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why cant you divide by zero, wont it just be zero?

my teacher asked this but i dont know why you cant divide by zero

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While zero shows up at the end of 10, 20, 30..etc it has a value there. But zero alone is not a true number..in my opinion because zero is the absence of value. Alone its not a number but the absence of a number and a place holder on the number line.

Rich, are you saying zero is not a number? I strongly disagree. Not only is zero a number, it is one of the most important numbers, perhaps THE most important. 

zero is the answer to the sum of a number and its additive inverse.  In other words, the definition of an additive inverse is a + (-a) = 0

I wish you could "like" these comments, because if I could, I would "like" Robert's.

Gady, you teacher asks a good question. Imagine, if you decide to give something that you separate easily. Got that? Now let's say there were it could be split into parts where where each person got one. This makes sense when you are dealing with whole persons right. But it does not make sense when there are zero persons. This is what Math teachers will call undefined. Take for example

  • if it is for tow people, you divide that in half,
  • one person you give the whole thing,
  • so for zero people, what do you do?

Well you can't give any portion sense you don't know what portion to give for zero and it can't be the whole thing or a portion of it, so they determined long ago to agree this is undefined (not define).

So if anyone asks, what do you get when you divide by zero, just say the Math people said it is undefined or not defined. They will say back, that is good or 

I could never do it either. Tai (the Math guy)

Pardon the misspelled word tow versus two. The spelling auto-corrected and there is no edit button once you post. Just be aware of that when you add your answers. And watch your grammar, since the editing application is not here for a complete system for grace and mercy towards us tutors.

I am a Math Major from Cal Poly SLO town and enjoy working with youth and adults and even poking a little fun at our circumstances. Math, learning and life should be a fun experience for the student and the mentor.

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19 Answers

1)Let us try to use a nonzero number divided by 0 . For example, 5 divided by 0: 5/0=? so 0*?=5 Since any number multiplied by 0 equals to 0. Hence, there is no number that solves the equation. Hence, the value of a nonzero number divided by 0 remains undefined.

2)Let us try to do 0 divided by 0 . 0/0=? so 0*?=0, Any number multiplied by 0 equals to 0, so ? can be any number. Hence, the value of 0 divided by 0 is still undefined.

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I particularly like this explanation. It is short, simple, and correct. There is enough explanation for a student to learn what they need to learn and continue onto discovering more about this problem later in calculus when limits are dealt with more theoretically. 

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It is known that 0 times anything equals 0:

0 x 6 = 0
0 x 2 = 0
0 x 9999 = 0

With that said, for the sake of argument, let's assume we can divide each side by 0, and we get:

6 = 0/0
2 = 0/0
9999 = 0/0

So which is it? 

Since dividing a constant by zero does not give you a single answer each time, it is accepted that you cannot divide by zero, and the answer is undefined. 

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Dividing each side by zero in itself is undefined for each side! You cannot use this rational for concluding that 0 x 6 / 0 = 6???

Arthur, Give Kevin a little poetic license! I was a little startled at first to see that as well, until I reviewed the qualifier "for the sake of argument" and "let's assume."  That covers a lot of hypothetical territory, and gives the explanation a solid example to disprove. I try to avoid using "cannot" with students, as it sets up a poor mental achievement environment.  I even consider the book explanation of dividing by zero to be prefaced with something like,
"Well, you CAN divide by zero all day, but any answer you get won't do you much good, because it is not any distinctive value, and therefore it is undefined."  

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Well, I explain it with money.  Take a dollar.  Divide it into piles of 50 cents, how many piles do you get? 

Good, 2. 

So 1/0.5 = 2

Now, divide it into quarters, how many? Good, 4. Now we are getting somewhere.

1/0.25 = 4

Yes, let's speed this up a little bit.  Divide it into pennies, what do you get?  Yes, I know you aren't stupid, sorry if this line of questioning is a little condescending.

