pi and grahams number

[mollykins]

on qi and in a book i'm reading, it said that Grahams number is so big that all the material in the universe couldn't write it, but what about computers, they've made pi into a number that takes up over a terabyte of space, so could they, if they knew what grahams number was, writie it on computers? or do they count as the material in the universe that could write it down, but steven fry said that there wasnt enough material in the universe the make enoguh ink?

[joggerjayne]

Maybe a computer could work out the number, but could never display it or print it.

[dizmouk2009]

Least we know the last 10 numbers in Grahams numbers are 2464195387. That makes it fun.

[Jumbuck]

pi is a transcendental number and so does not have a cut-off point i.e. it has an infinite number of digits after the decimal point. The latest calculation of its value stopped at the number of digits that they decided to stop at for reasons of storage etc but one can never calculate it's value exactly because that would take an infinite amount of time.

Graham's number is not a transcendental number (as far as I know) and thus has a size - although it is very big. The difference is that we know that pi has a value lying between 3 and 4 and is thus well within our everyday experience of size whereas Graham's number is far beyond our experience.

In theory if you started saying all the digits of Graham's number then, at some far point in the future, you would reach the final digit but you could never do the same with pi because there is no end to them.

[chakka35]

It is true that there is not enough material in the universe to express Graham’s number by conventional means, so Donald Kuth devised special notation where 3^3 means three cubed and 3^^3 means 3^(3^3) i.e 3^27.

3^^^3 = 3^^(3^^3) which would, if expressed conventionally give a tower of exponents 7,625,597,484,987 layershigh!

Now consider the number 3^^^…^^^3, in which there are 3^^^^3 arrows – an unimaginably large number.

Then the number 3^^^…^^^3 in which the number of arrows is that previous number.

Carry on doing this, making the number of arrows in 3^^^…^^^3 equal to the previous number until you are 63 steps from 3^^^^3. That is Graham’s number.

(Thanks to the Penguin Dictionary of Curious and Interesting Numbers)

[vascop]

So what's the point of this Graham's number'? I could invent a number using say 5 where Graham's number uses 3 and get an even bigger number, and call it vascop's number - gee I'm famous!
It all seems totally ridiculous to me.

[chakka35]

Sorry vascop, I should have explained. Of course you can think up numbers bigger than Graham's, since numbers go on to infinity.

Graham's is the largest number which has ever been used or needed in a mathematical problem. Don't ask me to amplify but it is "an upper bound, derived by R L Graham, from a problem in a part of combinatories called Ramsey theory".

[mollykins]

Infinity comes in idfferent sizes though, its been proven.

it was in the same book as grahams number and pi, 'the hair of the dog'.

I can't find the page but its something like there are different sets of numbers that don't link together for somereason like normal numbers, fractions and decimals and their infinties are different, somehow.

[chakka35]

Not so much different sizes, mollykins, but different colours as you might say.

There's an infinite number of odd numbers, which is strange because that infinite number omits the infinite number of even numbers, the infinite number of fractions, the infinite number of numbers ending in 7, and so on.

In fact there is obviously an infinite number of different 'colours' of infinite numbers!

[bigcatpants]

keep in mind that you're saying 'material in the universe', meaning all the little atoms that make up computer parts and such. Another method of attempting to envision how graham's number cannot possibly be every denoted by amount: A Planck Length is a unit of distance that has been said [though it is untrue] "any smaller distance essentially makes no sense". This is smaller than your protons, electrons, and as far as I know, any other common subatomic particle. if you were to get a 1x1x1 cube of planck length, you'll have the smallest unit of volume. Now, if you were to measure the volume of the universe in Planck Cubes, that wouldn't be near Graham's Number. Measuring the biggest thing in the universe [the universe itself!] using the smallest unit of measure doesn't even amount to Grahams number!

Try thinking about infinity now. ;)