Why do we generally round 5's up instead of down?

As an example, the number 15, rounded to the nearest tens, rounds to 20. I understand it's arbitrary, as 10 and 20 are equidistant from 15, I just wonder if there's any discernible logic behind the convention of rounding up. Even something like, 'It just feels more natural', would probably satisfy me. Just had to ask :)

• It just feels more natural :) Mar 5 '13 at 21:40
• Because we are generous. Mar 5 '13 at 21:41
• I guess cause 10, 11, 12,13,14 are rounded to 10, that are 5 numbers, and 15, 16, 17, 18, 19 are rounded to 20. Which are even 5 numbers Mar 5 '13 at 21:50
• For more information on rounding methods I recommend volume 2 of Donald Knuth's The Art Of Computer Programming, where section 4.2 goes into some detail on rounding methods and how important they can be to managing error propagation. Mar 5 '13 at 22:16
• Among other reasons, it makes matters a bit simpler. If I want to round $1.5002$ to the nearest integer, I only have to look at the tenths place. If $5$'s rounded down, I'd have to look as far as the ten-thousandths place to make sure my number was strictly closer to $2$ than to $1$. Mar 5 '13 at 22:25

WE do this because when you round, generally it is with decimals that are rational or long. When there is 5.000000000000000001, it is better to round up. So, lets say you have the square root of 26 (5.09901951359) It's close to 6 than to 4. When rounding, you can't have 5.090919534243123 whatever to round down because decimals cant go down. What Brett Frankel said: Among other reasons, it makes matters a bit simpler. If I want to round 1.5002 to the nearest integer, I only have to look at the tenths place. If 5's rounded down, I'd have to look as far as the ten-thousandths place to make sure my number was strictly closer to 2 than to 1. Normally, this is only seen in higher level math though, where there ARE no straight integers and instead many irrational numbers. To round pi to the nearest thousandth 3.142 In reality, pi is 3.1415925635, after .5 there are many numbers that increase the value to higher than five, inherently making it closer to 10.

• +1 for pointing out a practical reason that isn't arbitrary. Apr 26 '17 at 9:08
• See this Q&A Jun 20 '21 at 22:31

For small numbers like $15$, it may feel closer to $20$. We have a certain (vague) logarithmic appreciation of numbers, so $15$ feels farther from $10$ than from $20$. As the numbers get larger, this becomes less important. Even $25$ doesn't feel to me much closer to $30$ than $20$. But you have to do something. Sometimes you round to evens, which has the advantage of not accumulating errors if you add up a lot of them.

• I think you're onto something with that. A logarithmic sense of scale. Supposedly our sense of hearing operates on some kind of logarithmic scale as well.
– ivan
Mar 5 '13 at 21:55
• Douglas Hofstadter had a nice Scientific American article on that. He suggested that one should judge numbers on magnitude up to something around 10,000 (and I would argue differences between numbers under 20 are even larger) then by logs up to $10^12$ or so, then by log of log, and so on. So if you see a pile of 800 of something and guess it is 1000, that is as good as looking at a pile of a billion and thinking it is 100 billion... Mar 5 '13 at 21:59

It isn't always. A popular rounding method called banker's rounding rounds 15 to 20 but 45 to 40.

But one reason it might be rounded that way is that round(x) is often implemented as $\lfloor x+1/2\rfloor.$

• In the case you would implement it as $[x+1/2]$, I would rather round it down, seems more natural to go to take the part that already suits your needs Mar 5 '13 at 21:49
• Wouldn't that (the implementation) be confusing cause for effect? That banker's rounding is a cool idea. Now I'm going to have to read the whole wikipedia page on Rounding...
– ivan
Mar 5 '13 at 21:50
• @ivan: I don't think so -- I think that the convenience of that implementation leads people to use it even when they have other choices. I know I've used it for that reason before. Mar 5 '13 at 21:51

Wikipedia does have a nice article on this problem. Hope it helps in any way.

• Yep, I just ended up there :)
– ivan
Mar 5 '13 at 21:52

There is actually a practical reason for the rule that we round up rather than down when we are exactly at the mid point, which counters the argument one usually hears that it is an arbitrary choice. This is provided in Bradlee's answer but I'm posting one of my own to expand on it.