1/0.01 = 100

Now do you agree that if we had a coin that was really small, and you had to collect 100 of them to buy 1 penny, then you could make even more piles with that doller? And let's call that thing a puny.  So 100 punies = 1 penny, and that 10,000 punies equals 1 dollar.  Sooooooooo

1/0.0001 = 10,000

And we could keep going on, and dividing that number 1 by smaller and smaller numbers, and then I'm going to ask you to divide that dollar up into piles of zero.  

How many piles do you get?

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Good, sequential logical response.
Thanks George. I think a kid who asks this question is trying to find the words to ask this question: I think I conceptually understand addition, subtraction, multiplication and sometimes division. I get how dividing 10 apples into 1 pile gives me 10 apples. But what I don't understand is dividing by a number les than 1 (in absolute value). If I understood that, I think I'd get why you can't really divide by zero.

Here is another idea -- 8/2 also means how many times can you subtract 2 from 8 to get to 0.  

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Of course you can divide by zero, and I'm glad you're willing to consider this question. The issue is in what context could it ever mean anything? Given the many answers included here, there are many examples where doing so doesn't make sense, but does this mean you can't 'ever' meaningfully divide by zero? No. I just don't know of any case where it would make any sense to do so. That's why mathematicians call it undefined.

And no it doesn't equal zero to divide by zero, I couldn't imagine in any possibly conceivable sense that dividing by something 'unrelated' would guarantee something completely 'unrelated'.... (sounds like magic actually, Haha).... Just suffice it to say if zero means "no amount", then what does dividing something by "no amount" mean? that's all it is. Later in mathematics it represents something much more interesting though.

This is a great way to point out that math is actually a language: numbers and groupings of numbers represent something, something conceivable, even at their most abstract.

 

Without going to the unnecessary point of considering advanced mathematics (calculus and beyond), division is just asking the question "how many times do you have to add the divisor (the number in the denominator) to get the dividend (the number in the numerator)??"

For example, 6 ÷ 3 = 2 because you have to add 3 to itself two times in order to get 6.

Now consider 6 ÷ 0....  (or any other non-zero number divided by 0).  How many times do you have to ADD zero to get 6?  There is no valid answer - not even "infinity" works, because if you add zero to itself "infinity times" then you still end up with zero, not the number you are trying to get to.

Note that this is a very simplified answer that is intended to give a simple conceptual view, not a strict mathematical view.  Specifically, it is NOT true that 0/0 = 1, despite the fact that you could add 0 to itself 1 time to get 0.  You could argue that any answer is possible - that is, I can add 0 to itself 10 times and still get zero.  Thus, 0/0 could be 1, or it could be 10, or pretty much any other number.  In fact, that is exactly what happens, and you need more advanced math to determine the true answer in any particular case (it can be a different answer for different problems.)

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Michael, "Without going to the unnecessary point of considering advanced mathematics (calculus and beyond), division is just asking the question...." This is a great answer and the first paragraph pretty much summs up this discussion LOL

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To answer this, first let's consider some other cases. 

6 divided by 6 = 1. 

6 divided by 3 = 2.

6 divided by 2 = 3.

6 divided by 1 = 6.

Let us pretend these are apples.  So in the first case, I am asking to divide 6 apples into 6 groups.  How many apples are in each group?  Well, only 1.  If I put 1 apple into 6 different spots on my kitchen table, I will have 6 groups of apples.

I then take the 6 apples and divide them into 3 groups.  When I divide them out over the kitchen table, I then find I have 2 apples in each group.  So I have 3 groups of 2 apples.

For the third option and fourth option above, I am separating the apples into 2 groups of 3 and then just 1 group.  So all 6 apples end up in just 1 group.

We then ask ourselves what is 6 divided by 0.  I want to take my six apples and divide them into zero groups.  Well... I can't just make the apples disappear.  I am still stuck with 6 apples.  But if even if I have all six apples in one spot on the table, there would still be 1 group of apples, not 0 groups of apples. 

There is no way to evenly divide something up into 0 parts.  You could say, I'll form parts that are smaller than "1"... say... 0.5 ... or 0.25 or 0.125 ... getting smaller and smaller like on a ruler when you are dividing up 1 inch into pieces.  However, you will always be dividing it up into SOMETHING..

