# mathematics: 5 - 0 = 5 ; 5 x 0 =5 **URL:** \ **Category:** Factual Questions\ **Created:** [March 28, 2004, 1:03am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835 "2004-03-28T01:03:02Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![river](https://avatars.discourse-cdn.com/v4/letter/r/76d3ee/32.png) [@river](https://boards.straightdope.com/u/river)\ **Post date:** [March 28, 2004, 1:03am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/1 "2004-03-28T01:03:02Z")
first, this is going to get into mathematics that i’m not going to know or understand. so really, this is a doofus going into it. establishing this, let’s dive. 5 - 0 = 5 if you have five apples and take away none - you will still have five apples. 5 x 0 = 0 if you have five apples but multiply it by nothing - you would have gained nothing. so here is my question, why is it that for subtraction we are left with the amount we started with as the answer but for multiplication we rely on the fact that we gained nothing? what i’d like is this: to not have the mathematics here be seen like a theory of convenience (as in, because x 0 = 0 would work better in mathematical equations). maybe the fault here is using the apples as an example? are they irrelevant? any help? ---
**Author:** ![GorillaMan](https://avatars.discourse-cdn.com/v4/letter/g/50afbb/32.png) [@GorillaMan](https://boards.straightdope.com/u/GorillaMan)\ **Post date:** [March 28, 2004, 1:06am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/2 "2004-03-28T01:06:58Z")
The apples are fine as an example: 5 x 0 = ? 5 x 1 = 5, because one set of five apples totals five. 5 x 2 = 10, because two sets each of five apples totals ten apples. 5 x 0 … no sets of five apples results in no apples. so 5 x 0 = 0. ---
**Author:** ![leenmi](https://avatars.discourse-cdn.com/v4/letter/l/f1d935/32.png) [@leenmi](https://boards.straightdope.com/u/leenmi)\ **Post date:** [March 28, 2004, 1:09am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/3 "2004-03-28T01:09:30Z")
You are looking at it wrong. In the first example, you have a set of five from which you subtract nothing leaving you with a set of five. In the second example, you have five sets of nothing, leaving you with nothing. 5 X 3 is the same as 3 X 5. Either 5 sets of 3 or 3 sets of 5. A x B = B X A 5 X 0 = 0 X 5. It’s not just five multiplied zero times. It’s also zero multiplied five times. Which is zero, and makes more intuitive sense when spoken. Five times nothing is just a lot more nothing. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [March 28, 2004, 1:10am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/4 "2004-03-28T01:10:30Z")
Here’s one way to look at it. Take one you know, like 5 x 2. You can read that as “What is the answer when you add five twice?” Likewise, 5 x 20 is “What is the answer when you add five twenty times?” So, 5 x 0 is “What is the answer when you don’t add any fives at all?”. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [March 28, 2004, 1:11am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/5 "2004-03-28T01:11:51Z")
It’s because zero is the identity element for addition (among the reals), but not for multiplication (that is one). In terms of apples you could rephrase it as how many apples in total you’d have if you have zero groups (I use the word non-technically) of five apples. The answer is obviously zero apples. ---
**Author:** ![leenmi](https://avatars.discourse-cdn.com/v4/letter/l/f1d935/32.png) [@leenmi](https://boards.straightdope.com/u/leenmi)\ **Post date:** [March 28, 2004, 1:16am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/6 "2004-03-28T01:16:03Z")
> [@MC Master of Ceremonies](#): > > It’s because zero is the identity element for addition (among the reals), but not for multiplication (that is one). > > In terms of apples you could rephrase it as how many apples in total you’d have if you have zero groups (I use the word non-technically) of five apples. The answer is obviously zero apples. Great answer. Clear as a bell. He had his identity element wrong. It’s so obvious now. :rolleyes: ---
**Author:** ![Martin\_Hyde](https://avatars.discourse-cdn.com/v4/letter/m/47e85d/32.png) [@Martin\_Hyde](https://boards.straightdope.com/u/Martin_Hyde)\ **Post date:** [March 28, 2004, 1:16am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/7 "2004-03-28T01:16:18Z")
Another thing you need to realize is, multiplication is just an easier way of dealing with long strings of additions. 5 x 2 = 2+2+2+2+2 = 10 Or (due to the communitave property of multiplication): 2 x 5 = 5+5 = 10 So 5 x 0 = 0+0+0+0+0 = 0 Five zeroes = zero. Or using the communitave property of multiplication: 0 x 5 = Zero fives, which is zero. Multiplication basically just tells you how many sets there are of something: 5 x 2, there are five sets of two or two sets of five. 10 x 1,000, there are 1,000 sets of ten or ten sets of 1,000. 5 x 0, there are zero sets of five, or five sets of zero. Both equalling zero. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [March 28, 2004, 1:21am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/8 "2004-03-28T01:21:19Z")
> [@leenmi](#): > > Great answer. Clear as a bell. He had his identity element wrong. > > It’s so obvious now. :rolleyes: [http://mathworld.wolfram.com/Group.html](http://mathworld.wolfram.com/Group.html) ---
