Lying awake with a (for me) rare bout of insomnia the other night, I started to think about math formulae (normally a good way to induce unconsciousness). However, I got to the formula for an area of circle, which is of course pi times the square of the radius, and had a thought. We know that pi is an infinite decimal, so any calculation made using pi must by definition be inaccurate. Okay, if you are using pi to 20 decimal places this will be by a miniscule amount, but nonetheless it is still inaccurate. Dumb question - does this mean that in theory there could be a more accurate method of calculating the area of a circle and we just haven't figured it out yet, or are we 'doomed' to be 'inaccurate' by using pi? And talking of infinity - are all infinite groups equal? All whole numbers form an infinite set. But there again, so are all possible pairs of whole numbers. In e.g. a finite set of 1000 numbers, there are a lot more possible pairings of whole numbers (1+2, 1+3, 1+4 ... 999+1000) than there are single numbers (1,2,3 ... 1000). Of course all infinite sets are by definition infinite, but there is a strong intuitive sense that the set of e.g. all possible pairs of whole numbers is a lot bigger than the single numbers. Please does anyone know a good (non-technical) source that can explain this?