# 0/0 Anything divided by zero is undefined

In summary, while there is an exception to the first two rules- zero divided by anything is zero- the third rule still applies- anything divided by itself is 1.
Anything divided by zero is undefined (or infinity?), zero divided by anything is zero, and anything divided by itself is 1. so what is zero divided by zero? it seems to me that this must be an exception to at least to of the above rules.

0/0 is also "undefined" but many texts use the term "undetermined" to distinguish that it is a special case: 1/0 is "undefined" because saying 1/0= a would be the same as saying 1= a*0 which is not true, for any a. 0/0 is "undetermined" because saying 0/0= a would be the same as saying 0= a*0 which is true for all a. It is "undefined" but for a different reason- there is no one answer.

The distinction is especially important in dealing with limits. If I need to find the limit of f(x)/g(x) (as x goes to some number, a, f(x), g(x) have limits separately at a) and I naievly take the limits of f(x) and f(y) separately, three things can happen: if the limit of g is non-zero, I just form the fraction to get the limit of the fraction. If the limit of g is zero and the limit of f is not, then the limit does not exist ("undefined"). If the limit of g and f are both zero- then I have to look more closely, the limit may not exist or it may be any number.

Your statement :'zero divided by anything is zero' is not a reliable statement, it is merely a 'rule of thumb' - applicable in most circumstances, but not all.

HallsofIvy said:
0/0 is also "undefined" but many texts use the

Doh! Beat me by seconds! (But personally, I think my answer is more accurate and succinct.)

Look at the$$_ {limit x\rightarrow1} \frac{x^2-1}{x-1}$$

... it seems to me that this must be an exception to at least to of the above rules.
...or the rules are wrong. In the second and third rules, the "anything"s are what's left after applying the first rule; they are not strictly "anything"s. Besides, when you talk about "anything", it is important to specify what set "anything" belongs in. Surely, "horses" and "laziness" don't belong in the set you have in mind.

Gokul43201 said:
... Surely, "horses" and "laziness" don't belong in the set you have in mind.

talking of horses, here's one way of understanding what's going on in terms of apples and oranges - if you had 10 apples and wanted to give an equal no. of them to 5 people, then the number of apples each one gets is 10 /5 =2 oranges.

Extending this, if you had 0 oranges and wanted to give an equal no. to 5 people, the the number each one would get is 0/5 = 0 oranges.

further, if you had 0 oranges to begin with, and wanted to equally distribute that to 0 people, then you can't talk of any specific answer to this problem - hence you say the number is undefined.

hope that helps.

rainbowings said:
talking of horses, here's one way of understanding what's going on in terms of apples and oranges - if you had 10 apples and wanted to give an equal no. of them to 5 people, then the number of apples each one gets is 10 /5 =2 oranges.

Extending this, if you had 0 oranges and wanted to give an equal no. to 5 people, the the number each one would get is 0/5 = 0 oranges.

further, if you had 0 oranges to begin with, and wanted to equally distribute that to 0 people, then you can't talk of any specific answer to this problem - hence you say the number is undefined.

hope that helps.

While this works perfectly for an illustrative example of why any definition of division by zero is "undefined", I think it is misleading.
The actual reason why division by zero is undefined is because undefined is to be taken literally as meaning "It is not defined". This is because the operation of division on the real numbers is defined only for pairs of numbers where the second of the pair is not zero. In other words, ( this is becoming a favorite statement of mine ), asking what "1/0" is, is essentially asking what $\sqrt{a \hspace{3} mouse}$ is. It's sillyness because it's something that is outside the scope ( or domain ) of the operator.
Obviously, my version isn't an intuitive one. Your view is like, "Don't stick your finger in the light socket because you could die". Mine is like, "Don't stick your finger in the light socket because I told you not to." Your's is more likely to prevent people from trying. The problem occurs when someone does. If they don't die, and tell their friends, we're in for a heap of trouble. I can still say, "Ok, so you didn't die. I still told you not to"

The way I like to think about it stems more from linear algebra.
For a number a, define 1/a as the number such that a* (1/a) = 1. There is no
number b that exists such that 0*b = 1, since 0*x=0. If b did exist, it would be
1/0, but it doesn't, and division by 0 is the same as multiplication by 1/0, but
1/0 doesn't exist, so what you've got is a heap of gibberish. All of this occurred
to me in relation to non-invertible matrices.

Lots of people argue about 0/0, but nobody seems to care if the inverse of a
matrix doesn't exist. One can always think of a number as a 1x1 matrix, clears
things up for me at least.

robert Ihnot said:
Look at the$$_ {limit x\rightarrow1} \frac{x^2-1}{x-1}$$

I don't understand what does this have to do with it. Could you please explain? the answer is obviously 2.

$$\frac{Sin(x)}{x}$$

at x=0 above is defined as 1, hence 0/0=1 by logic deduction

AntonVrba said:
$$\frac{Sin(x)}{x}$$

at x=0 above is defined as 1, hence 0/0=1 by logic deduction
You are incorrect.

$$\frac{\sin(0)}{0}$$

Is not defined, as division upon 0 is not defined. However:

$$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$$

There are many limits where you get a situation of looking at the limit of function as it approaches 0/0 and as it goes you get anything like -pi2/e or complex infinity.

## 1. What does it mean when a number is divided by zero?

When a number is divided by zero, the result is undefined. This means that the mathematical operation cannot be completed and there is no answer.

## 2. Why is dividing by zero undefined?

Dividing by zero is undefined because it violates the fundamental rules of mathematics. In basic arithmetic, division is defined as the inverse of multiplication. However, there is no number that can be multiplied by zero to give a non-zero result. Therefore, division by zero is not possible.

## 3. Can't the answer be infinity?

No, the answer cannot be infinity when dividing by zero. Infinity is not a number, it is a concept that represents an unbounded quantity. When dividing by zero, there is no quantity or number that can be used to represent the answer.

## 4. Are there any exceptions to dividing by zero being undefined?

No, there are no exceptions to dividing by zero being undefined. This is a fundamental rule in mathematics that applies to all numbers, whether they are positive, negative, or fractions.

## 5. Is it ever useful to divide by zero?

No, it is not useful to divide by zero. As the result is undefined, it does not provide any meaningful information or solution. In fact, dividing by zero can lead to errors and inconsistencies in calculations and equations.

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