## Thursday, 28 June 2012

### What can you count to?

I was talking with my Dad a few nights ago and he mentioned a child in his school and her proclamation that she could count to 100. While worthy of a pat on the back for a 6 year old it's probably not something many adults would list in their skill set. 100 was a massive number to that little girl and the largest number she knew of at that time, but how high can I or any adult for that matter count?

To start I figured ask any adult to count as high as they can in standard dictionary numbers, nine times out of ten they'll answer "Hundred, thousand, million, billion, trillion.....ummm". Most people get stuck at a trillion even though if you know some simple shape names you can work out what comes next in the sequence quite easily. After trillion comes quadrillion, quintillion, sextillion, septillion, octillion and so on, but even these numbers are unimaginably small compared to some numbers we've actually managed to name.

Probably the most well know giant number thanks in part to a bit of promotion from Google is a Googolplex although it isn't even the biggest number, that honour falls to Graham's number which is unimaginably larger than a Googolplex. For this post though I'm going to focus on describing just how big a Googolplex is.

An easy way to comprehend just how big a Googolplex is would be to write it down. Unfortunately that just isn't physically possible, for reasons that I'll explain but before I continue it's important to know that a Googolplex can be expressed by or 10googol, basically a Googol followed by a Googol of zeros.

To begin, say we could physically write a Googolplex, how long would it take? Quite a long time, in fact a good bit longer than your life expectancy. Writing two digits per second it would take about 1.51×10^92 years to write a Googolplex which is about 1.1×10^82 times the age of the universe.

What if we were to print the digits of a googolplex in unreadable, one-point font? That wouldn't work either, it would take about 3.5×10^96 metres to write a googolplex in one-point font. The observable universe is estimated to be 8.80×10^26 meters so the distance required to write the necessary zeroes is 4.0×10^69 times as long as the estimated universe. To look at it another way, an average book of 60 cubic inches can be printed with 5.0×10^5 zeroes, that's 5 characters a word, 10 words a line, 25 lines a page and 400 pages total. The observable universe contains 6.0×10^83 cubic inches. So using that size of a book and stuffing the universe with multiple copies we could only fit 5.3×10^87 zeros, far short of a Googolplex.

In fact there are only about 2.5×10^89 elementary particles in the observable universe so even if we were to use an elementary particle to represent each digit we'd run out of particles well before reaching a Googolplex.

I thought it might be possible to write a Googolplex down in a format which doesn't take up physical space, a computer hard drive. If we used a single byte to store the digits could it be expressed in a format that could theoretically be viewed? Theoretically, yes we could eventually have a hard drive big enough to store a Googol of zeros but we aren't even close to making one this century. For example, in 2009 the estimated size of the internet was put at 500 exabytes (I'm not going to explain how big an exabyte is but all you need to know is it's very very big). If we used each byte contained in those exabytes to represent a part of a Googolplex we'd only have 5.0x10^20 zeros, nowhere near the 10^100 zeros needed.

To answer my original question, how high can I or any other person count?

It isn't a question of intelligence or even knowing the names of numbers, it's actually a question of how long you can stay alive. If we take a hypothetical man that lives to a ripe old age of 91 and assume that he began counting when he was 5, averaged 2 seconds a number (big numbers take a long time to speak) and continued until he was 90 he'd only have reached 1,340,280,000. So if you ever meet a 91 year old man that claims he has counted to 1,340,280,000 he's either wasted his entire life or he's a liar. He's probably a liar.....