# What does -3.5 round to?

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

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1. ### coyoteNew commenter

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It seems to me that the answer shouldn't depend upon the arbitrary position of the origin.

3.5 rounds up to 4.

Subtract 7, and it becomes -3.5 ... and it should still round up to 4 - 7, which is -3.

I'm not too taken with the idea of rounding "to the left" or "to the right" on a number line. When you're dealing with negatives I far prefer to use the vertical (y-axis) number line rather than the horizontal (x-axis) number line.

Work on the y-axis and the meaning of the proposition "any .5 rounds up" is plain to see. And it means that -3.5 rounds "up" to -3

3. ### BillyBobJoeEstablished commenter

I have to say it would never have occurred to me to round to anything other than -4. It seems to so obviously follow from the normal rules for rounding (5,6,7,8,9 round to the same thing, 0,1,2,3,4 to something else). To do otherwise breaks the algorithm of rounding-by-checking-the-next-digit which is a very useful shorthand. If we're going to have an arbitrary convention of this type it makes sense that it should be both consistent AND easy to apply.

But BBJ ... rounding to -3 IS the result of a convention which is both consistent and easy to apply.

The issue is between two equally consistent and equally easy to apply conventions.

One is "round .5 to the next integer which is larger than it"; the other is "round .5 to the next integer which has a greater magnitude".

The confusion arises because we learn about rounding at a time when we only ever work with positive numbers, and the distinction between "the next integer which is larger" and "the next integer which was a greater magnitude" is a distinction difference, because 4 > 3. We therefore tend to be sloppy and insufficiently precise as to which of the two formulations is the actual convention we are applying. Our chickens then come home to roost when we start working with negative numbers as the distinction DOES now have a difference, since -3 > -4

5. ### BillyBobJoeEstablished commenter

Well no, it's not consistent. Rounding -3.5 to -3 means that you're grouping 5 with 0,1,2,3 and 4, rather than with 6,7,8,9 as you do with positive numbers. That means the process requires more thought and, as other have mentioned, means that in applied contexts you get different answers depending on which direction you take to be positive.

coyote likes this.

0 doesn't need rounding. There are 9 digits which may need rounding: 1, 2, 3, 4, 5, 6, 7, 8 and 9.

5 needs to be grouped with EITHER 1, 2, 3 and 4 OR with 6, 7, 8 and 9.

You cannot escape the result that whichever group you put it with will have 5 digits to the other group's 4.

What you have described is not an inconsistency; and I do not see that the additional thought required in applied contexts is any different or more onerous from the additional thought required in applied contexts to address the point that 3.5 may round to 4, but 3.5 - 7 does not round to 4 - 7.

The choice is between "rounding away from zero" (your preferred solution) or "rounding to the next larger number" (my preferred solution). It is just that, however - a matter of preference.

But if you are going to go with "rounding away from zero" then PLEASE do no call it "rounding UP", as that is bond to cause confusion on the part of those who struggle with the concept that -3 is a larger number than -4.

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7. ### BillyBobJoeEstablished commenter

Such confusion would be good, seeing as -3 is clearly a smaller number than -4. It is also a higher number. Numbers get larger the further away from zero they are. This is a consistent concept across natural numbers, integers, real numbers and complex numbers.

8. ### Apple101Occasional commenter

This is true but students need to be careful as most will think -4 is smaller than -2. If it was money in a bank, they would consider less to be smaller. You are right but we must be mindful

9. ### rachel_g41Established commenter

I worked for the Met Office for a while. They (and I think it's a general convention in meteorology) round to the odd - so both 2.5 and 3.5, whether positive or negative, would round to 3.

It makes odd numbers more likely as max/min temperatures, if you're ever thinking of betting on such a thing.

10. ### PiranhaStar commenter

Probably not very important, but if you round -3.5 to -4, then the actions of rounding and taking the modulus can be done in any order without affecting the result. My preference would be for -4 anyway. The convention of looking at the next digit only gives this result. I also think that -1 is a better approximation for -0.5 than 0 is. However, whatever we do is just a convention, and I have not seen a definitive ruling. It is worth remembering that, with continuous data, the probability of a result being exactly -3.5 is zero.

11. ### CerviniaOccasional commenter

Round 3.04 to the nearest whole number.

Sure. 3. But it's the 4 in the 10s to the minus 2 column that I just rounded off, NOT the 0 in the 10s to the minus 1 column. If those 4 hundredths hadn't been there to begin with, I would have just had 3.0 which doesn't ROUND TO 3 ... it IS 3

13. ### Waterloo97

Hello all,

Officially wading in in an attempt to settle this debate because some of the reasoning that I am seeing is not, I have to say making a great deal of sense.

-3.5 rounds to -4. To say that it rounds to -3 as this is a corresponding direction on a number line to rounding 3.5 to 4 creates further implications for other numbers which are clearly nonsense. If we insist on negative rounding occuring in the same direction on a number line as positive rounding, then -3.8 would round to the right to give -3 and -3.2 would round to the left and become -4. Clearly -3.8 should round to -4 because it is moving in the same direction away from zero as 3.8 does when rounded to 4. While it seems to be rounding to a 'smaller' number it is in fact rounding to a number with greater magnitude.

