A while ago L asked me what the biggest number I knew was. There are times when I start to regret that I gave her a semi-truthful answer. The mathematically accurate answer is, of course, that there is no biggest number, but this didnt satisfy L. So instead I told her that the biggest number I knew was a googol. (The reason that this is only semi-true is that I actually also know what a googolplex is, but that is too big a number for me to contemplate for more than an instant). A googol, for those who dont know, is 10 to the power 100 (written as 1 followed by a hundred zeros). Or, to put it differently, one hundred thousand trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion.

A googol is, therefore, a really big number. I sort of know this, but I still get flummoxed when I think about it. And I think about googols rather more than I used to, because L has taken to using the term. She, of course, is as the stage where big number encompasses anything from about 50 upwards. She is also currently very keen on hypothetical questions. So just yesterday I got what happens if we saw a googol of butterflies and every one was a different colour? And also (since we were going to feed the ducks): what if there were a googol of ducklings in the river?

I could provide a roughly coherent answer to both of these. Firstly, a lot of the butterflies would look the same colour, because there arent that many different colours. And a googol of ducklings wouldnt fit in the river or even Hitchin. But afterwards, of course, I got to thinking about how many butterflies would be the same colour and how much space a googol of ducklings would take up. And that way lies madness (and mathematics, which tends towards the same thing). After some calculation, I came up with some very approximate answers.

Truecolor graphics gives about 16 million (= 1.6×10^{7}) different colours and approaches the limit at which the eye can detect differences. Therefore youd be left with around 10^{92} butterflies of each minutely different colour. Which is a lot of butterflies.

As for the ducklings. Assuming that you have small ducklings, they might have a footprint of around 5 cm x 5 cm. Since ducklings are quite happy squashing up together, you could then get 400 to a square metre. A square kilometre is a million square metres, so you could get 4×10^{8} ducklings into that area. Then it starts to get hairy (or feathery). The earths radius is around 7000 km, so its surface area (4 pi x(radiius squared)) is about 6×10^{8} square kilometres. If you covered the earths surface in ducklings, youd therefore have around 2×10^{17} ducklings. For a whole googol of ducklings, youd therefore need 5×10^{82} Earths (which is five hundred million, trillion, trillion, trillion, trillion, trillion, trillion, trillion).

At this point my mind goes woozy and I realise that I should have told L the biggest number I know is a trillion. All I hope is that she never asks me how many numbers there are between 0 and 1…

Just pray she never hears about Graham’s Number…

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There’s an old answer to this, from Milo and the Phantom Tollbooth which I do heartily recommend for curious children. The largest number you know is actually that number (in this case a googol)… plus one. And then one more, and one more… And so on till you run out of zeros in your notation. And this is maybe the only useful way to approach the concept of infinity?

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I’ve tried the ‘there’s always a number one bigger’, but L’s not yet entirely convinced by that one. And I still have to work out whether to tell her my biggest mathematical secret when she’s older: that there is more than one size of infinity.

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