Dividing by 0

Dr. Math FAQ || Classic Problems || Formulas || Search Dr. Math || Dr. Math Home

#### Why can't you divide by 0? Why is 0/0 "indeterminate" and 1/0 "undefined"? Why is dividing by zero "illegal"?

Here, in their own words, are some explanations by our 'math doctors'. Follow the links to read the full answers in the Dr. Math archives.

Division by zero

Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related.
```   12 divided by 6 is 2   because
6 times 2 is 12

12 divided by 0 is x   would mean that
0 times x = 12
```
But no value would work for x because 0 times any number is 0. So division by zero doesn't work.

- Doctor Robert

My teacher says you can't divide a number by zero. Why?

Let's look at some examples of dividing other numbers.
```10/2 = 5    This means that if you had ten blocks, you could
separate them into five groups of two.

9/3 = 3    This means that if you had nine blocks, you could
separate them into three groups of three.

5/1 = 5    Five blocks could be separated into five groups
of one.

5/0 = ?    Into how many groups of zero could you separate
five blocks?
```
It doesn't matter how many groups of zero you have, because they would never add up to five since 0+0+0+0+0+0 = 0. You could even have one million groups of zero blocks, and they would still add up to zero. So, it doesn't make sense to divide by zero since there is not a good answer.

If you know a little bit about multiplication, you could look at it this way:

```10/2 = 5    This means that 5 x 2 = 10

9/3 = 3    This means that 3 x 3 = 9

5/1 = 5    This means that 5 x 1 = 5

5/0 = ?    This would mean that the answer x 0 = 5, but
anything times 0 is always zero.
```

- Dr. Margaret

Why can't you divide a number by 0?

For one thing, when you divide one number by another, you expect the result to be another number. Look at the sequence of numbers 1/(1/2), 1/(1/3), 1/(1/4), ... . Notice that the bottoms of the fractions are 1/2, 1/3, 1/4, ..., and that they're going to zero. If there's a limit to this sequence, we would take that number and call it 1/0, so let's see if there is.

Well, the sequence turns out to be 2, 3, 4, ..., and that goes to infinity. Since infinity isn't a real number, we don't assign any value to 1/0. We just say it's undefined.

But let's say we did assign a value. Let's say that infinity is a real number, and 1/0 is infinity. Then look at the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), ..., and notice again that the denominators -1/2, -1/3, -1/4, ..., are going to zero. So again, we would want the limit of this sequence to be 1/0. But looking at the sequence, it simplifies to -2, -3, -4, ..., and it goes to negative infinity. So which would we assign to 1/0? Negative infinity or positive infinity? Instead of just assigning one willy nilly, we say that infinity isn't a number, and that 1/0 is undefined.

- Dr. Ken

When something is divided by 0, why is the answer undefined?

The reason is related to the associated multiplication question. If you divide 6 by 3 the answer is 2 because 2 times 3 IS 6. If you divide 6 by zero, then you are asking the question, "What number times zero gives 6?" The answer to that one, of course, is no number, for we know that zero times any real number is zero not 6. So we say that division by zero is undefined, for it is not consistent with division by other numbers.

- Dr. Robert

Because there's just no sensible way to define it.

For example, we could say that 1/0 = 5. But there's a rule in arithmetic that a(b/a) = b, and if 1/0 = 5, 0(1/0) = 0*5 = 0 doesn't work, so you could never use the rule. If you changed every rule to specifically say that it doesn't work for zero in the denominator, what's the point of making 1/0 = 5 in the first place? You can't use any rules on it.

But maybe you're thinking of saying that 1/0 = infinity. Well then, what's "infinity"? How does it work in all the other equations?

Does infinity - infinity = 0?
Does 1 + infinity = infinity?

If so, the associative rule doesn't work, since (a+b)+c = a+(b+c) will not always work:

1 + (infinity - infinity) = 1 + 0 = 1, but
(1 + infinity) - infinity = infinity - infinity = 0.

You can try to make up a good set of rules, but it always leads to nonsense, so to avoid all the trouble we just say that it doesn't make sense to divide by zero.

What happens if you add apples to oranges? It just doesn't make sense, so the easiest thing is just to say that it doesn't make sense, or, as a mathematician would say, "it is undefined."

Maybe that's the best way to look at it. When, in mathematics, you see a statement like "operation XYZ is undefined", you should translate it in your head to "operation XYZ doesn't make sense."

- Dr. Tom

What is the value of 0/0? (Is it really undefined or are there an infinite number of values?)

There's a special word for stuff like this, where you could conceivably give it any number of values. That word is "indeterminate." It's not the same as undefined. It essentially means that if it pops up somewhere, you don't know what its value will be in your case. For instance, if you have the limit as x->0 of x/x and of 7x/x, the expression will have a value of 1 in the first case and 7 in the second case. Indeterminate.

- Dr. Robert

Whenever I try to divide a number by 0, I get an error on both my computer and calculator. Please explain to me why this can't be done.

Here's a little experiment for you to try on your calculator. Observe the output when you try the following set of calculations:
```       1/1
1/.1
1/.01
1/.001
1/.0001
1/.00001
etc...
```
until your calculator can't go any further or you get tired. You should notice that the answers continue getting larger and larger.

Another way of thinking of it is to imagine filling a box with apples. Say a box can hold 100 apples. Now try filling it with apples that are half the size of these apples. You can put 200 in the box. Now imagine a special, magic apple that takes up no room at all. How many can you put in the box?

Well, the answer is... there is no answer! That is why mathematicians refer to numbers that are divided by 0 as "undefined." Some people tend to think of them as being infinite, but this isn't exactly true. There simply is no answer.

