## what is 1 divided by 0?

A thread on webmaster world poses a question I had several months ago but never found a clear answer for.

The question is simple, but the answer is more complicated.

What is 1 divided by 0?

According to Google’s calculator, it’s 0, but I think they’re right out. The two answers I had found were infinity and undefined (or error).

As an experiment, I decided to try it on all the calculators I could get my hands on:

The unix desk calculator (dc): divide by zero
Gnome’s calculator (gnome-calculator): inf
KDE’s calculator (kcalc): error
My HP 48G: Error: infinite result
My palm pilot: error

So that’s 3 votes for error and 2 for infinity.

Next, I tried various programming languages:

Perl: Illegal division by zero
PHP: Division by zero
Bash shell: division by zero
C/C++: division by zero
Java: / by zero
Python: ZeroDivisionError: integer division or modulo by zero

With all 6 being errors, it looks like programmers have decided you’re not allowed to do it.

That makes a total of 9 votes for undefined (or error) and 2 votes for infinity, so undefined is the big winner.

If you’re still not satisfied, you can find more on the subject by searching Google for divide by zero.

1. Undefined is definitely the way to go here, since defining a number for the result of division by zero pretty much destroys very important mathematical properties and makes possible proofs that all numbers are the same, and other nasty things.

The only context in which infinity is a useful answer is when you’re taking the limit of n/x where x approaches 0. In this case, x never actually IS 0, and the value goes up towards infinity as x get smaller and smaller. This still isn’t a fully general solution, though, since negative values of n will go to negative infinity and n = 0 is indeterminate, i.e. there’s not any answer that you can determine.

When dealing with computer hardware, IEEE 754 (the most common floating point spec for ALU implementations) defines n/0 in terms of limits, so you have the possibility of +inf, -inf, and NaN. Integer math (on x86 at least) triggers an interrupt on division by zero.

Comment by Levi on April 21, 2005 @ 9:28 am
2. Put that equation into a real world scenario: I have one orange I want to divide it into zero pieces.
Answer: You would never want to do that, so why do it?

Am I over simplifying it?

Comment by jason on April 21, 2005 @ 3:16 pm
3. that’s recalculus!

Comment by whaleman on April 21, 2005 @ 11:16 pm
4. Well lets see… I think we shold have all learned this is Trig and Pre-cal… so the answer is undefined… b/c like Levi said… if you use infinity as the answer than you are dealing with screwin up the whole mathamatical platform… b/c you are giving reason to beleive that all numbers are one in the same… so I think the Smart thing to do is go with undifined as the answer…

Comment by Sunshine Harrington on April 22, 2005 @ 8:28 am
5. dan, I don’t think it’s actually an ‘alternate question’, it’s a question for which you might use the limit to find a reasonable answer that matches your intuition. It must be noted that you can’t actually treat infinity as a real number, or you end up with the same issues discussed earlier and a mathematical system that is nonsensical.

All this mess surrounding zero is part of why the Greeks and their intellectual followers fought against it for so long. Though it’s hard to believe now, for a long time there was no such thing as a number zero. Even when computations involving a number zero were introduced in the East, it was considered a heretical idea in the West due to philisophical implications. Eventually the utility of it won out, and now if you try to tell people that at one point no one knew about zero, you get a lot of funny looks. I’m probably getting a funny look from you now, in fact, but it’s true!

Comment by Levi on April 22, 2005 @ 9:14 am
6. When you divide by zero on a PowerPC chip (G4, G5, etc) you get zero. Since these chips are his holiness steve jobs’ true choice, we may have a divine answer here. (This is an important different from x86 to watch out for when writing portable code.)

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Comment by ammon on April 22, 2005 @ 10:14 am
7. I get the Seinfeld reference, Dan, but am surprised to see you use it here. And woah/whoa — per m-w.com, at least. (I was thinking whoah, but defer to the dictionary.)

I love the philosophical 1/0 debate going on here. Good stuff.

Comment by DavidH on April 22, 2005 @ 2:19 pm
8. It’s undefined and it specifically addresses the “persistent but misguided reader who insists on asking “What happens if I do divide by zero?”:

http://mathworld.wolfram.com/DivisionbyZero.html

Comment by Justin on April 23, 2005 @ 4:29 pm
9. think of deviding by zero like this take (one)devided by(nothing)
(two)devided by (1) equals 2 beacause your seperating 2 objects into groups of 1 each and the answer is the amount of groups you end up with which is 2 but if you devide by zero your seperating 2 objects into groups containing no objects thus you end up with infinity its not alreals its the largest # ever which is unreasonable so thats why it seams its undefined its both really just like a quadratic function there are 2 answers its undefined and infinity

Comment by helper on May 20, 2005 @ 10:32 am
10. um, it’s quite obvious.

take a look at this:

1 divided by 0.1= 10
1 divided by 0.01=100
1 divided by 0.001=1000

Therefore as the denominator becomes smaller, the result of the equation becomes greater. Hence, by dividing a number by 0, the result becomes infinite.

