The Highest Number

What is the highest number there is? Let's try and find out by working our way up. To save myself from having to write down a lot of digits, let me first explain the scientific notation:

• 1 × 10² = 1 × 10 × 10 = 100
• 1 × 10³ = 1 × 10 × 10 × 10 = 1000
• 1 × 10⁴ = 1 × 10 × 10 × 10 × 10 = 10000

Basically, powers of 10 can be used to define the number of zeros that follow the ‘1’. So here's some large numbers you might know: a million, a billion, a trillion, a quadrillion... These can be written respectively as 10⁶, 10⁹, 10¹² and 10¹⁵. You might knew these first few, but that list actually goes on for a while. Take for example a vigintillion, which is 10⁶³. Are there any higher?

Of course! How about the googol? It is notated as 10¹⁰⁰. That's a ‘1’ with a hundred zeros! Is this the largest number? Nah, we can put at least the centillion (10³⁰³), septuagintacentillion (10⁵¹³) and the millinillion (10³⁰⁰³) on the table.

We need to do better than this, so let's push it into crazy territory. A googolplex is 10^googol — yes, that's 10 to the power of a googol. A ‘1’ with a googol zeros. 10¹⁰¹⁰⁰. Fine! It is a 1 with 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 zeros. So as you can see I just wrote out a googol. I cannot write out a googolplex: there is not enough room in our universe to do so. See, the number of atoms in the observable universe is estimated to be around 10⁸⁰. This means that writing down a googolplex requires far more zeros than there are atoms in the universe!

The smallest definable distance is the Planck length: 10^-35 or ten undecillionth of a meter. This is many, many times smaller than an atom, even a proton. In fact, anything smaller than the Planck length has no meaning in physics. There's an estimated 10⁸² cubic meters in the observable universe. Divide that by the Plank length cubed, and you will find there are 10¹⁸⁷ Planck volumes in the universe. In comparison to a googleplex, that's still nothing. I can safely say that a googolplex is unimaginably more times larger than the total volume of the observable universe, expressed in the smallest possible unit of space!

If a googolplex is that astronomically big, could it be the highest number? No way — there are far, far higher numbers. Like Graham's Number. That number is so ridiculously large that I cannot even write it down using the scientific notation, like I did a googolplex. Even using ‘power towers’ (10 to the power of 10 to the power of 10 to the power of 10...) is no good. There would be too many levels! Yet, this number was used in a mathematical proof. Not by writing it down though. Even Graham himself does not know how many digits it has. Or what the first digit of it is. Only some of the rightmost digits be can be extrapolated. Still, if you thought the scale of the googolplex was absurd, Graham's Number is exponentially more so.

So is this it then? Hah! Not even close. You might have already realized that if I claimed this to be the highest number, you could just add 1 to it. And ta-da, you would have your new highest number. But we could add 1 to that again as well...

The truth is, there is no such thing as the highest number. Because you can always increase that number and the math would still work. It is this realization that led to the concept of infinity, or ∞.

In the mathematical sense, infinity is the highest number, yet it cannot be assigned a value. Those freaky mathematicians still manage to do calculations with it though!
Infinity has some bizarre properties too. But those are better saved for another post...