The Ridiculousness of Significant Figures

In my Physics lesson today we were posed with a seemingly simple question, but it is one to which it is very difficult to provide a solution. How can we calculate the mass of the earth? Upon hearing this you would expect the problem to be solved through the use of different high tech machines. However, the true answer is much simpler, it is through a bit of simple mathematics, if the ancient greeks can do it with rulers alone i'm sure that proves that it doesn't require any expensive equipment. The way they did this was rather ingenious really, they dug pits at a known distance apart from each other along the equator, they then waited until the sun lit up the bottom of the pit (i.e. when the sun is directly above the pit at midday) and then contacted the people at the second pit and told them to start timing. They would time up until the point that the sun shone into the bottom of the second pit and stop timing. This would seem to be an insignificant piece of information, knowing how long it takes to shine light in two different pits, but actually it provides us with the key to answer the question of the mass of the earth. Let me explain how:

Let's say that the pits are 5 km apart, if it took an hour for the light to get from one pit to the other (these being completely random numbers) then we know that 5 km would be one twenty-fourth of the earth's circumference. This is because as there are 24 hours in the day and it takes one hour to spin 5 km this therefore means that in 24 hours the earth would spin with a total circumference of 5 x 24 = 120.

With this data we can then use 2zr (there is no pi sign here so I shall use z) to work out the radius of the earth. I happen to know that this value is 6.4 x 10^6 m as the values that I made up were obviously not going to be correct. 

Now that we have the radius it is relatively simple (ok, it's not that simple) to work out the mass, to do so we must rearrange the equation F = (G x M1 x M2) / r^2 which is Newton's Universal Law of Gravitation. G being the universal gravitation constant, M1 being the mass of obejct 1, M2 being... well you can guess what it is, r being the distance from the centre of one object to the centre of the other and F being the force between them. In order to gain all of the necessary data to calculate the mass I must first assume that I am calculating the force between me and the centre of the earth (therefore providing the two seperate masses), weighing 64 kg myself this provides M2, F is 64 x 9.81 = 628N, G is 6.67 x 10^-11N m^2 kg^-2 and r being 6.4 x 10^6m. Now that I have all the necessary data I have to rearrange Newton's law to give me M1 = Fr^2 / GM2 ... M1 = 628 x (6.4x10^6)^2 / (6.67 x 10^-11) x 64 = 6.03 x 10^24 kg. So finally I have the mass of the earth! It is not the precise mass but it is within 10% of the true value, however, due to the number of significant figures it is impossible to comprehend how heavy that would be, we know how much 10 kg is even 100 kg but when you get into times ten to the power of twenty four it is somewhat harder to imagine this mass. It would be like saying to someone how much is that television? 50000 pennies please.