What is the largest real number one can write within 200 characters?

[micromass]

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:

9999

 

[ShayanJ]

##{\LARGE e^{\tan(\frac{\pi}{2}-10^{-9.7})} }##

 

[Borek]

Knuth's notation:

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.

 

[axmls]

Well keeping in line with the above post, just do this:

x!, x is Graham's number.

That would be taking Graham's number factorial 176 times.

 

[micromass]

Well keeping in line with the above post, just do this:

x!, x is Graham's number.

That would be taking Graham's number factorial 176 times.

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

 

[axmls]

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

How about this then?

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.

 

[micromass]

How about this then?

x↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑x, x is Graham's number

Yeah, that'll do.

 

[micromass]

Yeah, that'll do.

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

 

[Borek]

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.

 

[axmls]

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.

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.

 

[micromass]

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.

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

 

[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.

 

[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!

 

[micromass]

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!

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.

 

[ShayanJ]

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.

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

 

[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!

 

[micromass]

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!)

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.

 

[axmls]

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

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.

 

[ShayanJ]

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.

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:

##{\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.

 

[PeroK]

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

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

Is better.

 

[PeroK]

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.

What about:

##y+1## Where ##y## is the largest number expressible in 200 characters.

 

[Borek]

Time to get a bag of popcorn.

 

[ShayanJ]

What about:

##y+1## Where ##y## is the largest number expressible in 200 characters.

No, we won't have enough popcorns for this kind of thing! Because it will break the rule and will render the thread endless.

 

[micromass]

What about:

##y+1## Where ##y## is the largest number expressible in 200 characters.

You didn't read the rules, did you?

- Attempting to replicate (forms of) Berry's paradox is not allowed.

 

[PeroK]

You didn't read the rules, did you?

Is that Berry's paradox?

 

[micromass]

Is that Berry's paradox?

It's in the same spirit, yes https://en.wikipedia.org/wiki/Berry_paradox

 

[Cruz Martinez]

This is a nice site which gives some of the backround necessary to follow this thread: http://waitbutwhy.com/2014/11/1000000-grahams-number.html

 

[Vanadium 50]

How about 7, only with a font size of 9.999 x 10^92?

You did ask for the largest number, and not the number of greatest possible magnitude.

 

[Student100]

What about https://en.wikipedia.org/wiki/Kruskal's_tree_theorem#Friedman.27s_finite_form?

 

[Ben Niehoff]

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

Is better.

Why stop there?

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

 

[Borek]

It is insane. As long as we deal with numbers that can be expressed either directly or using logarithmic scales I have no problems comparing orders of magnitude (and guessing which number is larger, or trying to somehow evaluate their values). But I fell so hopelessly lost when it comes to hyperoperations

 

[epenguin]

The product of all the numbers except this one that have been and will be posted on this thread raised to the power of that same number that same number of times.

 

[Borek]

all the numbers except this one that have been and will be posted

No way.

- No references to earlier posts allowed.

But you can take a product only of all numbers that WILL be posted to stay in accordance with the rules

 

[micromass]

But you can take a product only of all numbers that WILL be posted to stay in accordance with the rules

And since I will post $0$, that is not a good idea.

 

[Ben Niehoff]

And since I will post $0$, that is not a good idea.

The product of the factorials of (the absolute values of) all the numbers that will be posted, then. :D