# Why is 5 always rounded up? **URL:** \ **Category:** Factual Questions\ **Created:** [May 18, 2001, 12:34am UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774 "2001-05-18T00:34:46Z")\ **Posts on this page:** 8\ **Page:** 2
**Author:** ![Kamandi](https://avatars.discourse-cdn.com/v4/letter/k/5daacb/32.png) [@Kamandi](https://boards.straightdope.com/u/Kamandi)\ **Post date:** [May 18, 2001, 10:29pm UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/21 "2001-05-18T22:29:14Z")
OK. I’ve sweated blood over this and I think I’ve got it. [This link](http://www.uop.edu/cop/psychology/Statistics/Rounding.html) and [this one](http://www.ask.com/main/metaanswer.asp?metaEngine=directhit&origin=0&MetaURL=http%3A%2F%2Fask%2Edirecthit%2Ecom%2Ffcgi%2Dbin%2FRedirURL%2Efcg%3Furl%3Dhttp%3A%2F%2Fwww%2Esteve%2Do%2Ecom%2Functwa%2Ftwa34%2Ehtml%26qry%3DWhat%2Bis%2Bthe%2Brule%2Bfor%2Brounding%2Bnumbers%253F%26rnk%3D5%26cz%3D9c07fe1d2dd908bd%26src%3DDH%5FAsk%5FSRCH&qCategory=ref_&metaTopic=NIST+Guidance+for+calculations+and+rounding+numbers&ItemOrdinal=4&logQID=2C0A9A33F00C4049BD917E6AFDE845C1&sv=203&back=http%3A%2F%2Fwww%2Eask%2Ecom%2Fmain%2Faskjeeves%2Easp%3Fask%3DWhat%2Bis%2Bthe%2Brule%2Bfor%2Brounding%2Bnumbers%253F&ask=What+is+the+rule+for+rounding+numbers%3F) state the rule “if rounding 5, round to the nearest even number”. But they don’t explain why. Which way to round 5 only becomes significant if you are rounding and subsequently processing a lot of numbers, as **DMC** illustrates. Imagine we’re rounding one significant digit off of 100 randomly selected numbers and then summing them. 8.4 becomes 8, 13.9 becomes 14, etc. We would expect 10 of the randomly selected numbers to be X.5. Of those, five would be preceeded by an even number and five would be preceeded by an odd number. Rounding all X.5’s **up** and then summing would result in a sum that was 10 x 0.5 = 5 units **more** than rounding all X.5’s **down**. Rounding to the nearest even number would result in a sum halfway between these two results. I don’t know if that’s clear or not, but I tried it and it works. I am such a pedantic prick. Who answers his own posted questions? Why did I even bother to post the question? I couldn’t figure it out yesterday, today it’s clear. I am an ass. ---
**Author:** ![Daemon](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@Daemon](https://boards.straightdope.com/u/Daemon)\ **Post date:** [May 19, 2001, 6:50am UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/22 "2001-05-19T06:50:58Z")
From [Snopes](http://www.snopes2.com/business/bank/salami.htm) ---
**Author:** ![KneadToKnow](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kneadtoknow/32/3999_2.png) [@KneadToKnow](https://boards.straightdope.com/u/KneadToKnow)\ **Post date:** [May 19, 2001, 10:48am UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/23 "2001-05-19T10:48:11Z")
> [@](#): > > \*Originally posted by Philster \* > 0,1,2,3,4 /5 /,6,7,8,9 > > Wanna re-think that Knead? Um … no. Your illustration serves only to underscore my original point which was that 5 is _more than halfway_ between 0 and 9. Please note that between 0 and 5 there are 4 digits and between 5 and 9 there are 3. Ergo 5 is closer to 9 than it is to 0. I would further like to go on record as saying that I do not dispute the issue of whether or not there are times when people choose to “round” 5 down or even always “round” 5 towards an even number and call it something like “banker’s rounding.” What I dispute is whether that is indeed “rounding.” I maintain that it is not. It is, instead, an aribtrary “rule” which has no basis in number theory and exists only to pacify the queries of people who ask questions like the OP (no offense intended) because they are suffering under the misimpression that 5 is halfway. C’mon folks, we’re trying to fight ignorance here, aren’t we? ---
