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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