Dividing by ZERO….there is one case where you can divide by zero, and that is in electricity. Let’s say you plug in a lamp, your iPhone, or computer into the wall socket; that provides energy in the flow of electrons to your device. Electricity does not flow until a ‘LOAD’ or resistance exists AND a path of current to flow. One can argue if you do not have a complete circuit for voltage or current to flow that voltage/current does not exist because the energy is ‘Potential’. But if you take a piece of metal, form it into a ‘U’ shape, and put that into a wall socket, THEN tell me voltage does not exist! So how does voltage know in an instant that there is a load, and then it flows to power up the device? This is the only example which I have come up with which I can divide by zero and demonstrate a practical use. It’s a good chance your teacher won’t know the answer to this question unless they are a physics teacher but this is an example where math and physical science differ. BTW; if you really want to expand this discussion relating to electronics and math look up the term ‘Infinity’ and I will leave this subject on that note.

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I will add this; if you really want to understand math find a way to use it in conjunction with electronics, physics, or chemistry, and questioins like these can be more expansive in class discussions.

Kristoffer, good comment but I'd argue you're not 100% correct - the reason an electric current does not flow when there is no load, has nothing to do with the electricity "knowing" whether or not there's a load.

It's due to the resistance of air being too high (and the voltage too low) for a current to pass through it - connecting a wire greatly lowers this resistance and offers a viable path for current to flow. Current will always take the path of least resistance. 

Increase the voltage high enough, and it'll be high enough to ionize a path through the air, and a current will flow to the nearest low-resistance object (ie you'll essentially create a spark).

Essentially from V = IR, or I = V/R... if R is very large compared to V, then I is very close to zero. 

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Gady, you teacher asks a good question!

Imagine, if you decide to give something that you can separate easily. Got that? Now let's say that it can be separated easily, where each person can get one portion. This makes sense when you are dealing with whole persons right. But it does not make sense when there are zero persons. This is what Math teachers will call undefined. Take for example:

  • if it is for two people, you divide that in half,
  • if one person, you give the whole thing,
  • so if for zero people, what do you do?

Well you can't give any portion, if you don't know what portion to give by dividing by zero and we know it can't be the whole thing or a portion of it. So they (the Math people of long ago) determined to agree this is undefined (not defined).
So, if anyone asks, what do you get when you divide by zero, just say the Math people said it is undefined or not defined. They may most likely say back, that is good or I could never do it either. That is when you can smile and laugh and breathe till they ask another question that stumps you.

 

Have a great Math day!!!

From Tai W (the Math guy from Modesto, California)
PS: disregard my comment above. I should have taken my time but thought there would have been an edit button after I clicked add comment. Did revise, herein, so here we go again as a Mentor note to myself and others (live a little and learn a little, hope I live a lot and learn a lot to share!). Learning, Math and Life should be fun and personal because it makes it easier to remember in a personal universe!

Another way to see why this is true: try dividing by a very small number. For example: 2/0.001 = 2000. Now make that denominator even smaller: 2/0.0001 = 20,000. And smaller: 2/0.0000001 = 20,000,000.

As you can probably tell by now, as the denominator gets smaller and shrinks towards zero, the quotient (the answer) grows larger, towards infinity. So technically speaking, you can say that dividing anything by zero = infinity, and that's "undefined" (can't put a number on it) in math.

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Infinity is NOT the same as "undefined".  

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You cannot divide by zero because zero has no multiplicative inverse:  No real number multiplied by zero equals any number other than zero.  This is clearer when you realize that any number multiplied by zero must be zero; therefore, you cannot divide by zero (it is undefined).

Here is a graphical way to think of it.  

Suppose you wanted to divide the number 1 by 0.  In other words, you want to find y = 1/x when x = 0.  Well, try graphing the function y = 1/x.  You will notice that to the left of x = 0, the value of y approaches negative infinity.  Meanwhile to the right of x = 0, the value of y approaches positive infinity.  Since 1/0 can't be both negative infinity and positive infinity, it is undefined.

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Your argument falls short if y = 1/x2 - they both go to positive infinity.

