Describe the biggest number (10^10^10^10^1000)

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Vytron
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Describe the biggest number (10^10^10^10^1000)

Postby Vytron » Mon Mar 02, 2015 7:33 am UTC

Yet another big number thread. This one only allows number that can be described in some way.

I.e. you know how the number looks like, so you can compare a number that was described with any other number that was described, and tell which one is bigger. If you can't tell if it's bigger or not, then it's an invalid number for the thread.

Examples:

10^100

^Can be described as a 1 followed by 100 zeroes, so it's fine for the thread.

Graham Number

^Even though you can describe how this number is constructed, you can't describe the number itself (only the last digits of the number are known, not how it starts.)

2^15456978

^That one can't be described, i.e.:

3^246811

11^354

Can you tell which ones are bigger without a calculator?

So they're invalid entries.

This thread can still be very big by using previously described numbers to build bigger ones, but here's a difference:

10^10^100

Description: A 1 with a string of 0s where the number of 0s is a 1 followed by 100 zeroes.

It's fine with a straightforward description.

2^2^2^2^2

Description: 2 to the power of 2, 4 times.

^Invalid, you aren't meant to say how to construct the number, but how the actual number looks like (so describe the resulting 200352930... and such number.)

5469876145

^Has implicit description (doesn't need a description because you can see the number in full.)

In other words, posting the result of 2^15456978, 3^246811, or 11^354, or describing Graham's number is fine, but not just directions of how to arrive to the number (e.g. a number must be imaginable, like a 1 with some number of 0s afterwards, but must not need to be calculated).

I'll start with:

1
Last edited by Vytron on Sat Mar 21, 2015 2:15 pm UTC, edited 7 times in total.

GoogologyMaster
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Re: Describe the biggest number

Postby GoogologyMaster » Mon Mar 02, 2015 11:50 pm UTC

10,000

The number can be described as the number of dots shown below:
10000dots.png
10000dots.png (6.83 KiB) Viewed 2887 times

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Quercus
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Re: Describe the biggest number

Postby Quercus » Tue Mar 03, 2015 12:06 am UTC

My number can be described as the number of dots shown in the previous post, with the following addition:

Code: Select all

.

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Re: Describe the biggest number

Postby Vytron » Tue Mar 03, 2015 12:35 am UTC

20,000

My number can be described as the total dots if you put GoogologyMaster's picture from 2 posts ago in front of a mirror, and count them all.

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SirGabriel
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Re: Describe the biggest number (20k)

Postby SirGabriel » Tue Mar 03, 2015 3:20 am UTC

28,092

My number can be described as the total number of words in the Johannine literature in the Greek New Testament.
This description doesn't meet the requirements, but since the number contains an implicit description, it doesn't really need to.

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SDK
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Re: Describe the biggest number (20k)

Postby SDK » Tue Mar 03, 2015 3:36 am UTC

100,000

This is a one, followed by two zeroes, followed by a comma, followed by three zeroes.
The biggest number (63 quintillion googols in debt)

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Reecer6
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Re: Describe the biggest number (20k)

Postby Reecer6 » Tue Mar 03, 2015 4:23 am UTC

10,000,000,000

A one followed by a number of zeroes equal to one followed by a zero.

If I were to come up with some kind of, I dunno, notation for this, it would be 1(1(1))

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SirGabriel
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Re: Describe the biggest number (20k)

Postby SirGabriel » Tue Mar 03, 2015 1:27 pm UTC

Okay, let's move beyond implicit descriptions.
Let f(x) equal the number formed by stringing together every counting number from 1 to x (for example, f(12) = 123456789101112).
My number is f(17). For comparison, this is just over a septillion.

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Re: Describe the biggest number (20k)

Postby patzer » Tue Mar 03, 2015 3:30 pm UTC

Vytron wrote:
2^15456978

^That one can't be described, i.e.:

3^246811

11^354

Can you tell which ones are bigger without a calculator?


Um, yes? A quick Fermi estimate gives:

11^8 = 2*10^8(roughly) (214358881 if my mental arithmetic is correct)

So 11^24=10^25

and 11^354=10^368 (ish)

2^10=10^3 (ish)

2^15456978 = about 10^4700000

3^4=81; 3^6=729; 3^8=6569

ergo 3^17=10^8 (ish)

And 3^246811 = 10^130000 (ish)

They're thousands of orders of magnitude different!

So, uh, 2^15456978 is biggest.

Was this a trick question or something?

Anyway, my number is 31415926535897932384623433.

sorry I'll stop showing off now...
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Re: Describe the biggest number (20k)

Postby SirGabriel » Tue Mar 03, 2015 5:25 pm UTC

Using the same function as my previous post: f(100)

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Re: Describe the biggest number (20k)

Postby Quercus » Tue Mar 03, 2015 5:37 pm UTC

Borrowing SirGabriel's function: f(f(3))

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Re: Describe the biggest number (20k)

Postby Reecer6 » Tue Mar 03, 2015 7:26 pm UTC

Those calculations weren't QUICK though, were they?

g (x) is a function that strings together f(1), f(2), and so on until f(x). So, g(3) = 112123.

