Wednesday, 25 June 2014

Why does 5 round up? – The maths of rounding and approximating

I was taught from a young age, as I’m sure you were too, that 5 and numbers greater than 5 round up, whilst numbers smaller than 5 round down. But why?

The essence of rounding numbers is to get an accurate, long value such as 6.72 and make it an easy, simple value like 7. In order for the final number to be as close as possible to the original, whilst still being simple, you must find the nearest ‘nice’ (which depends on the context) number. The important word here is nearest.

This means that if you were rounding 6.72 you would chose 7 as that is closer to 6.72 than 6. In the same way, 6.42 would round down to 6, as 6.42 is close to 6 than it is to 7. It follows that 6.49 would round down while 6.51 would round up, 6.499 down and 6.501 up, 6.4999 down and 6.5001 up. The division is clearly going to be at 6.5, but that is equidistant from both 6 and 7 so which way does 6.5 go?

The answer lies purely in convention, from a purely mathematical perspective it should do neither, but then again rounding isn’t a purely mathematical concept. There are many different methods of sorting this, from the odd/even rule to the 5 always rounds up to the view that nothing in real life would give a value of exactly five. It also depends on context, however, for example if the maximum voltage for a circuit is 4.5V you should round that down to 4V rather than face the danger of going above the exact value. On the other hand, if you need a minimum fuse of 3.5A you should round that up to 4A in order for it not to fuse when a normal current goes through it (I admit my knowledge of electronics is limited). All in all, the general convention is that 5 rounds up, albeit with little mathematical basis.

Rounding also leads to other interesting fallacies and mistakes. Recently someone asked me whether 4.46 would round up or down. Immediately I said 4, with my opinion that 4.46 is closer to 4 than 5. However, their argument was that 4.46 would round up to one decimal place to give 4.5 and that would then round up (as we have decided that convention dictates 5 rounds up) to 5. This shows the issue with rounding a rounded number, which can cause numerous differences such as in this case the large difference between 4 (a lovely, even, perfect square) and 5 (a nasty, odd, prime number). In these situations, it is important to go back to the trick of, is it closer to 4 or 5?
Another fallacy caused by rounding is the practice of performing calculations on a rounded number which is perfectly shown in this classic joke:
Museum goer: How old is this dinosaur?
Tour guide: 70 million years and 2 weeks
Museum goer (shocked): How do you know?
Tour guide: When I first started working here the manager told me the dinosaur was 70 million years old. And I’ve been working here for 2 weeks.

This shows the issues with false precision, which in other words is taking a rounded number literally.

In conclusion, although rounding is essential in the real world, it is easy to make mistakes and important to question the essence behind the mathematics of rounding.

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