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What does -3.5 round to?

Discussion in 'Mathematics' started by Elfrune, Oct 10, 2016.

  1. Elfrune

    Elfrune New commenter

    Don't know if anyone can settle a debate. What does -3.5 round to, to the nearest whole number. I'm getting a lot of -3's and -4's and no-one knows quite which is right!
  2. drpallad

    drpallad New commenter

    It can be either which explains your mixed response. In the traditional round up method, as in the direction of travel on a number line, you would round up to -3. You can also apply the half round down rule and round down, that direction of travel again, to -4.
  3. CarrieV

    CarrieV Lead commenter

    youmakemesmile likes this.
  4. Apple101

    Apple101 Occasional commenter

    I would normally say -3. In essence there isn't a correct always answer because there are many ways to round depending on which system you use.

    I would say it doesn't matter as long as you do the same all the time.
    helenjbell9 likes this.
  5. coyote

    coyote New commenter

    We've had this one before, but it was a long time ago.
    It definitely matters.
    To answer this question, you need to understand why the default method is to round 3.5 to 4:
    My favourite explanation is that 0 1 2 3 4 round "down" while 5 6 7 8 9 round "up".
    Following this logic, -3.5 must round to -4 with "down" actually means towards 0 on the number line. Certainly this is the correct reasoning if a quantity might be positive or negative, depending on your modelling assumptions (happens a lot in M1 for example).
    But of course when dealing with real life data other rounding rules are sometimes needed and these can vary a lot.
    Maths_Shed likes this.
  6. Maths_Shed

    Maths_Shed Occasional commenter

    I'd agree with -4 as it's a continuation of the concept of directed numbers.
    mathsmutt likes this.
  7. armandine2

    armandine2 Established commenter

    you owe the bank £3.50 - you should round to £4 not £3
    mathsmutt likes this.
  8. colinbillett

    colinbillett Occasional commenter

    There is a convention of dealing with half-way, when it doesn't really matter, of going for the even number. So I'd say -4, but does it matter?
    mathsmutt likes this.
  9. coyote

    coyote New commenter

    This is a very interesting question. I have use OP's question - or a variation of it - as an interview question for PGCE students:
    "A student asks you what -3.5 rounds to, to the nearest whole number. How do you respond?"
    I wasn't actually interested so much in their conclusion, as in whether they could actually engage with the question and discuss it sensibly.
    mathsmutt and colinbillett like this.
  10. Apple101

    Apple101 Occasional commenter

    As a discuss question it has some use as there are several different explanations from different rounding systems depending on which one you prefer to use. I also think its kinda pointless but I can how its useful for that purpose.
  11. neddyfonk

    neddyfonk Lead commenter

    That reminds me of a famous computer programming fraud where in the payroll system the programmer rounded down all the halfpennies and put them all onto his own account. The totals for bank transfer looked ok so no one twigged.
    bevdex and colinbillett like this.
  12. adamcreen

    adamcreen Occasional commenter

    I think that was the plotline from Superman II
    bevdex and colinbillett like this.
  13. colinbillett

    colinbillett Occasional commenter

    If you invested one of those halfpennies now, by the time you got to the Restaurant at End of the Universe...
    bevdex and mathsmutt like this.
  14. ElPintor

    ElPintor New commenter

    .5 should always round up, regardless of whether the number is positive or negative. So -3.5 should round to -3. Show your pupils a number line to demonstrate the concept if they are confused. To round to -4 would be rounding down, as of course it indicates a smaller quantity or measurement. You need to go up from .5 ie to the right on your number line or towards the greater number.
  15. aleaiactaest

    aleaiactaest New commenter

    Would that logic not suggest -3.6 should round "up" to -3 when it's clearly closer to -4?
  16. Apple101

    Apple101 Occasional commenter

    As stated earlier you can either round towards the right of the number line so -3

    Towards the left -4

    Or towards zero -3

    All ways have merit you just have to be consistent
  17. crashMATHS

    crashMATHS New commenter

    If we consider this as a higher level question, then there is no real answer. Let's take the positive case of rounding 3.5; then we see that there is no 'nearest' whole number to 3.5 - 3 and 4 are equidistant from 3.5. I mean if you took a load of computer programmes and used a round function, then you'd get some that say 3 and some that 4, based on the system they use.

    Rounding itself is a poorly defined mathematical concept, as it depends on the system that you choose to use. This is why the ceiling and floor functions are defined in mathematics to eliminate this ambiguity.

    The usual method used in schools and known to most people is that 3.5 should be rounded up to 4 - and that is the system I personally use myself, i.e. if I had to round 3.5 to 1 significant figure, I'd write down 4. The general rule thus is to "round away from 0 when the tenth digit is greater than or equal to 5" and so using this system, we should define a rounding function as I have described in the picture attached.

    Using this function, -3.5 would round down to -4. This is what I believe concurs with the system adopted by exam boards.

    Though, I really should say this case makes very little difference in the broader field of maths when we consider what it means to 'round' - it's just personal choice really - and where advanced maths comes along, we use other functions to make it clearer what exactly we want to obtain.

    Attached Files:

    Last edited: Oct 24, 2016
  18. ElPintor

    ElPintor New commenter

    -3.6 would round to 4, yes, because it is clearly closer. As for the debate over the .5, it depends largely on the context. Without the context, the question is pretty difficult to answer. The convention I stated (rounding -3.5 to -3, rather than -4) is the one utilised in meteorology. Half always rounds "up", as per convention. In another context, it may round the other way (ie. down to the -4). It may appear that the rule does not follow "logically", but that is due to the "specialness" of the halfway point distinction. However, as others have stated, it is difficult to select an overall correct answer, as it really depends on what the need for the rounding is in the first place.
  19. mathsmutt

    mathsmutt Star commenter

  20. cach9801

    cach9801 New commenter

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