My number is g(26), which is a, bit more than 330 digits long. The previous was around 250, I think.

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Re: Describe the biggest number (20k)

Postby SirGabriel » Tue Mar 03, 2015 7:32 pm UTC

Reecer6 wrote:Those calculations weren't QUICK though, were they?

g (x) is a function that strings together f(1), f(2), and so on until f(x). So, g(3) = 112123.

My number is g(20), which is a, bit more than 200 digits long.


A bit more than 200 digits isn't enough. f(f(3)) = f(123) > 10^260

Edit: Reecer6 edited his post while I was in the process of replying to it.

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Re: Describe the biggest number (20k)

Postby SirGabriel » Tue Mar 03, 2015 7:46 pm UTC

h(x) is a function that strings together g(1), g(2), and so on until g(x). So h(3) = 1112112123
i(x) is a function that strings together h(1), h(2), and so on until h(x). So i(3) = 111121112112123

My number is i(9), which, if I'm not mistaken, has nearly 500 digits.

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Re: Describe the biggest number (20k)

Postby SDK » Tue Mar 03, 2015 9:47 pm UTC

10^500

This is a one followed by 500 zeroes with commas interspersed accordingly.
The biggest number (63 quintillion googols in debt)

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SirGabriel
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Re: Describe the biggest number (20k)

Postby SirGabriel » Tue Mar 03, 2015 10:47 pm UTC

9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

This is the number 9 repeated 1000 times.

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Re: Describe the biggest number (20k)

Postby Reecer6 » Tue Mar 03, 2015 11:47 pm UTC

I'd do i(11) because it's only 6 digits longer than that, but I don't think a difference that small is allowed by the rules.

i(20) has over ni... just under ten thousand digits, so I'll put that as my number. Why the big difference? So it's clear enough that it's bigger than a 9 repeated a thousand times.


You know what? I can just post this. This is i(11).

xkcd 40.png


And this is the previous number.

xkcd 41.png
xkcd 41.png (6.12 KiB) Viewed 2696 times


Mine is clearly bigger than it!

To describe these numbers without using an indecipherable function...

Start wrting 1s. Keep putting down 1s until you get four. The four is for the number of iterations the function's gone through, so you can repeat this at higher levels. Once you have four, put down the next higher number - a 2. Now, start this over again but with three 1s. After the third 1, put a 2 again. Keep going like this, reducing numbers of 1s, until you have written four 2s. Now, put down a 3. The pattern repeats as it did with the 2, starting with three 1s. Stop doing this once you get to the number you plugged into the function. Here, it's 11.

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Re: Describe the biggest number (20k)

Postby faubiguy » Wed Mar 04, 2015 3:06 am UTC

My number is a 1 followed by the same number of 0s as there are dots in this picture:

Image

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SirGabriel
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Re: Describe the biggest number (20k)

Postby SirGabriel » Wed Mar 04, 2015 3:52 am UTC

10^(10^10)

A one followed by ten billion zeroes.

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Re: Describe the biggest number (20k)

Postby GoogologyMaster » Wed Mar 04, 2015 8:47 pm UTC

10^10^20

one followed by 100,000,000,000,000,000,000 zeros

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Re: Describe the biggest number (20k)

Postby Vytron » Thu Mar 05, 2015 12:05 am UTC

@patzer: Okay, you're describing how to compare such numbers without a calculator, so you could send numbers like 2^15456978 as long as you describe how they're bigger like that (this applies to everyone.)

I send:

2x10^10^20

In scientific notation, this is a 2 followed by 100,000,000,000,000,000,000 zeros.

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Re: Describe the biggest number (2x10^10^20)

Postby GoogologyMaster » Thu Mar 05, 2015 12:13 am UTC

10^(10^20+1)

1 followed by 100,000,000,000,000,000,001 zeros, five times larger than the previous entry

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Re: Describe the biggest number (2x10^10^20)

Postby Vytron » Thu Mar 05, 2015 12:35 am UTC

6x10^(10^20+1)

In scientific notation, a 6 followed by 100,000,000,000,000,000,001 zeros.

Yes, count them, this is:

6

SIX

Times larger than the previous entry. Like, whoa, man!

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Re: Describe the biggest number (2x10^10^20)

Postby Reecer6 » Thu Mar 05, 2015 1:00 am UTC

My number is 1 followed by 13 zeroes, which is then followed by the number one digit smaller than it, and then the number smaller than that, and so on. So, the first 40 digits of my number are 1000000000000099999999999999999999999998. My number is, more or less, repeating 13 digits 10,000,000,000,000 times. Well, it'll be less than that, since it'll get smaller towards the end, but that will only affect the last tenth, so it isn't too important.

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Re: Describe the biggest number (2x10^10^20)

Postby Vytron » Thu Mar 05, 2015 1:15 am UTC

Sorry, I don't think that's valid, if you can't count how many digits it has it's an "unimaginable number."

For instance, I have no idea if a number that repeated 13 digits 10,000,000,000,001 times would be bigger or smaller, or exactly how many times surpass it.