Take the the following three numbers that we want to round to 1 decimal place:

0.149
0.150
0.151

In the case of 0.149 we only ever need to look at the hundredths place. It makes no difference what appears to the right of that because the number will always be smaller than 0.15, so we know from looking at that place alone that we will round down to 0.1.

With 0.150 and 0.151, we look at the hundredths place and see we have a 5. There are two possibilities for what might appear in the places to the right: all zeroes, or a non-zero digit. If it's all zeroes, we round up to 0.2 because of the rule. If there is a non-zero digit, then we round up, because the number is greater than 0.15. Either way we round up, which means that we don't actually need to bother looking at any places to the right of the hundredths place to determine our action.

What if the rule was that we round down? In the case of 0.149, we still only need to look at the hundredths place for the same reason as above. With 0.150 and 0.151 though, looking at the hundredths place isn't good enough. If the places to the right of it are all zeroes then we will have to round down to 0.1 because of the rule, but if there is a non-zero digit then we will have to round up because it is greater than 0.15. Because there are two possibilities, we have to actually check.

With a rounding up rule we have one procedure that works for all cases: look at the place to the right of the one you are rounding to. With a rounding down rule we need a special procedure when we find a 5 in that place: check each place to the right until you either reach the end of the number or find a non-zero.

Note that this logic doesn't apply to negative numbers however unless you invert the rule (i.e. round down for the digit 5 in a negative number).

I guess, the convention is , for $.5$, you move to even one.

Here is my thought, but it's just a theory.

In the case of 15 rounding up or down to 10...

15 is 50% more than 10...

...but 15 is 25% less than 20.

While they're equidistant in value, 15 is more relative to 20 than it is to 10.

5/20 has less value than 5/10.

• These ideas seem to be contained already in Ross Millikan's answer. Apr 20 '15 at 13:16

Rounding numbers make more sense with decimal. So a 3.14 can be 3.10 and 3.16 can be considered 3.20, That is just done to have more easier calculations. Considering 15 as 20 may have a adverse effect on your calculations.

Shall 15 be rounded to 20 or 10?

Depends on where 15 is

A) Just 15. Treat as is.

B) 46.4215. Can be treated as 46.4220

C) 9.5. Treat as is.

Many textbooks will say 20. What I personally follow is I start looking at the next place following 15. For eg.

if the number is 3.1567, that becomes 3.157 or 3.16 or 3.20. But if it is just 2 decimal (3.15), the easier approach is to "upgrade" because in my 5th grade teachers own words "...you already climbed half the steps of the staircase, easier would be to upgrade..."

PS: He still teaches and I am an engineer now. :)

I think of 10 intervals in the decimal number system:

$$0-1, 1-2, 2-3, 3-4, 4-5,$$ $$5-6, 6-7, 7-8, 8-9, 9-10.$$

For example, why 1.5 should be rounded up to 2, not down to 1?

Because it is in the 6th interval:

$$[1-1.1), [1.1-1.2), [1.2-1.3), [1.3-1.4), [1.4-1.5),$$

$$[1.5-1.6), [1.6-1.7), [1.7-1.8), [1.8-1.9), [1.9-2).$$

Note: In various contexts, the criteria can be defined to round down or up regardless of the closeness. Say, 1.2 chairs are broken imply 2 chairs are broken!

Another reason could be that if you only look at the first $$k$$ decimals after the point and it ends with $$5$$, then the most likely outcome is that what goes to the right isn't all zeros.

I randomly realized that 1.5 is no further away from 1 than it is from 2 just now and was compelled to ask the same question. After reading some of the answers, it occurred to me that the convention, it seems to reason, arises purely for the sake convenience. You see, necessarily, the more digits a decimal has, the larger the value is or becomes (if we of course disregard trailing 0s). For example, no matter how many digits you append to the end of 1.5 (e.g. 1.54, then 1.549, then 1.5491, and so forth), the value always increases and never decreases. Therefore, rounding up is the most convenient way to approximate without further compromising integrity or precision. In other words, doing it this way ensures we can disregard the greatest and or consider the least amount of information while maintaining a stable level of precision.