In short, round the modulus of the number and then reapply the positive or negative sign.

This is not so. The principle of rounding to the nearest integer takes precedence, so that -3.8 will always round to -4 and -3.2 will always round to -3.

The discussion is about how to round those negative numbers which are exactly equidistant from the integers either side (i.e. the exact .5s), and three approaches have been suggested:

(1) rounding "up" according to the modulus (i.e. away from zero whether the number is positive or negative)

(2) rounding "up" according to the magnitude (i.e. always in the same direction on the number line whether the number is positive or negative)

(3) rounding to the odd number

For completeness' sake, even though nobody has mentioned it, one could equally have a convention of rounding to the even number.

At the end of the day it is a matter of convention, not of any immutable mathematical inevitability - as has already been pointed out.

15. ### BillyBobJoeEstablished commenter

Except that magnitude and modulus, in the context of numbers, are the same thing, so your (1) and (2) are synonymous rather than juxtaposed. It's not a matter of inevitability, but it is a matter of consistency and having conventions that follow logically from each other rather than requiring different rules for different situations.

Um ... no. Statements (1) and (2) are not synonymous at all. If you want to quibble about choice of words, then please suggest another word besides magnitude to express the concept that -3 > -4 and I will happily use that in statement (2).

And, I repeat for the umpteenth time, there is no more lack of consistency or requirement for "different rules for different situations" in conventions (2) and (3) than there is in convention (1).

Convention (1) requires that a positive 0.5 be rounded up to a larger number, and that a negative 0.5 be rounded down to a smaller number. Expressed that way, it is inconsistent, and requires different rules for different situations. But if you express it as "round away from 0" then it is consistent and does not require different rules for different situations. However, once you express it this way it is difficult to find any logical requirement for this convention.

You can play the same games with both conventions (2) and (3). Express them one way and they provide a consistent rule which does not "require different rules for different situations"; but re-express them in a different way, and they can be made to appear inconsistent, or to require different rules for different situations.

Let me put it this way. Your rule of rounding according to the modulus can be re-expressed as "ignore the positive or negative sign at the beginning of the number; pretend it's positive even if it's not; round it up; then stick the positive or negative sign back in front of it". The obvious question which this approach prompts is "but WHY should I start by ignoring the negative sign at the front of -3.5?" There are a number of answers you can give to this, but none of them boils down to "because logic demands it". All of them, rather, boil down to "because the convention I have chosen to adopt demands it".

17. ### BillyBobJoeEstablished commenter

No, it's that it works regardless of the convention (e.g. for direction) that you're using. A rule that results in different answers to the same suvat question depending on the direction taken as positive is just plain stupid. In a great many situations positive and negative are reversible, and it's logical for rounding to be consistent under such a reversal.

"Just plain stupid" is strong language, Billybobjoe; and unwarranted in this case. The fallacy in your logic lies in the leap from "this is true for a great many situations" to the proposition that it is therefore logical to have a rule which proceeds upon the basis that it is true in all situations. That is not a logical progression at all.

The convention which I favour can be expressed by a single universal rule which is true for ALL integer values of x, including zero, as follows:

"The set of number values which round to the integer x is defined by the inequality (x - 0.5 ) < or = x < (x + 0.5)"

[Apologies for my inability to produce the conventional < or = symbol ... ]

I struggle to see why a rule which can be so clearly and succinctly expressed in a way which will be universally true should be "just plain stupid". Indeed, I think you would struggle even to justify the epithets "plain stupid", "just stupid", or indeed "stupid" simpliciter. "Plain", on the other hand, seems appropriate. It is a very plain, clear rule, which is easy to express, easy to understand, easy to justify and easy to apply.

The alternative convention which you favour needs to be expressed thus: "the set of values which round to the positive integer x is defined by the inequality (x - 0.5) < or = x < (x + 0.5); the set of values which round to the negative integer x is defined by the inequality (x - 0.5) < x < or = (x + 0.5); and the set of values which round to the integer x where x = 0 is defined by the inequality (x - 0.5) < x < (x + 0.5)"

I may be doing you a slight disservice here in that I suspect it is possible to re-express the first two parts of your rule as a single inequality using modulus notation - and do please feel free to supply that inequality if you wish. The point remains, however, that no matter how you express your rule in terms of inequalities, it is going to have an exception for the set of numbers which round to zero; whereas my rule is a single rule of universal application to all integers including.

Now, I do not say that it is "just plain stupid" to prefer a convention which can only be expressed by a rule which needs to make an exception for zero over a convention which can be expressed in terms of a single rule of universal application; but I do venture to suggest that you are, perhaps, overstepping the mark just a teensy weensy bit in declaring it "just plain stupid" to prefer a convention which can be expressed in terms of a single rule of universal application over a convention which can only be expressed in terms of a rule in which it is necessary to make an exception for zero.

19. ### SpiritwalkernessStar commenter

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