**Author:** ![river](https://avatars.discourse-cdn.com/v4/letter/r/76d3ee/32.png) [@river](https://boards.straightdope.com/u/river)\ **Post date:** [March 28, 2004, 1:23am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/9 "2004-03-28T01:23:24Z")
> [@leenmi](#): > > It’s not just five multiplied zero times. It’s also zero multiplied five times. Which is zero, and makes more intuitive sense when spoken. ohhh this one makes a lot of sense. yes, thank you. i never thought of it that way. it works for 0 - 5 = -5 too which is just precious to savor. > [@](#): > > Five times nothing is just a lot more nothing. this comes in handy too. ---
**Author:** ![river](https://avatars.discourse-cdn.com/v4/letter/r/76d3ee/32.png) [@river](https://boards.straightdope.com/u/river)\ **Post date:** [March 28, 2004, 1:24am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/10 "2004-03-28T01:24:49Z")
and thank you everyone! ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [March 28, 2004, 1:27am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/11 "2004-03-28T01:27:01Z")
Not to mention the limit of 5x as x-\>0 ---
**Author:** ![vasyachkin](https://avatars.discourse-cdn.com/v4/letter/v/c5a1d2/32.png) [@vasyachkin](https://boards.straightdope.com/u/vasyachkin)\ **Post date:** [March 28, 2004, 1:59am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/12 "2004-03-28T01:59:17Z")
thank god he didnt have to look at binary logic where 1+1 = 0 and 1-1 = 0 as well … ---
**Author:** ![c\_carol](https://avatars.discourse-cdn.com/v4/letter/c/e79b87/32.png) [@c\_carol](https://boards.straightdope.com/u/c_carol)\ **Post date:** [March 28, 2004, 3:21am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/13 "2004-03-28T03:21:58Z")
> [@river](#): > > ohhh this one makes a lot of sense. yes, thank you. i never thought of it that way. it works for 0 - 5 = -5 too which is just precious to savor. 🙂 I think you’re starting to see why we math nerds like math. Pretty cool, isn’t it? ---
**Author:** ![cornflakes](https://avatars.discourse-cdn.com/v4/letter/c/9d8465/32.png) [@cornflakes](https://boards.straightdope.com/u/cornflakes)\ **Post date:** [March 28, 2004, 3:47am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/14 "2004-03-28T03:47:27Z")
> [@river](#): > > 5 - 0 = 5 > if you have five apples and take away none - you will still have five apples. > > 5 x 0 = 0 > if you have five apples but multiply it by nothing - you would have gained nothing. To put it simply: 6 - 0 = 6 If you buy a six-pack and don’t drink any - you still have six beers. 6 x 0 = 0 If you buy no six-packs - you’re out of beer. ---
**Author:** ![Dogface](https://avatars.discourse-cdn.com/v4/letter/d/9d8465/32.png) [@Dogface](https://boards.straightdope.com/u/Dogface)\ **Post date:** [March 28, 2004, 4:35am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/15 "2004-03-28T04:35:04Z")
Did Schoolhouse Rock ever do a zero? (or a one, for that matter) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 28, 2004, 4:42am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/16 "2004-03-28T04:42:06Z")
The formal definitions of addition and multiplication rely on the notion of the successor of x, denoted as x’. What exactly that means is irrelevant, but we define 1 = 0’, 2 = 0’’, and so on and so forth. Addition is defined with x + 0 = x and x + y’ = (x + y)’. Multiplication is defined with x \* 0 = 0 and x \* y’ = x + (x \* y). From this, it’s easy to see that x + 1 = x’. **vasyachkin** , are you talking about a boolean algebra, or **Z** [sub]2[/sub]2? Be careful: there is no - operator in a boolean algebra. ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [March 28, 2004, 4:49am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/17 "2004-03-28T04:49:42Z")
> [@Dogface](#): > > Did Schoolhouse Rock ever do a zero? (or a one, for that matter) Zero my hero ---
**Author:** ![Bambi\_Hassenpfeffer](https://avatars.discourse-cdn.com/v4/letter/b/bbe5ce/32.png) [@Bambi\_Hassenpfeffer](https://boards.straightdope.com/u/Bambi_Hassenpfeffer)\ **Post date:** [March 28, 2004, 4:52am UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/18 "2004-03-28T04:52:29Z")
> [@Dogface](#): > > Did Schoolhouse Rock ever do a zero? (or a one, for that matter) Of course! [My Hero, Zero](http://www.schoolhouserock.tv/My.html). > [@](#): > > What’s so wonderful about a zero? > It’s nothing, isn’t it? > > Sure, it represents nothing alone. > > But place a zero after 1 > And you’ve got yourself a 10. > See how important that is? > When you run out of digits, > You can start all over again. > See how convenient that is? One of my favorites. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [March 28, 2004, 2:45pm UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/19 "2004-03-28T14:45:45Z")
Another thing to consider is that 5 x 1 = 5. If 5 x 0 = 5 was also true, then you could cancel out the fives in both equations and prove that 1 = 0, an obvious fallacy. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [March 28, 2004, 11:59pm UTC](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835/20 "2004-03-28T23:59:30Z")
> [@c\_carol](#): > > 🙂 I think you’re starting to see why we math nerds like math. Pretty cool, isn’t it? I dunno about you, but I avoid numbers whenever possible. [Next page](https://boards.straightdope.com/t/mathematics-5-0-5-5-x-0-5/236835.md?page=2)