Similarly for y = -1/x2 (which at x = 0 is -1/0 = 1/-0 = 1/0) - they both go to negative infinity.  These functions (1/x, 1/x2, -1/x2) don't explain what division by zero should be (if it existed).  They just demonstrate some of the inconsistencies you encounter when you try it.

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Let's say that you have 20 apples, but you can't find anyone who wants them.  Mathematically, the answer would 20/0.  Dividing by zero is like saying that no one cares about the thing you are trying to divide.  If no one cares, why bother?   

Picking up where Kevin S. left off:

What he demonstrated was 0/0 is indeterminate (all real numbers are equally good values for this expression) so we call it indeterminate.

Dividing anything else like 2 or -7 by zero makes things worse and we call the answer "undefined". Why?

Assume there is a real number x such that x=a/0 where a is not zero. By definition of division, we have a=0x=0, a contradiction. Hence there is no such x.

There are branches of mathematics where division by zero is defined as infinity. Those branches have severe limitations in solving most real world problems.

Let us define division by zero and see what happens.

X/0 = Y

Now if we accept that any number times 0 is 0, we have problem.

(5)0=0
(4)0=0
Therefore (5)0=(4)0, right?

But if dividing by zero is allowed, then

(5)0/0=(4)0/0

5=4

This is not a very useful result. To make consistent set of mathematical laws while defining division by zero, we have to throw out a ton of useful laws, such as defining multiplication by zero and the identity property of division: A/A=1. The benefit of defining zero does not outweigh the costs.

 

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I'm curious to know what branches of mathematics you are referring to.

Michael, division of a non-zero complex number by zero is defined as complex infinity, though this is only slightly different from saying it is undefined.

Some implementations of interval arithmetic define 1/0 as infinity.


I think there is at least one other branch but the name escapes me at the moment.

Interval arithmetic - not heard of it previously, but looking it up, I can certainly understand why they define 1/0 as infinity...    If I'm reading right, they actually  define 1/[a,0] as [-8, 1/a] and 1/[0,a] as [1/a, 8] which makes perfect sense given that we are working with boundaries instead of specific numbers.  However, in my admittedly novice opinion, this is more a consequence of the range-bound nature of this type of math than any deviation from the commonly held interpretations of what it means to divide by 0.  Still, a very interesting read - thanks for the tip!

yeah ok...  FYI for some reason, this silly editor turned my infinities in to 8's.  sorry!  just read all of the 8's as infinity.

Ok, well I've sort of decided that I like the idea that 1/0 is infinity better than that you can't do it after reading this and thinking about it some more, but here's my question:

You said let's assume we have this number such that X/0 = Y.  And so that means we have found a multiplicative inverse for 0 right? So does that mean that 5*0/0 is the same thing as 4*0/0 or does it mean that

5/0 * Y = 5 and 4/0 * Y = 4?

 

 

Charles, the statement "X/0=Y" does NOT imply that we have found a multiplicative inverse for 0, even under the assumption that we could divide by zero, because a multiplicative inverse of a number is that which when multiplied by the number results in 1, not just any number X.

Robert's answer is trying to show that dividing leads to contradictions in basic math.  He tries to do this by assuming that 0/0 = 1 (as this is true for every other non-zero number: A/A = 1), thus resulting in the absurd result that 4=5.  Since this is clearly not true, the assumption that led to this result, namely that 0/0 = 1, must be false.

Bottom line: 0/0 is undefined.  You can say that any non-zero number divided by 0 is positive or negative infinity, and that's ok if you are trying to get a basic handle on it, but even that isn't the true answer.

 

Now I realize this is a very elementary explaination for why you can not divide by zero but if you go back to the elementary school explaination of fractions then this is what you get. Lets assume we have a pie to be cut up then what does 1/2 mean : that pie is to be cut up into two (equal) parts and 1/2 says you have one of those two parts.  1/0 would mean you have a pie that is cut up into zero parts and you have one of those parts. The zero denominator would deny the existence of the pie and that contradicts something that we know is true (there is a pie to be cut up) therefore we can not divide by zero. :o)

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Depending on what level of mathematics you are using, the proper answer to the question would be to use limits. The idea of Limits is the precursor to Calculus.

