Writing out a googolplex in long form

[Hypercube]

I was just wondering; is there any way, within the realm of what is practically possible, to write out the number 'googolplex' in its long form? I mean, writing it down on paper is obviously impossible; it would probably take more than a googol years to do so, no to mention that the universe isn't even close to being large enough to accommodate the amount of paper that would be required.

 

But modern computers are capable of extraordinarily rapid calculations and the like. Is there any way for a computer to render the googolplex in its full form? Even in principle?

 

Oh, for anyone who doesn't know, a googolplex is 10^10^100; 1 with a googol zeroes after it.

[Sisyphus]

In a computer? Seems like binary would just make it harder.

 

However, in base googolplex, the long form is simply "10."

[Hypercube]

No; I'm talking about writing it out like, 100000000000000000000000000............0

 

Just like you would write 'a million' as 1000000.

[Schrödinger's hat]

Umm, not really. Unless you mean using some kind of compression technique, then it'd be no different to writing 10^10^100Well, let's generalise the idea of 'write' to mean arranging some things which we will take as representations of 0s and 1s.

 

Additionally, I shall take the mass of the observable universe as the largest semi-reasonable constraint on our ink-supply (more universe may become visible later, and expansion will put a lot of stuff out of our reach forever etc, but this the only reasonably upper bound I can think of)

 

There's about 10^50 kg of mass we can observe (give or take a few orders of magnitude from error and whether or not you include dark energy/dark matter).

There definitely won't be enough atoms to write it, photons will get away all the time, so I wouldn't really regard that as writing. So lets say we can convert the whole universe into positrons and electrons (I'm ignoring parity, colour and all sorts of other conserved quantities, but these would only put tighter constraints on it)

mass of 10^-30 each

 

This gives us 10^80 0s to work with. Still only about a billion trillionth of what we need to write down a googolplex. This is only approximate, and maybe you can eek out a few more orders of magnitude, but with any sensible definition of the term 'write in long form' the answer is no.

[Mr Skeptic]

Sure. All you have to do is blank a large enough medium (instantly writing all the zeros), and then put the 1 at the proper location. You'd need a medium that can hold that number of zeros though.

[CaptainPanic]

Sure. All you have to do is blank a large enough medium (instantly writing all the zeros), and then put the 1 at the proper location. You'd need a medium that can hold that number of zeros though.

I think that Schrödinger's hat just explained that the universe itself wouldn't be a big enough medium to do that.

[Mr Skeptic]

I think that Schrödinger's hat just explained that the universe itself wouldn't be a big enough medium to do that.

 

Use photons, or part of the universe that extends beyond our observable universe, to make up for the missing 10^20 bits.

[Xittenn]

Wouldn't it suffice to have a base system represented by atoms, their isotopes, electron configuration and incorporating entanglement in some fashion? Light could be used to 'flip-flop' electron configurations in atoms given a proper medium to buffer such a state. I'm pretty sure there are far more compact forms than electron-positron pairs binary that wouldn't be too difficult a constraint; hypothetically speaking of course.

 

Converting bases with these large numbers is a problem and I would have to write my own software should I wish to present any real data on the matter.

Edited March 4, 2011 by Xittenn

[Schrödinger's hat]

Skeptic, it's not a missing 10^20 bits, it's a missing 10^20 times as much.

If you extend the definition of write beyond what I had, photons would work reasonably well, but you'd have to have a high enough frequency so they didn't overlap (or be very very patient with your transmitter)

Xitten, again, it depends on what you mean by write. There are probably more than a googolplex ways of rearranging even small systems if you use every single degree of freedom, but I interpreted using such a system as much the same as writing 10^10^100.

I interpreted write as getting a thing we call a 1, and a thing we call a 0, then laying them out side by side. My idea of laying them out was predicated on them not flying of at the speed of light, but I suppose if you didn't mind that then photons are just dandy for the task, and radio-wave frequencies would probably give you the extra numbers you need with the mass available.