(the number you're sending is valid, but not its description)

It's important that the number you send is not "fuzzy", but that it's some concrete amount.

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Re: Describe the biggest number (2x10^10^20)

Postby Vytron » Thu Mar 05, 2015 1:17 am UTC

(Or you could, you know, send a number that repeats 13 digits 10,000,000,000,000 times, where those digits are 9999999999999, and it would be fine)

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Re: Describe the biggest number (2x10^10^20)

Postby SirGabriel » Thu Mar 05, 2015 3:43 am UTC

10^(10^100)

1 googolplex: a 1 followed by a googol of zeroes, where a googol is a 1 followed by 100 zeroes.

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Re: Describe the biggest number (2x10^10^20)

Postby Vytron » Thu Mar 05, 2015 4:20 am UTC

7x10^(10^100)


7 Googolplexes: A 7 followed by a googol of zeroes, where a googol is a 1 followed by 100 zeroes.

This is 7 times larger than the previous entry.

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Re: Describe the biggest number (2x10^10^20)

Postby GoogologyMaster » Thu Mar 05, 2015 12:36 pm UTC

10^10^101

Googolplex^10, which is a one followed by as many zeros as a one followed by 101 zeros.

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Re: Describe the biggest number (2x10^10^20)

Postby SirGabriel » Thu Mar 05, 2015 1:02 pm UTC

f(10^100)

The number of digits in this number can be described as two nines followed by ninety-seven eights followed by two nines followed by a zero.

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Re: Describe the biggest number (2x10^10^20)

Postby Vytron » Thu Mar 05, 2015 1:55 pm UTC

I think GoogologyMaster's number is higher.

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Re: Describe the biggest number (10^10^101)

Postby SirGabriel » Thu Mar 05, 2015 2:12 pm UTC

f(10^100) > 10^(9.98*10^101) > 10^10^101

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Re: Describe the biggest number (10^10^101)

Postby Vytron » Thu Mar 05, 2015 2:16 pm UTC

But in your description you say:

"The number of digits in this number can be described as two nines followed by ninety-seven eights followed by two nines followed by a zero."

The number of digits that you describe are:

2 (the two nines)
+
97 (the 8s)
+
2 (the other two nines)
+
1 (the zero)

For a total of 102 digits.

While the description of GoogologyMaster's number:

"a one followed by as many zeros as a one followed by 101 zeros."

Means it has more than 102 digits (it has 1000... digits where those 0s are 101.)
Last edited by Vytron on Thu Mar 05, 2015 2:18 pm UTC, edited 1 time in total.

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Re: Describe the biggest number (10^10^101)

Postby GoogologyMaster » Thu Mar 05, 2015 2:18 pm UTC

Vytron wrote:But in your description you say:
2 (the two nines)
+
97 (the 8s)
+
2 (the other two nines)
+
1 (the zero)

For a total of 100 digits.

This should be 102 digits (2+97+2+1=102)
Edit: It got corrected.

Anyway, I enter the number
10^10^200 - 1
The number of digits in this number is a one followed by 200 zeros, all digits are 9s.
Last edited by GoogologyMaster on Thu Mar 05, 2015 2:19 pm UTC, edited 1 time in total.

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Re: Describe the biggest number (10^10^101)

Postby Vytron » Thu Mar 05, 2015 2:18 pm UTC

Edited right after you quoted.

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Re: Describe the biggest number (10^10^101)

Postby SirGabriel » Thu Mar 05, 2015 7:12 pm UTC

I did not say that my number has 102 digits, I said it has 998888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888990 digits.
The number of digits in the number of digits of my number is 102, just as with GoogologyMaster's number.

My new number is 10^10^200 - 1/(10^10^200)
The number of digits before the decimal and the number of digits after the decimal in this number are both a one followed by 200 zeroes, all digits are nines.

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Re: Describe the biggest number (10^10^101)

Postby GoogologyMaster » Thu Mar 05, 2015 8:50 pm UTC

Now I have a great idea. Suppose we had a sequence with some number of 0's and then zoom out so we can see more 0's. In this sequence, I zoomed out by 2x to see 4x as many zeros.

ffdfos.png


My number is 10^4^1000 which is Level 1000 in this sequence. (imagine zooming out 1000 powers of two)

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Re: Describe the biggest number (10^10^101)

Postby SirGabriel » Thu Mar 05, 2015 10:50 pm UTC

GoogologyMaster wrote:My number is 10^4^1000 which is Level 1000 in this sequence. (imagine zooming out 1000 powers of two)


I don't know about you, but I can't quickly estimate 4^1000 in my head.

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Re: Describe the biggest number (10^10^101)

Postby GoogologyMaster » Fri Mar 06, 2015 12:46 am UTC

4^1000 = 4^(2*500) = 16^500
So this number is bigger than 10^10^500.

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Re: Describe the biggest number (10^10^101)

Postby WarDaft » Fri Mar 06, 2015 4:33 am UTC

10^10^10^10

A discrete 10-hyper-hyper-hyper-cube.
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