When I was 3, 99 was the biggest number I knew of. One day I was reciting the numbers at the playground (?!) and stopped at 99 with an air of finality. This other dude said, “…aaand hundred!”, and then continued with “Hundred one, hundred two,…”, etc. This felt weird. He was wandering in uncharted territory, yet the numbers could be constructed with what I already knew. All I needed was the magic word ‘hundret’ or whatever he said! After ‘One Hundred and Ninety Nine…”, I was expecting him to stop, but this wizard went on with “…Two Hundred, Two Hundred and One,…”, etc. This felt weirder. Not immediately, but sometime later when I was ruminating over multiplying and adding numbers I already knew with ‘Hundred’. To have a number as big as Ninety Nine Hundred and Ninety-Nine was breathtaking.
Now, this is not entirely true. Some of it is (like the counting in playground part). Rest is fiction that’s fabricated to serve as a good introduction to…wait for it…Googology – The Study of Big Numbers! If you think we’ll end this article with giants like Googleplex or Graham’s Number, oh boy did you underestimate the size of significant large numbers. Things are gonna get pretty big, pretty soon.
Starting from the Bottom
0 is the number with the smallest absolute value. Next natural number is 1, which by the way, is infinite times bigger than 0. In between the two of these, the additive identity and the multiplicative identity respectively, are some cool really small ones.
Take Googolplexianminex for example, which is 0.000…0001 with (googolduplex-1) zeroes after the decimal. Googolduplex is like googolplex which is 1 followed by googol zeroes, but has 1 followed by googolplex zeroes, and googol is our plain old friend with one followed by a hundred zeroes.
To get an idea of how small this number is, let’s try to get an idea of a googol. Googol isn’t hard to write. If you google googol, google displays googol immediately in the first page itself. Let’s say inflation-adjusted richest man ever, 400 Billion Dollars worth Mansa Musa, donates each of his bucks to one human being. Although we’re going beyond the scope of reproductive potential of our species with this example, we’ll pretend that’s not a problem, for now. Each human being now hands over all the micro-organisms inhabiting their gut microbiome to King Musa. This way he’ll go broke but have 100 Trillion times 400 Hundred Billion microbe to work with.
Remember how Googol had 1 followed 100 zeroes? Musa now has 4 followed by 24 zeroes number of microbes. If each of these travel to the edge of the Universe and back and then repeat that every Second as many times as the age of Universe (in seconds) and manage to add up their odometers, we’ll be looking at (4×10^24)x(~8×10^26)x(~4×10^17) metres. A hefty ~10^72 metres. That’s a lot of travel, even though we’re technically breaking the barrier of speed of light every second!
Mind You, 10^72 is not 72% of Googol, but rather 0.000…0001% with 25 zeroes. Huh, we left on a mission to study big numbers. We’re still in between 0 and 1, trying to make sense of Googolplexianminex. In the process we encountered Googolduplex, then Googolplex, then Googol and after epic multiplications, we’re still at (10^(-25))% of it. Therefore, I believe this wouldn’t be a nice time to introduce you to a Googoltriplex or, as you might reason out, a number with 1 followed by a googolduplex of zeroes!
How to grow?
This particular question gets thrown around a lot over the internet, mostly in banner ads in shady sites. But, we’re asking a specific question. How to grow a number? Rephrasing, what are some functions that yield large numbers?
If you count from 1 to 10, with each number a second, it takes you 10 seconds, duh. Counting to hundred takes 100 seconds or about 2 minutes, although in reality it would take more as bigger numbers contain more syllables. In a utopian world, however, counting to 1000 will take a bit more than quarter of an hour with one number a second. A million shall take about one and half week. A Billion will take 31 years and a Trillion will..well, it’ll take 317 centuries, but I’m having a gut feeling that you’ll be facing a time constraint here.
So, now let’s start counting every millisecond. Or what if we go down to micro, or nano, or pico, or femto, or atto, or zepto, or yocto? Yeah, counting in yoctoseconds sounds cool, because this way you’ll be able to count a Trillion in a picosecond! A quadrillion in a nanosecond. A quintillion in a microsecond, sextillion in millis, septillions in seconds and then skipping over octillions, nonillions, decillions and settling with an undecillion in 317 centuries. Not a bad deal if you ask me.
This way of counting one number after the other is very linear. Even when we try to jump from a number a sec to a number in a millisec, it’s still exponential at best. In the field of googology, exponentiation is being cheap. We need tetrations and pentations and only then will our life be merry.
Segue to Graham’s Number
You repeat addition to multiply, repeat multiplication to exponentiate, so we’ll repeat exponentiation to tetrate. Exponentials grow quickly, but tetrations are even quicker. Knuth’s arrow notation comes in particularly handy at this moment. A arrow B is defined as simple exponentiation. A double-arrow B is therefore tetration. It means, A raised to A B number of times or A^A^A^…^A^A^A, B times! Triple arrow starts to make us go crazy. 2 arrow 4 is just 16, 2 double arrow 4 becomes a tower of 2s with 4 2s in tower or 2^2^2^2 which is, 65536. 2 triple arrow 4 is just a tower of 2 with 2 double arrow 2s. That is, 2^2^2…^2^2^2 with 65,536 times!
Playing the same game with 3s, we get 3 arrow 3 as 27, 3 double arrow 3 as 9.6 Trillion and 3 triple arrow 3 as well, it’s big. But we don’t care, let’s instead name 3 quadruple arrow 3 as g1. Then g2 becomes 3 arrow arrow … arrow arrow 3 with g1 arrows, g3 becomes 3 arrow arrow … arrow arrow 3 with g2 arrows and so on. Finally, when we reach, g64 or G, we stop and call it the Graham’s Number.
Why though? The reason Graham’s number is famous and not Graham’s number plus one or something is because this particular number solves a problem in combinatorics. The universe is considered to have a volume of ~10^80 metre cubes and the smallest possible unit volume, plank volume equals ~10^(-105). Fitting one digit in each in each plank volume of the Universe does no justice to Graham’s number. In fact, fitting googolduplex digits to each plank volume of googoltriplex universes (multiverse?) still falls miraculously short of the goddamn Graham’s number.
The only way to make yourself forget about Graham’s number and calm down is by going crazy with TREE(3). Which is so big, so so big, so so so big, so so so … so so big (with Graham’s Number of “so”s), that, well, nothing actually, its really very big. It props up from a fast-growing function in graph theory. Replace that 3 with 4 or Graham’s Number and you get an even bigger number. Thinking about this gives me a similar feeling to that time in childhood when I was trying to comprehend ninety-nine hundred ninety-nine, even though, I’m not quite sure whether that really happened.
Till now, we were in the domain of computable numbers. This could have been written down like 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 (googol), if we had enough space. But beyond these exist the domain of uncomputable. These are basically functions which churn out humongous numbers.
Busy beaver numbers are a great entry point to this topic. At this time, we are no longer hoping to define numbers as the tag uncomputable mean that these are you know, uncomputable. Some other examples include the largest named numbers in professional mathematics, the Rayo Numbers. Others intruding into the list are BIG FOOT, Oblivion, Sam’s Number and finally, infinity (although it’s better to treat infinity as a concept rather than numbers).
We discussed a lot, but all these numbers were exactly 0% of infinity! Therefore, the title wasn’t a clickbait. Cheers!