Then it's just a matter of bandwidth, and how patient you are.

I guess there would be entropy constraints too (you're getting a high entropy system -- the universe -- and putting it into a very low entropy state).

My thought exercise was largely intended to portray the difficulty of writing such a thing, not theoretical impossibility.

[khaled]

let's see memory size needed to keep this number in it long form,

 

if the googolplex contains 1 next to googol zeroes, [math]{10}^{100}[/math] digits

 

and since we can encode digits in Hexadecimal Base (4 bits), in other words, 1 Byte for every 2 digits

 

needed memory size = [math]\frac{{10}^{100}}{2}[/math] Bytes

 

= [math]\frac{{10}^{100}}{2 \times 1024}[/math] Kilo Bytes

 

= [math]\frac{{10}^{100}}{2 \times 1024^2}[/math] Mega Bytes

 

= [math]\frac{{10}^{100}}{2 \times 1024^3}[/math] Giga Bytes

 

.. it seem after some calculations, that If we have a memory unit (assume X-Byte) = [math]1024^{33}[/math] Bytes

 

then we can represent a googolplex digits on a memory medium of Size 3 X-Bytes, where

 

[math]3 \times 1024^{33} > \frac{{10}^{100}}{2}[/math]

 

References:

 

http://en.wikipedia.org/wiki/Googolplex#Size

 

http://www.wolframalpha.com/input/?i=%2810^100%29%2F%282*%283*%281024^33%29%29%29

Edited March 6, 2011 by khaled

[Xittenn]

If all attempts are being made to optimize would binary coded decimal truly be appropriate?

[khaled]

If all attempts are being made to optimize would binary coded decimal truly be appropriate?

 

that's what I've done above, BCD, every digit (0~9) can be coded using 4 bits (0000 ~ 1001),

 

but it needs [latex]3 \times 1024^{33}[/latex] memory size ...

 

But even with this optimization, at the mean time .. it's impossible !

Edited March 7, 2011 by khaled

[John Cuthber]

A googolplex is represented by 10 if you use the googolplex as the number base.

On the other hand, I haven't space here for all the symbols you need.

[RVJ]

Bringing life to an old topic:

There is a website that allows you to fully visualize what it is you're asking.

 

http://www.stanford.edu/~nitsche/cgi-bin/numbers.pl?googolplex

 

If I were you, I wouldn't waste my time waiting for your computer to load all of those zeros.

I'm sure it would be possible to write a code that renders numbers in real time if you used some sort of loop (rather than saving a googolplex worth of zeros).

[Delta1212]

it would probably take more than a googol years to do so

Well, since there are only a googol zeroes in the number, that would require writing it a pace just a very tiny fraction faster than one digit per year.

[EdEarl]

I think it can be done if we use another base, such as 10^100 base.

[Ron Bert]

Visualizing a googolplex (1 with all those zeroes) is our attempt as humans to reach out and touch something larger than ourselves. It is still a finite number in the googolfold of space in time. In the search for the largest number, endless zeroes may be added to the void or realm we attempt to understand. To physically represent this 'googolplex' number, one could consider volume over the conventional linear number frame reference that we are all more familiar with. Compactness could also be considered in representing the area involved, as has been suggested by using sub-atomic particles or photons as character counters (still mostly space if gravitationally loose; density yet another variable). Ah, counting sands in the sea of space. In fact, I may suggest that at times we all try to think too hard. I believe the larger question, in our ability to cope with these huge numbers, is found in the limits of what we can perceive. Directional focus would also have to be considered in that of an inward or outward (or both) nature. How much of the (chosen frameset) universe would we have to fill with zeroes to complete this answer. There is more than a googolplex of possibilities. Perhaps a googolplex is best seen in the palm of each of our hands as we all hold the sky. There is an incomplete rendition at http://www.googolplex.com.

Edited May 27, 2013 by Ron Bert