that I personally find very remarkable.

The first piece of information is from the Wikipedia article on the number "googol" (which equals 10^{100}, i.e. the digit 1 followed by one hundred 0's), see https://en.wikipedia.org/wiki/Googol:

By Archimedes's calculation, the universe of Aristarchus (roughly 2 light years in diameter) fully packed with sand, would contain 10^{63} grains. If the much larger observable universe of today were filled with sand, it would still only equal 10^{95} grains. Another 100,000 observable universes filled with sand would be necessary to make a googol.

I find this remarkable because, according to my intuition, 10^{95} doesn't "feel" that much larger than 10^{63}, yet it is astronomically larger. And one would have to fill 100 000 of our observable universes with sand to reach one googol worth of sand grains.

Similarly:

Carl Sagan pointed out that the total number of elementary particles in the universe is around 10^{80} (the Eddington number) and that if the whole universe were packed with neutrons so that there would be no empty space anywhere, there would be around 10^{128}.

This again shows just how much larger 10^{128} is than 10^{80}.

To name large numbers, one can either use the short scale or the long scale, see https://en.wikipedia.org/wiki/Long_and_short_scales:

The short scale is now used in most English-speaking and Arabic-speaking countries, in Brazil, in the former Soviet Union and several other countries.

Countries where the long scale is currently used include most countries in continental Europe and most French-speaking, Spanish-speaking, except Spanish-speakers born into an English-speaking culture (e.g. Puerto Rico because of its influence from English-speaking United States) and Portuguese-speaking countries, except Brazil.

Afrikaans uses the long scale. Hence a billion ("biljoen") equals 10^{12} (not 10^9) in Afrikaans, and 10^9 is a "miljard" in Afrikaans (translated into English as "milliard" when using the long scale).

 

[Johnatan56]

That's why milliard, billiard, trilliard, etc.
Same thing for German.
Millionen, Milliarden, Billionen, Billiarden, Trillionen, Trilliarden, etc.

 

[MirageF1]

Sorry to burst your bubble so to speak....there's the Googolplex which puts them all to shame.

A googolplex is the number 10^googol, or equivalently, 10^(10100).
Written out in ordinary decimal notation, it is 1 followed by 10^100 zeroes, that is, a 1 followed by a googol zeroes.

 

[Techne]

TREE3 is also a pretty big number :

 

[Techne]

 

[MirageF1]

TREE3 is also a pretty big number :

I'd say.It dwarfs even a Googolplex ...and Graham's number.

But..AMAZINGLY... it is FINITE(believe it or not!!!!),and does have an upper bound although currently unknown.

Larger than a Googolplex...

10^googol

Where a googol is 10^100
- i.e. 1 followed by 100 zeros
so a googolplex is 1 followed by 10100
zeroes
It is realtively trivial to create a number bigger than a googolplex :

• googolplex + 1
• 10 * googolplex

Are both bigger than a googolplex and in fact there are an infinite number of numbers bigger than a googolplex.
Don’t get me wrong - a googolplex is a massive number - and it would be impossible to write it out in the space available in the Universe, or within time the universe is expected to exist, but for a truely massive number - look at Graham's number - Wikipedia ; a number so big it cannot be written using the 10^x notation I used above. Mathematicians had to invent an entirely new ‘up-arrow’ notation to be able to write out Graham’s number and similar.

 

[Humberto]

Hehe... I can beat you both... How about TREE(4).

 

[Humberto]

It's amazing that TREE(3) is so large, considering that TREE(1) = 1 and TREE(2) = 3.