Contest: Large number

1. Mar 25, 2016

micromass

Staff Emeritus
This is just for fun. No prizes. I just want to see what people come up with.

The rules are this:
- You need to write in your post a real number (reminder: real numbers are finite by definition).
- The person who writes the largest real number wins
- Your usable characters are limited to 200 characters.
- You need to present your number between CODE brackets so I can count the characters
- Standard mathematical functions are accepted, everything else must be referenced or explained (and the explanation falls within the 200 character limit.
- No references to earlier posts allowed.
- Attempting to replicate (forms of) Berry's paradox is not allowed.

For example:

Code (Text):

9999

Last edited: Mar 25, 2016
2. Mar 25, 2016

ShayanJ

Code (Text):
${\LARGE e^{\tan(\frac{\pi}{2}-10^{-9.7})} }$

3. Mar 25, 2016

Staff: Mentor

Knuth's notation:
Code (Text):
9↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑9
Edit: I see someone has edited the post - note I wasn't sure if the Knuth's notation counts as standard, so I have not used all possible up arrows, leaving place for the "Knuth's notation" name. If if counts as standard, then obviously there should be 198 up arrows.

4. Mar 25, 2016

axmls

Well keeping in line with the above post, just do this:
Code (Text):
x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!, x is Graham's number.
That would be taking Graham's number factorial 176 times.

5. Mar 25, 2016

micromass

Staff Emeritus
I'm pretty sure Borek's is bigger though...

6. Mar 25, 2016

axmls

Code (Text):
x↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑x, x is Graham's number
Now it would be fun to figure out whose is bigger. Borek's has 23 extra arrows, but at the same time, mine uses Graham's number as a base.

7. Mar 25, 2016

micromass

Staff Emeritus
Yeah, that'll do.

8. Mar 25, 2016

micromass

Staff Emeritus
Or does it? If Borek has more arrows, then he should still have a higher number...

9. Mar 25, 2016

Staff: Mentor

I am so happy I am not the one to estimate which one is larger :)

But I am not convinced you are right about !!!!!... meaning - !! is a double factorial, which is not the same thing as factorial of factorial.

10. Mar 25, 2016

axmls

Ah, you're right. I figured parenthesis would be wasteful, but I realize it's not the same thing.

Now, micromass, are you sure more arrows necessarily means a larger number? After all, each Graham's number is calculated from an unfathomable number of arrows.

11. Mar 25, 2016

micromass

Staff Emeritus
Yes, you are correct. I'm sorry. I forgot how Graham's number was defined exactly... Yours should definitely be larger then.

12. Mar 25, 2016

axmls

Yeah, the number of arrows in the second step of 64 steps in calculating Graham's number is larger than the number of Planck volumes in the universe. I've never been so scared of a number.

I actually read an article a while back--can't remember where--by a mathematician who was going to write a satire piece declaring Graham's number to be an upper bound for the integers, effectively making it possible to proof many theorems from number theory by exhaustion.

Then he realized that Graham's number is so big, it really wouldn't make a difference at all when it comes to proofs.

13. Mar 25, 2016

ShayanJ

I chose -9.7 in my number because that was the biggest I could get wolfram alpha calculate the value of the expression with, so people could know what it is. Going deeper and deeper into the negative numbers, makes the value of the expression larger and larger. So I can write ${\LARGE e^{\tan(\frac{\pi}{2}-10^{-10^{100}})}}$, but...well...who knows what this number is and whether its bigger or smaller than the numbers suggested by others!

14. Mar 25, 2016

micromass

Staff Emeritus
Definitely smaller. It is very easy using Taylor expansions to compute the number of digits of this number. The numbers posted by others are so large, you'll need special notations to compute the number of digits.

15. Mar 25, 2016

ShayanJ

So what about ${\LARGE e^{\tan( \frac{\pi}{2} - 10^{- 10 \uparrow^{100} 10} )}}$ ? (where $\uparrow^n$ means repeating $\uparrow$ n times!)

16. Mar 25, 2016

ShayanJ

Actually you can nest all of these notations together to get bigger and bigger numbers, there is no limit. It seems the restriction to 200 characters is exactly because of this, so people actually should give the biggest number possible that can be represented using 200 characters. But in math you always can invent shorthand notations so even that won't place a limit and I guess this proves that this thread won't terminate.(If it was a computer program...as a thread, it will bore the participants and die out!)
But I like it, Its nice to see people try!

17. Mar 25, 2016

micromass

Staff Emeritus
Nono, since the shorthand notations must be explained within the 200 character limit. This will ensure there definitely is a largest number expressible in 200 characters. After all, there are far less than $100^{200}$ numbers expressible this way. So there definitely is a largest, even though it is hard to find which one it is.

18. Mar 25, 2016

axmls

Graham's number uses more arrows than there are Planck volumes in the observable universe for the first step in calculating it. Then it takes the result of that number, and that's the number of arrows for the next step. It repeats that 63 more times, and then you've got Graham's number.

However, with your notation, we could easily say, for instance $$x \uparrow ^{x!} x$$ and it would seem the 200 character limit allows us all the freedom to go arbitrarily large.

By the way, don't try to comprehend the size of that number if $x$ is Graham's number. You might go comatose.

19. Mar 25, 2016

ShayanJ

Yeah, that seems to be a large number. But that still doesn't prove its bigger than the number I posted!
Anyway, I can do this:
Code (Text):
${\LARGE e^{\tan( \frac{\pi}{2} - 10^{-x \uparrow^{x} x} )} \uparrow^x x}$
Where x is the Graham's number and $\uparrow^a$ means repeating $\uparrow$, $\left \lceil a \right \rceil$ times.

20. Mar 25, 2016

PeroK

${\LARGE e^{\tan( \frac{\pi}{2} - 99^{- 99 \uparrow^{999} 99} )}}$

Is better.