Mathematical Constants

It's graham's number

I recently had the pleasure of meeting Graham's Number, the Guinness Record largest named number -- one with a "practical" purpose, anyway, its being used in a mathematical proof.

It is shown to be an upper bound on a not unreasonably stated word problem (and it also relates to the coloring of the corners of hypercubes of dimension n, but that's so hard to picture):

"Take any number of people, list every possible committee that can be formed from them, and consider every possible pair of committees. How many people must be in the original group so that no matter how the assignments are made, there will be four committees in which all the pairs fall in the same group, and all the people belong to an even number of committees."

Some uncommon notation is need to express it with a formula. Just as multiplication is a generalization of addition, and exponentiation is a generalization of multiplication, the arrow function is a generalization of exponentiation.

For instance:

3↑3 is the ordinary 3-cubed, 27.

3↑↑3 is 3 raised to (3 raised to 3), 3↑27, a good-sized number, 7,625,597,484,987.

3↑↑↑3 is 3↑↑(3↑↑3), or 3↑↑7,625,597,484,987, a very large number. That is: 3 raised to (3 raised to (3 raised to...)))

7,625,597,484,987 times.

3↑↑↑↑3 is 3↑↑↑(3↑↑↑3). Big. Very big.

Still with me? 3↑↑↑↑3 is the starting point for defining Graham's number.

G1 is 3↑↑↑↑3.

G2 is 3↑↑...G1 total arrows...↑↑3. Yikes! But, we've only just begun.

Generally, Gn is 3↑↑...Gn-1 total arrows... ↑↑3

Finally, Graham's Number is G64. That is: 3↑↑...G63 total arrows...↑↑3. It makes me feel woozy. I think I'm gonna hurl.

The next whole number larger than a googol is (googol plus 1).

-- There is also the number whimsically named a "googolplex", defined as 10googol ,

or ' 1 ' followed by a googol zeros.

-- There is also the number named a "googolplexian", defined as 10googoolplex ,

or ' 1 ' followed by a googolplex zeros.

-- There are also numbers with names that are much larger than these, but I don't

know anything about them.

-- There's no such thing as the "largest" number. There might be such a thing as

the largest number with a unique name, but if you choose a number, then no matter

how large it is, I can always add ' 1 ' to your number and make a larger one.

Avogadro's number is the number of "elementary entities" (usually atoms or molecules, ions, electrons, protons etc.) in one mole. Its value is 6.0221415 × 1023.

There are 6.0221415 × 1023 atoms per mole of atoms. This number is known as Avogadro's number.

Compare this to: There are 6 bottles in one six-pack, 12 eggs in one dozen, 12 dozen eggs in one gross (144) etc. They are just numbers.

Note: A mole is a useful quantity, because the mass of 1 mole of a substance will have the same 'grams' as the atomic mass. Example Helium with 2 protons and 2 neutrons, has atomic mass = 4. One mole of Helium atoms have a mass of 4 grams.

If you had a mole of sheets of paper stacked up, it would be 125 BILLION times taller than the distance between Earth and the moon! Now that's a big stack of paper! A mole of water molecules, on the other hand, is about is about 1/2 a fluid ounce of water!

The atoms in a water molecule (H2O) has an atomic mass of (1 + 1 + 16 = 18), so 1 mole of water is 18 grams. With the density of liquid water at nearly 1 gram/milliliter, this mole of water has a volume of 18 milliliter = 0.61 fluid ounce.

So for example, if you have 1 mole of atoms, you have 6.02 x 1023 atoms.

Or, if you have 2 moles of atoms, then you have 1.204 x 1024 atoms.

In other words, to convert from moles of atoms to number of atoms, multiply by 6.02 x 1023.

If instead you have 1.8 x 1023 atoms, then to find the number of moles, just divide:

1.8 x 1023 ÷ 6.02 x 1023 = 0.299 moles.

In other words, to convert from number of atoms to moles, divide by 6.02 x 1023.

Avogadro's number is 6.02*10^23.