In a previous post I argued that it is hard to picture a billion, or a billionth. Because a billion is a pretty big number. However, in science we need to deal all the time with numbers much bigger than that, even if they aren’t easy to visualise. The fact that they aren’t easy to think about makes estimating quantities really difficult, even to within an order of magnitude.
Big numbers in science
For example, if you were asked how many stars are in the observable universe, it would be quite hard to get close to 70 billion trillion (source skyandtelescope.com) unless you had some prior knowledge. That number is 70 000 000 000 000 000 000 000. Or 7 x 1022 in standard form (which we previously joked should be used compulsorily). Now 7 x 1022 is 70 trillion times bigger than a billion, the number I couldn’t visualise in the first place!
Carl Sagan said that there are more stars in the sky than grains of sand in all the beaches in all the world. Is that true? Well, if a grain of sand has a volume of 1 mm3 (which would be quite course sand, and will almost certainly lead to an underestimate of sand grain numbers), then 1 billion grains of sand is about a cubic metre of sand. Or approximately a jumbo bag that you buy from the building merchant and have delivered by a lorry (truck). Now scale up the jumbo bag to fill all the beaches in the world. That’s a lot of bags. And even more grains. Jason Marshall the Math dude estimates the number of grains to be 5 x 1021. His process of estimating is persuasive, but you could easily follow similar reasoning and come to a different answer. That’s not a criticism of him, it’s a statement of how even well-arrived-at estimates can vary wildly when the numbers in question are so large.
So it is true, it seems that there are more stars in the sky than grains of sand. And both numbers are fairly similar to the number of molecules of water in a single drop. Just think about that for a second…
Big numbers in mathematics
And yet the numbers we have looked at are nothing compared to the size of numbers dealt with in maths. There is even a Wikipedia page called “Large Numbers”. Let’s start with a googol, which is not the worst estimate for the number of particles in the universe. It’s not that accurate, I’m sure, but will do for these purposes. What is a googol. Well if we were to turn every word in this paragraph into a zero and precede them with a ‘one’, we have a googol. There are a hundred words in this paragraph, so a googol is:
10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
It’s quite an inconvenient number to write like this, but very easy in powers of ten: [latex]10^{100}[/latex].
Look, we did it with five digits!
And what’s a googolplex? A googleplex is ten the the power of a google. It’s easier to write than it is to visualise: [latex]10^{10^{100}}[/latex]. It doesn’t look that much bigger than a googol, does it? After all, we just wrote it with only seven digits. But to write it out in full? For a googol, we transformed every word in a paragraph into a zero. For a googolplex, we would have to transform every particle in the universe into a zero.
A googol was invonvenient to write out in full, but easy to write in power notation (we just needed 5 digits). A googolplex was impossible to write out in full (unless all the particles in the universe turn into zeros) but still easy to write out in power notation, using just seven digits. A googolplex is much larger than the numbers scientists deal with on a daily basis. However, there is a number called Graham’s number (not the number in your phone under ‘Graham’) that is so big, that even in power notation, not only would it require rather more than the seven digits for a googolplex, but the digits required wouldn’t fit in the observable universe. It can only be written down by inventing a new notation for writing numbers.
I’m trusting Wikipedia for that statement, I really have no idea how big Graham’s number is… I’m never going to need it. After all, I struggle with a billion…