- Dr. Ethan

Why are operations of zero so strange? Why do we say 1/0 is undefined? Can't you call 1/0 infinity and -1/0 negative infinity? Why not?

1/0 is said to be undefined because division is defined in terms of multiplication. a/b = x is defined to mean that b*x = a. There is no x such that 0*x = 1, since 0*x = 0 for all x. Thus 1/0 does not exist, or is not defined, or is undefined.

You wish to introduce a new element (or maybe two elements), infinity, which you wish to append to the real number system. That is not prohibited. After all, that is how we got from natural numbers to integers (appending negative integers and zero), and from integers to rationals (appending ratios of integers), and from rationals to reals (appending limits of convergent sequences), and from reals to complexes (appending the square root of -1). What you end up with is not the real number system, however. Furthermore, if you wish to define the four operations + - * and / for this new system, you probably want them to be the same on real numbers, and just add on the definitions of things like infinity + r and r/infinity, for any real number r.

Some of these work fine. It makes sense to define:

```          infinity + r = r + infinity = infinity
(-infinity) + r = r + (-infinity) = -infinity
infinity + infinity = infinity
(-infinity) + (-infinity) = -infinity
infinity - r = infinity
(-infinity) - r = -infinity
r - infinity = -infinity
r - (-infinity) = infinity
infinity - (-infinity) = infinity
(-infinity) - infinity = -infinity
infinity * r = r * infinity = infinity  for r > 0
(-infinity) * r = r * (-infinity) = -infinity  for r > 0
infinity * r = r * infinity = -infinity  for r < 0
(-infinity) * r = r * (-infinity) = infinity  for r < 0
infinity * infinity = (-infinity) * (-infinity) = infinity
infinity * (-infinity) = (-infinity) * infinity = -infinity
infinity / r = infinity  for r > 0
(-infinity) / r = -infinity  for r > 0
infinity / r = -infinity  for r < 0
(-infinity) / r = infinity  for r < 0
r / infinity = 0
r / (-infinity) = 0

```
Where we get into trouble is with defining the following:
```               infinity + (-infinity)
(-infinity) + infinity
infinity - infinity
(-infinity) - (-infinity)
0 * infinity
infinity * 0
0 * (-infinity)
(-infinity) * 0
infinity / infinity
infinity / (-infinity)
(-infinity) / infinity
(-infinity) / (-infinity)
infinity / 0 = infinity
(-infinity) / 0 = -infinity

```
These expressions are called "indeterminate forms." These can all have a large range of different values, depending on exactly where the "infinity" parts came from.

As a result, the system you construct is not closed under addition, subtraction, multiplication, or division.

- Dr. Rob

I want to use 'divide by zero' to indicate a physically impossible task. What does the phrase actually mean?

Well, division by zero is not so much "physically impossible" as it is "in violation of mathematical axioms." You see, the phrase "physically impossible" implies a task that cannot be done, no matter the amount of exertion of effort, whereas the phrase "in violation of mathematical axioms" means that the operation contradicts certain basic assumptions regarding the system in question.

Numbers have certain properties and rules; for instance, we say that adding, subtracting, multiplying, and dividing two numbers will give another number. Subtraction is the opposite of addition, as division is the opposite of multiplication. Any number multiplied by zero gives zero.

There are several of these basic rules, called axioms, and in particular, the kinds of numbers we are familiar with, and do basic arithmetic with, form what mathematicians call a "field." In this field, these rules I have described are called "field axioms." (There are others as well.)

In essence, the field axioms lay down a set of rules, i.e., basic assumptions, about how to put numbers together to get other numbers. And so, division by zero can be shown to contradict these rules (this proof is usually taught in beginning algebra classes.)

Technically speaking, division by 0 is not impossible; rather, it is contradictory to assumption. As such, we disallow it as a valid operation on numbers. "Physically impossible" is a more fitting description of a phenomenon, such as the creation of a perpetual motion machine, or the decrease in entropy of a closed system. Division by 0 is not so much a phenomenon as it is a supposed construction, which is provably contradictory to a given set of rules and therefore not permitted within the system upon which the rules are imposed.

- Doctor Pete

When asked the probability that the 'sky will fall' one student responded '0/0'. What should we have told this 8th grade student about dividing by zero?

To find the probability of something happening, you find out how many different ways the thing you're looking at can happen, and then divide that by the total number of things that can happen. So you'd have zero, the number of different ways the sky can fall (if you really think it can't fall), divided by however many things you think CAN happen, which is probably some positive number. In any case, the denominator can't be zero, since that means that there are zero things that can happen.

The difference between 0/0 and 1/0 is sometimes complicated. For instance, if I look at these two sequences:

```  Sequence A: 5*1  5*1/2  5*1/3  5*1/4  5*1/5
----,------,------,------, ------, ...
1    1/2    1/3    1/4    1/5

Sequence B:  1     1      1      1      1
---, -----, -----, -----, -----, ...
1    1/2    1/3    1/4    1/5
```
The numerator and the denominator in Sequence A are both going to zero, so this sequence should be getting closer and closer to 0/0. But notice that every term in this sequence is 5. So the limit of this sequence is 5. And I could replace the 5 with any number I want, to get whatever limit I want for 0/0. So if you say that the probability of something happening is 0/0, you might be saying that it has probability 1, or .5, or anything at all.

In the second sequence though, notice that we could write it as 1,2,3,4,...which goes to infinity. The only way we can interpret 1/0 (or any nonzero real number over zero) is as positive or negative infinity. In general though, listen to the oft-heard words of advice: DON'T DIVIDE BY ZERO, but with the following addendum: if you do, make sure you're prepared to deal with what happens. In your case, you could have caused the sky to fall, because you said that the probability of it happening might be 1.

- Dr. Ken

More from the Dr. Math archives: