A million, a billion, a trillion … who cares?
I have a sneaky feeling that, for a significant proportion of the population, a big number is anything that ends in ‘illion’ and they do not appreciate the differences between them. This was nicely encapsulated in John Allen Paulos’s book Innumeracy where he quotes an American politician who said: ‘A million dollars, a billion, a trillion, whatever. It doesn’t matter as long as we do something about the problem’. A little thought about this reveals its absurdity. A million dollars would not buy a small apartment in New York so it’s unlikely to solve any significant problem in the US. Meanwhile, the entire GDP of the US is less than 20 trillion dollars so spending a trillion dollars to solve a problem is not really feasible without an economic catastrophe.
In the news, we regularly come across references to millions, billions and even trillions. But how many people truly have an intuitive feeling for what these numbers mean? This is an area where, as mathematics teachers, we can dip a toe in the water of politics and morality because, whatever your opinion on these matters, the necessity of basing your opinion on an objective appreciation of the facts is something we can contribute to the debate.
Several years ago, a friend of mine was teaching a mathematics class when he stopped and did a countdown before announcing: ‘I have now been alive for a billion seconds.’ The students were surprised to realise just how long a billion seconds is – almost 32 years. A million seconds meanwhile is merely between 11 and 12 days. A trillion seconds ago, the oldest known cave paintings were being painted by our ancestors. Depressingly, I am not that far off the 2 billion seconds mark now!
As teachers, I think we can contribute to this in various ways. From the very start, students should be encouraged to say numbers properly, eg 3058 is ‘three thousand and fifty eight’ not ‘three oh five eight’, so they begin to appreciate the importance of place value. They should also realise that adding an extra digit has a much bigger consequence than might at first be realised. There are 9 times as many n-digit numbers as there are all the numbers with less than n-digits. This is easy to demonstrate by looking at what happens when you move from 2 to 3 digits or 3 to 4 digits.
There is a great website Is that a big number? that deals with these issues in an interesting and engaging way and has examples for students of all ages. For example this link looks at the dimensions of St Paul’s Cathedral. The good news is that students from a young age find big numbers interesting and even funny. So bring them into the classroom whenever appropriate and we may well help to produce citizens who can critically analyse some of the statements we hear daily and that tend to go unchallenged.