**Author:** ![k\_i](https://avatars.discourse-cdn.com/v4/letter/k/8491ac/32.png) [@k\_i](https://boards.straightdope.com/u/k_i)\ **Post date:** [May 19, 2001, 2:15pm UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/24 "2001-05-19T14:15:02Z")
> [@](#): > > The issue isn’t whether 5 is halfway between 0 and 0, because we don’t have zeros at both ends of our numbering system. We only have 10 digits, 0 through 9, and 5 is more than halfway through that sequence, so it rounds up. What makes this confusing for some people is that rounding up means going to the next sequence, which starts with a 0 again. Wrong my friend. You are not trying to round to 9, but rather round to the nearest 0, thus recieveing 11 digits - 0,1,2,3,4, 5 ,6,7,8,9,“10” When trying to round, you add a digit to your base. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [May 19, 2001, 4:01pm UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/25 "2001-05-19T16:01:08Z")
3.5 is _exactly_ in the middle between 3 and 4. The absolute value of rounding up or down is _exactly_ the same. You are not rounding up to 3.9, you are rounding up to 4. I can’t believe you can get confused by something this simple. ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [May 19, 2001, 4:15pm UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/26 "2001-05-19T16:15:22Z")
> [@](#): > > \*Originally posted by \*\*\*KneadToKnow \*\* > You will probably notice if you look carefully that money you owe the bank gets rounded, but money the bank owes you gets truncated. At banks that I’ve worked for, I always round off. Zero thru four truncate; five thru nine round up. > [@](#): > > \*Originally posted by \*\*\*k\_i \*\* > Wrong my friend. You are not trying to round to 9, but rather round to the nearest 0, thus recieveing 11 digits - 0,1,2,3,4, 5 ,6,7,8,9,“10” > > When trying to round, you add a digit to your base. I don’t understand. How do you have a “10” in the last significant position? 😕 It’s simple really. You have 10 digits. Five of them round down - (0, 1, 2, 3, and 4). Five of them round up (5, 6, 7, 8, and 9). ---
**Author:** ![Kamandi](https://avatars.discourse-cdn.com/v4/letter/k/5daacb/32.png) [@Kamandi](https://boards.straightdope.com/u/Kamandi)\ **Post date:** [May 19, 2001, 4:29pm UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/27 "2001-05-19T16:29:54Z")
> [@](#): > > \*Originally posted by KneadToKnow \* > \*\*C’mon folks, we’re trying to fight ignorance here, aren’t we? \*\* Indeed. Your argument that 5 is closer to 9 than 0 is immaterial. The issue of the OP is whether 5 is closer to 10 than to 0, since the numbering system of interest is base 10, not base 9. Imagine you cut an apple **exactly** in half. Which half is heavier? By definition, each half would weigh exactly the same. Since we can write “one half” alternately as 50%, 50% of the apple would be in your left hand and 50% would be in your right hand. Since we started with a whole apple (or 100%), 50% is exactly the same distace from 0% as it is from 100%. By the same argument, 5 is exactly the same distace from 0 as it is from 10. _Not 9._ ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [May 19, 2001, 4:43pm UTC](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774/28 "2001-05-19T16:43:51Z")
I can’t believe we need to discuss this. The absolute value of the error introduced by rounding 3.5 down to 3 or up to 4 is _exactly_ the same: 0.5 Now I’d like to know what banks are the ones who would round down to 3 but up to 3.9 Maybe the people at cyberrebates did that… [Previous page](https://boards.straightdope.com/t/why-is-5-always-rounded-up/65774.md?page=1)