Consider this: you have a bag of sweets containing 10 whole sweets but lots of shrapnel in the bottom of the bag. The shrapnel could make up 3/4 of a sweet yet you'd still say you have 10 sweets. Then in this case, are you not rounding down even though you have more than .5? I don't think you can "prove" which way to round, people just do it to simplify a number, it's not actually a mathematical theory (or somthing).

That's an interesting point. Most of our arguments so far have been theoretical, but in practical situations it is usually obvious which method to use.

If rickyoswaldiow's sweety shrapnel could be put together to make 2 whole sweets, it would still only count as 10. So, the rule there is to ignore any fractions.

Another example would be that if a factory that sells cans decides to combine them into 6-packs, you would divide by 6 and round down, even if you had 5 spare.

Conversely, if a bus can hold 25 people then to work out how many buses you need to hold a certain amount of people, you would have to round up all the time, even if the last bus will only have 1 person on it.

But for the theoretical side of it, we should just say that 0.5 rounds up to 1 because it is convention and if you try anything else it will be seen as wrong, even if you don't believe it is. The end.

]]>So, apply the rounding method to suit the data.

]]><√x>=[√([√x]+x)], where [x] is floor[x].

Cool!

]]>Taka a random bill. Estimate it by rounding off the cents (or pence, or centavos or whatever).

Example: 3.45, 12.07, 6.68, ...

Rounded: 3,12,7 ...What method will have the least error? (There are 100 possible cent values from 00 to 99)

That depends on the prices. Most prices tend to be biased towards the high end to make people think things are cheaper than they are. eg, £3.99 etc.

So, that would mean that rounding would give a higher value a than the actual one, so to compensate it would be better to round £3.50 downwards. That doesn't prove anything though.

]]>If a number is .5 we just don't round it.]]>

Example: 3.45, 12.07, 6.68, ...

Rounded: 3,12,7 ...

What method will have the least error? (There are 100 possible cent values from 00 to 99)

]]>And in my mathematica help i read something very strange:

It rounds .5 to nearest even integer!!!]]>

If you think that 4.0 rounds down to 4, then it is also true that 5.0 rounds up to 5. So you would have:

4.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 5.0

4.0, .1, .2, .3, and .4 round down

4.5, .6, .7, .8, .9, and 5.0 round up

So there is still 5 numbers that round down, and 6 numbers that round up.

]]><m.00000> = m, so <m.50000> = m+1. I round it up.]]>

If it's student mark and you are good, you'll round it up.]]>