The limit of X/Y as Y=> 0 would be infinity.  If you simply imagine the value of Y to be smaller, and smaller, the value of X/Y becomes larger and larger, until when Y approaches 0, X/Y approaches infinity.

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Almost.

If X is a function of Y (X(Y) = f(Y)) then it is possible that Lim Y->0 X/Y is not positive infinity. For example, let X= e-Y-1. The limit would evaluate to the indeterminate form 0/0. In this case, using L’Hopital’s rule gives us 0 as the final answer.

Actually, for your example, L'Hopital would give us a result of -1, because d/dY (e-Y - 1) = -e-Y.  Thus,:

lim Y->0 (e-Y - 1) / Y = lim Y->0 -e-Y / 1 = -e-0 = -1.

That's not correct, as you can't do this if you try to divide 0/Y for y->0, you actually have a limit. If you want to use higher mathematics, the reason is simply that 0 is not part of the multiplicative group of the real numbers, which might not be too helpful in this forum.

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You cannot divide by 0 because there is nothing to divide. Zero is nothing.  Think logically - if there are 12 pieces in a Hershey bar and you are divying it up between six kids, then each will have 2 pieces.  Thereby leaving nothing (0) left should another kid come and want some.

 

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I can't believe you have 4 dislikes. Your answer is great. I'm a middle school kid who wants to get math but every time I raise my hand my teacher responds by invoking theoretical limits and using words like indeterminate, and then I feel some air of condescension coming on so I nod my head sheepishly and allow him to move on with class. Finally, someone answered the question I wanted to ask but couldn't find the words for. And now that I understand, I can articulate my actual question. Here goes: I get addition, subtraction and multiplication. And when we are talking about whole numbers, I get division too. But what I don't understand is dividing by fractions, can you help explain that to me a little more.

Charles (and Cheryl):

You certainly should not be put off by over-handed answers.  If your teacher can't explain a concept clearly enough, you have every right to ask someone else.  I applaud you for doing so.

I suspect the 'dislikes' are due to the fact that she characterized "zero" as "nothing" in much the same way that Rich H did in the comments to the question (at the top).  However, Zero most definitely is a "something".  

Even worse, the statement "...thereby leaving nothing left" is an exceptionally misleading statement, because it implies that the remainder of the division is what matters, not the divisor.  Although it may be a "clear" answer that is understandable, it is certainly not a good one.  (Apologies to Cheryl.)

As for your question regarding fractions, you should post it as a separate question and you will get much better response.

 

Ok but I think the tutor's job is to figure out what a kid is actually asking and answer that question. If a kid gets limits, then he would never ask this question.

Certainly agree, and that's what I try to do...   Take a look at my answer above - I answer the question using nothing but addition and counting.  

But to provide an answer that is "right for the wrong reasons" is not a good plan, and I will never condone it.

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I think this is best approached this way:

If zero is the divisor then what dividend or value produces a result or quotient which is the number of times zero can be divided by it?  Suppose the divisor is 1 then then quotient is equal to the dividend but for zero we cannot determine how many times a dividend can be divided by it so it is undefined.  

Art

 

Because it is not a divisor

Think of it tihs way:

If you have 15 / 3, you have 15 things and you are putting 3 things in a group. If you divide it this way, you will get 5 groups.

If you have 15 / 1, you have 15 things and you are putting only 1 thing in a group. If you divide it this way, you will get 15 groups.

If you have 15 / 0, you have 15 things, but you will HAVE to make at least 1 group because you already have stuff. It is impossible to divide something by 0.

You CAN do 0 / 0, because you have nothing.

You CAN do 0 / 15, because you have NOTHING right now, but IF you had something, you would put 15 things in the groups.

I hope that helps in the concetual thinking of not being able to divide by 0.

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Sue, it is not true that you can do 0/0.  This is what is known as an "indeterminate form", and although in Calculus we can determine the limit near this point, it still doesn't obtain a defined value.  The best answer for 0/0, like any other number divided by 0, is "undefined".

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