Comment by Anonymous coward on November 12, 2005 @ 4:15 am
11. this is ridiculas it is obveosly infinity ask any math proffessor

Comment by zafar on February 23, 2006 @ 12:01 pm
12. Um yeah, undefined and infinity are the same thing, especially in calculus. Usually limits that involve infinity are called undefined. And it is due to the limiting pocesss stated above about decreasing the denominator, hence the limit as the denominator approacjes zero is indeed an exponentially increasing number. Undefined just means it has NO numerical value or definition since infinity is not an attainable umber. But have fun with this physics equation: V=IR is the voltage due to a current and a resistance. Say yourn given a voltage of 120 volts and a superconducting wire (absolutely no resistance so its 0) and you want to find the current, is the current undefined due to V/R=I written as 120/0= infinity, nope its 120 amperes. There are equations in physics where division by zero is allowed. But then again usually things in physics are messed up.

I know its an old topic but I couldnt let it just hang as the answer had to be undefined, 0 or infinity but as all of them at once.

Comment by Dan on June 13, 2006 @ 12:30 pm
13. But that’s just it isn’t it? All numbers ARE one and the same. I know many of you don’t agree with this statement but I’ll say it anyway b/c its the most logical response:
Anything divided by 0, does prove all numbers ARE the same, because God invented the language of numbers, it is the language of God.
God is to divide by 0, which is why science is constantly re-writing and re-proving itself by trying harder and harder to precisely define something. All it does is repeat… for infinity or forever.

Comment by Anon on August 23, 2006 @ 12:03 pm
14. My question is give justification why 0 divided by 0 does not exists i want something that i can be delivered in front of a lay men means some sort of a real problem, which easily understandable i know that anything divided by zero is infinity as it is possible to give a real sort of answer but why 0 divided by 0 does not exists

Comment by asfar nisar on November 8, 2006 @ 12:04 am
15. Division by Zero is possible………..0/0=0……………..use simple multiplicative and additive identies/inverses with this eqn…e.tc and it may be proven for any intiger.
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Comment by Arcturus on December 8, 2006 @ 8:45 pm
16. JavaScript returns infinity as well.
http://geneburnham.5u.com/JScalc.htm

Comment by Gene on January 17, 2007 @ 7:49 am
17. Sorry, I was just giving an example of a language that had not yet been used.

Comment by Gene on January 17, 2007 @ 6:01 pm
18. Oh, you’re quite welcome.
P.S. I’m sorry about all the ads on that page, it’s just how free web hosting stays free…

Comment by Gene on January 18, 2007 @ 5:50 am
19. 1 Divided by 0 is infinity. Think about it like this example:
How many twos you need to get a twelve?- the answer is 6 (2+2+2+2+2+2)
And how many zeros you need to get a 1 – its infinity because 0 can never reach 1
(0+0+0+0 ….. can never add up to 1)

Comment by Anon on January 25, 2007 @ 7:42 am
20. 1/0=infinity, Why? Because group theory proves it, graphing proves it, logic proves it, and algebra proves it.
~~~~~Group Theory and Algebra
They say that multiplication for the REALS is a group.
In order to be a group 4 things must be true
1) closure
2) associativity
3) Identity
4) Inverse
The multiplicative identity is 1 because x*1=x
This means that x*y=1, y is the inverse of x. Every possible x needs a y… However
what if x is 0?
0*?=1
We say nothing right? Not the case, otherwise the real numbers wouldn’t be a group. 0*infinity=1
Thus also 1/0=infinity
~~~Graphing
Anyone ever seen a parabola? (y=1/x) As x get’s closer and closer to 0, y get’s closer and closer to…what? Oh yes Infinity.
~~~~~Logic
I have 5 apples, I’m making groups of 0 apples. How many groups can I make? Here’s you group of 0 apples, and here’s your group of 0 apples, and yours…..forever! Infinity.
~~~~~Bonus Logic
Our system is poor, imagine a system where not only the numerical value of the number was important, but also the history of the numeric value. Every number has a history, it’s just we don’t often understand it. For example, 10 kids are in a class. This number 10 has a history, in fact a history and creation on more than one way. One history may be that 3 kids came in first, then later 2 kids, then 5 kids. Even though 10 is 10, the origin of that 10 can vary. In the real world sense there’s infinite history’s for every number but in the math world we can limit this. So here’s our new system, one where 3*2=6, and that 6 is not identical 3+3’s 6. It’s origin must be carried with it. Just because our system doesn’t carry the history doesn’t mean this factor isn’t there, and can come into play when working with things like infinity. Now let’s apply this to infinity. We know that there are different types of infinity, for example the sum of all terms of 2x from (1,2,3…inf) is infinity but it is not as great as an infinity as one that’s from the sum of all terms 5x from (1,2,3…inf). And thus we see that the origin of the infinity is greatly important.
Thus infinity+1 is not = to infinity; they both are forms of infinity yes, but their history must be kept with them. so (infinity+1)-infinity=1 and not 0; Only by keeping the history of the infinity can we solve things back to our system with it. This is a concept we are not use to in math.
One final concept of importance in the “infinity, infinity+1, infinity*2, infinity^2” stuff. Think as infinity as an arbitrary point, one that we use merely for reference. And the number’s actually instead of looking like this (…-2…0…1/8…sqrt2…5…infinity) the infinity end should look a little more like this …infinity-1….infinity….infinity+1…) where even as you keep adding to 0 or minus-ing from infinity you’ll never reach the same value unless you minus infinity from infinity or add infinity to 0.

And when you keep the history for infinity, all the laws of modern math still work actually.

Comment by neonash7777 on February 3, 2007 @ 9:40 pm
21. There have been some god awful statements made here. I’m of the opinion that some people need to stick to b1tching about computer games on tech forums and leave the right people to discuss mathematics :)

Comment by Tom on February 22, 2007 @ 6:59 pm
22. think of it like this 1/0 is infinte because 0 goes into 1 as much times as you want it too so it goes infinite times.

Comment by alle on August 14, 2007 @ 6:04 am
23. ha i think thats funny because according to what is the standard answer to this problem, undefined and infinity are the same.

1 divided by 0 would be both infinity and undefined.
it means you are dividing or separating or cutting 1 into 0 pieces. if i had 1 apple and i wanted to cut it into 0 pieces, how long would i be slicing that thing? an undefined (“not limited” according to the dictionary) amount of times, or an infinite amount of times. same thing.

however error is my fav.
divide by zero and then check your answer if you want to see what i mean. =D

Comment by kc on October 3, 2007 @ 11:06 am
24. Google no longer even /tries/ to search for numbers / 0. Try it.

Comment by thejapanesegeek on February 8, 2008 @ 8:54 am
25. 1 divided by zero is universally impossible and forbidden.
There is an incredible (maybe true but not released in public)book called Deep Storm by Child that includes impossibe equations like a^3+b^3=c^3.

Comment by Wes C on March 19, 2008 @ 7:54 am
26. How many of you have taken a Calculus course? Limits, L’Hopital’s rule, etc..

Particularly L’Hopital’s rule

Comment by Steven on September 25, 2008 @ 7:15 am
27. there is an answer to this 0/0 is 0 where as 1/0 or 2/0,etc is infinity let us assume that the number line is not straight but curved nto a circle this circle has extensible radius 0 is at e part of the circle and +nos to the right -nos to the left a point on the other end of the circumference be infinity(remember defn) so if u say 78907648998754379 is the point of infinity the circle will adjust size and fit a bigger no.idea is to say +finity and-infinity is almost the same no.tending to 0 geometrically infinity will lie betwen 1 and 0 diameterically opposite.getting ideas tell me your response and i will tell u the fill amazing solution recently developed at the iapt which changes the entire no system!!!!!!!

Comment by debo on September 30, 2008 @ 8:33 am
28. yes i know about this what a incredible theory quantumised disburgities in number. uses laws of physics to prove mathematical modules what an explicit solution

Comment by drona on September 30, 2008 @ 8:39 am
29. If we all listen to what the “experts” say about what we are suppose to think in life…how boring life would be. Get back to the basics/think outside of the box and be open minded. 1/0 is obviously 1. If you take 1 “apple” and / it by 0…you’re still left with 1 apple. It didn’t disappear into thin air!!! People should start using their own common sense and not let others dictate to them on how to think. Who are they to make the rules???

Comment by maxie on October 23, 2008 @ 2:09 am
30. I like the idea of 1/0=inf.

HOWEVER!!!!!!!

if 0 = -0

then inf. = 1/0 = 1/-0 = -1/0 = -inf.

Therefore inf. = -inf.

IMPOSSIBLE!!!!

Comment by Fluxmaster on November 26, 2008 @ 2:41 pm
31. Zero is the amount of something you have when you have none of it.
Therefore, 1/0 is the amount of something you have when you have the multiplicative inverse of none of it.
Except it can’t be, because that would imply that 1/0 = 1/0, which it can’t, because that would imply that 0 = 1, correct?
I figure that we cannot say that 1/0 “is” anything. We have to define a new kind of equality that follows the rules for numbers that are not numbers (assuming we still want to deal with 1/0).
And maybe 0 DOES equal 1. Maybe under some circumstances. Maybe under some circumstances, all real numbers are the same, because you’re looking at math from the perspective of something that is infinitely larger than all of them. For that matter, maybe equality is relative. Maybe that’s why we can’t divide a number by zero; we’re trying to divide a number by something infinitely smaller than it, so it makes sense that a number infinitely larger than it would result, but such a number is also infinitely larger than any other number, so from the perspective of such a number, any other number is zero, and zero equals zero, so any number equals any other number. Of course, under any everyday circumstances, any number could not possibly equal any other number; only two symbols or equations that represent the same number could be equal… Augh… WTFAMIDOINGHEREI’MINFREAKINGNINTHGRADE.

Comment by InvaderKap on November 29, 2008 @ 9:44 am
32. x/0 = Error/Undefined

You cannot split any number 0 times. An example used above is cutting an apple to share between 0 entities. Its just not possible. And it is not infinity either because you cannot cut the apple for infinite entities, and even if you could then it would be x/âˆž…

Comment by Nodnarb on December 8, 2008 @ 5:44 pm
33. I suggest checking out ZERO by Charles Seife

POINT 1: 1/0 is undefined in elementary mathematics
UNDER CERTAIN CONDITIONS (as in the limit of 1/x) it can be seen as being what is called an unsigned infinity (neither positive nor negative).

Assuming 1/0=x

then by following normal mathematical rules
x*0=1

this is invalid, even for infinity, since multiplication is merely repeated addition.

if x=infinite

(0+0+0+0+0+0…)=1 doesn’t make sense Adding any number of zeros together equals zero. It’s a basic mathematical property. Infinite is only the answer if taken in certain situations. The problem cannot be generalized as having a single nice answer.

POINT 2: saying 0 is not equal to -0 makes no sense. 0 is neither positive nor negative, and cannot be given either value. Zero has no sign.

POINT 3: Because of point 1, technically, multiplication of the reals is NOT a closed system. Infinite is not a real number, so even that answer doesn’t work. A real number has the property of being written by integers at various place values (eg. 0.1 has a 1 at the tenths place) infinity cannot be written in this way.

POINT 4: 0/0 is not 0. Apparently, you can’t divide by zero (true in an uncomplicated abstract case such as this). anything divided by itself is also apparently 1. but zero divided by anything apparently equals 0. Which do you choose?

0/0=x
x*0=0

x can be ANYTHING. It is undefined not because it has no answer, but because EVERYTHING (but infinity, which cannot be treated as a regular number) is an answer. It doesn’t make sense to ask it as a question.

I remember seeing another link that says how you can’t say something like 1/0=Â§ (you would assume you could, since it SEEMS magical to ‘make up’ imaginary numbers(the square roots of negative numbers)) Hope this clears some things up!

Comment by Robert on December 23, 2008 @ 6:30 pm
34. 1/infinity=0
since,the difference between 1 and 0.999999999…..is not a whole number
1/0=1/1/infinity= 1Xinfinity/1=infinity

CAN TRY THIS………………..for SURE!!!!!!!!

Comment by nishant on April 26, 2009 @ 9:28 am
35. @Robert: Yes, as for that link that you remembered seeing, one reason that you can’t set 1/0 equal to an imaginary number: If you were to do that, then a simple proof that 1=0 would be possible:

1. 1/0 = S (definition of S)
2. S = 1/0 (reflexive property)
3. 1/0 = 1/0 (transitive property)
4. 0 = 0^2 (a simple fact)
5. (1/0)*0 = (1/0)*0^2 (algebraic multiplication [multiplication on both sides])
6. 1 = 1*0 (for any x, x*1/x=1, and for any x, x^2/x=x)
7. 1 = 0 (any number times 0 equals 0)

As we know, 1 != 0; at least it never has. In 100% of the known history of the known universe, nobody has ever obtained an apple and wound up with no apples in a smooth transition that could only be described as nature and similarity to the point of equality. Similarly, nobody has ever had an apple, got another apple, and naturally wound up with three apples. But you know, this stuff is all statistics. We don’t know anything for sure. Anything. Maybe there really is a God, and He’s an evil bastard who’s trying to trick us by making science look so reasonable, all so that He can have His sadistic fun watching us die to His hands afraid and confused instead of knowing that the end of the world has come; after all, how could we have known? He made it so painfully obvious that all our scientific discoveries were true, only to take them away from us, only to pull us out of our illusion and into a harsh reality where everyone suffered and died painful, gory deaths?

That was a major digression. Sorry about that.

Comment by InvaderKap on May 4, 2009 @ 5:13 pm
36. yeh wot ever of corese it is infinaty

Comment by chloe on September 9, 2009 @ 8:53 am
37. I think the answer lies in the fact that there is no such thing as nothing. So we cannot use it to divide something by. This calculation cannot be said to be true in real terms. We’re kind of making a mistake by performing this calculation even. If nothing was possible then I believe the answer would be infinite.:

Comment by Andrew on September 27, 2009 @ 2:41 pm
38. i think that dividing by 0 is infinity but it is a complicated subject and therefore should not be done.

Comment by wayne on December 9, 2009 @ 6:02 am
39. maybe, it’s proving that the universe, never become by itself. There must be a creator

1 divide by 0. 0+0+0+0+0….. = 1. Sorry, that’s really unlogical.

Comment by Anonymous on January 2, 2010 @ 12:43 am
40. im kidding 1 divided by 0 is completly and utterly impossible

Comment by dog on July 14, 2010 @ 12:01 pm
41. yes 1 divided by zero is completly and uterly in accurate and wrong who ever argues 2 that is completly in accurate

Comment by dog on July 14, 2010 @ 12:02 pm
42. definition

indefinant-the end of infinity
infinity-never ending or all known and unknown numbers negative and/or positive numbers

zero times infinity =0 zero times indefinant =undefined
zero devided by zero= infinty undefined devided by zero= indefinant
undefined devided by indefinant=0

thus whoever says infinty is not A number they are right its not A number its any and every munber reguardless known and unknown negative or/and positive

Comment by theblaxpaceman on December 4, 2010 @ 3:17 pm
43. i forgot to give the meaaning of undefind
undefine-never begining or none existed not even in thought

Comment by theblaxpaceman on December 4, 2010 @ 3:26 pm
44. “maybe, itâ€™s proving that the universe, never become by itself. There must be a creator

1 divide by 0. 0+0+0+0+0â€¦.. = 1. Sorry, thatâ€™s really unlogical.”

I agree with that, Its, well, “ERROR” because it will never be done. Lets say you had all the time ever, and all you had to do is get 0 to go into 1. You’d never do it, would you? Meaning infinity is wrong, because if you DID have infinite 0’s, you’d STILL never reach it. So I don’t really know how to explain my awnser, other than;Impossible, unreachable, ERROR, ect.

Comment by Jacob on December 19, 2010 @ 6:13 am
45. Also, i love how theres like 4 comments of people going “whatever its freakin’ infinity! RGHHGHFH (rage). Calm down. :S

Comment by Jacob on December 19, 2010 @ 6:31 am
46. Only number 1 can be divided by 0. The answer is infinity. The rest numbers divided by 0 is undefined.
Well there IS such thing as nothing in this world. The universe created from nothing. Remember?
0+0+0+0… with no end of zero (surely it’s hard to imagine) equals 1.
Remember, we confused because our brain always try to limits everything. We talks about infinity here, something that can only believed by the believers. One last example : 0.(9) equals 1. If you finally can accept that, then you must believe 1/0=inf.
The rest numbers/0=undefined.

Note that (9) means repeated 9 to infinity.

Comment by Adonis on February 18, 2011 @ 9:39 am
47. 0 * Infinity = 0, therefore Infinity is not enough

Comment by Brent on February 1, 2012 @ 2:38 pm
48. 1 divided by 0 = error or undefined or infinity (not equal to zero)
for me; the answer is (0 remainder 1), if we check it by multiplying
0 x 0 = 0 + 1 and the answer is 1.
another… if the problem is 15/15 the answer is 1, if we check
1 x 15 = 15, thus 15/1=15 coz 15×1=15.
Thanks…

Comment by Elgin on July 26, 2012 @ 3:44 am