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Why does X divided by zero not equal X? After all, X is not being divided by anything.

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By that logic, X times zero would be X, since X is not being multiplied by anything. It is not always valid to think of zero as "nothing," or at least it is not always valid to think of it as "not anything." Certainly it is something. –  Qiaochu Yuan Nov 21 '10 at 1:54
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not anything = null and not zero. –  ykombinator Nov 21 '10 at 6:06

5 Answers 5

up vote 12 down vote accepted

Think of division as an "inverse" operation to multiplication.

For instance, let us look at $\frac{a}{b}$ means.

Let $x = \frac{a}{b}$. What it means is when $x$ is multiplied by $b$ it gives $a$.

So, if you want to make sense of $\frac{a}{0}$.

Say $\frac{a}{0}$ is $y$.

So, we want $y$ such that $y \times 0$ gives us $a$.

But we know that $y \times 0 = 0$, $\forall y$.

So there is a logical inconsistency here.

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if division is the inverse of multiplication and Y times zero equals zero then Y divided by zero should also be zero but it isn't. this logic doesn't hold up to the foundation its built on. –  RandomDownVoteVictim Aug 7 '13 at 1:19
    
@StrandedPirate: "Y divided by zero should also be zero" - why? –  Johannes Kloos Sep 10 '13 at 19:57
    
@StrandedPirate If $a \times 0 = b$ gives that $b = 0$, then $\frac{b}{0} = a$ doesn't necessarily mean that $a = 0$. Rather, the operation $\frac{b}{0}$ would only be allowed when $b = 0$ and $a$ could be anything, following from the original statement of $a \times 0 = 0$. In practice, the value of $\frac{x}{0}$ is in general undefined and can sometimes be calculated as a limit ($\frac{x}{0}$ is approaching a value) based on how you ended up with the statement. –  nitro2k01 Mar 4 at 13:03

I would say that if you're dividing by 1, then you're "not really dividing", which is why $X/1=X$. Dividing by $0$, however, is undefined. Here's a way to see this based on basic properties of arithmetic. Suppose we could define, say $3/0$ as a real number $r$. Then multiplying both sides of the equation $3/0=r$ by $0$ yields $3=0\cdot r$. But $0\cdot r=0$ for all real numbers $r$, so this is impossible.

Informally, dividing $X$ by $n$ tells you how big to make the pieces of pie if you have to serve them to $n$ people. This means that $n$ pieces of size $X/n$ give a total amount of $X$. But if $n=0$, then no matter the size of each piece, $0$ of them can never give a total amount of $X$. (If $X=0$, then the reasoning becomes more intricate.)

(This reminds me of another confusion involving $0$ and language. I have seen a temptation to replace "the solution is 0" with "there is no solution," and vice versa among precalculus or beginning calculus students.)


Added:

Perhaps this problem of contrasting dividing by $0$ and "not dividing by anything" can be further clarified by comparison with what happens when adding and subtracting. When you add $0$, you're "not adding anything", and sure enough $X+0=X$ holds for all $X$. Similarly for subtraction: $X-0=X$, and here the interpretation "not subtracting anything" works out. This works because $0$ is a neutral element with respect to addition (by definition). Changing operations to multiplication, $0$ is no longer the neutral element; $1$ is. Multiplying by $1$ is what translates to "don't change anything": $X\cdot 1=X$. Thus "undoing" multiplication by $1$, naming dividing by $1$, also doesn't change anything. But since $0$ no longer has this innocuous behavior with respect to multiplication, things similarly change for division. It changes drastically, because multiplication by $0$ becomes something that can't be undone.

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Related to that, I've seen students confuse $\{\}$ with $\{0\}$... –  J. M. Nov 21 '10 at 1:46
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The difference between zero and an undefined (or NULL) result is a common source of confusion in the programming world too –  friedo Nov 21 '10 at 4:23

I just want to mention that in certain situations it is perfectly valid to think of something nonzero divided by something zero as "infinity." For example, the slope of the line between the points $(a, b)$ and $(c, d)$ is $\frac{d - b}{c - a}$, unless $c - a = 0$ and $d - b \neq 0$, where this expression stops making sense. However, the corresponding line is vertical, and it is perfectly reasonable to think of its slope as being "infinity." More precisely, the space of slopes is a gadget called the projective line $\mathbb{P}^1(\mathbb{R})$ and if all one cares about is the geometric structure of this space (that is, one is not too concerned with adding and multiplying) it is perfectly valid to think about "infinity" in the same way as any other element, since they're all taken to each other by projective transformations. These ideas are very important in certain branches of mathematics, such as algebraic geometry.

A more concrete application is that the tangent function can be thought of as taking values in $\mathbb{P}^1(\mathbb{R})$ instead of in $\mathbb{R}$, and then it no longer has "singularities"; in fact, it defines a homeomorphism from the circle to $\mathbb{P}^1(\mathbb{R})$. Doing the same thing over $\mathbb{C}$ allows you to think of meromorphic functions as functions to the Riemann sphere $\mathbb{P}^1(\mathbb{C})$, and this idea also has many fruitful applications.

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+1. Tangential to your post, but I'm tempted to mention it because it is another common confusion among beginning students involving $0$ and language. We may tell students that the line between $(a,b)$ and $(a,d)$ does not have a slope (rather than saying that the slope is infinite), whereas the slope of the line between $(a,b)$ and $(c,b)$ is $0$. But to some beginners, the distinction between "no slope" and "slope $0$" is not always clear. –  Jonas Meyer Nov 21 '10 at 2:15

The best way to think of it is check out what happens when you divide by the numbers near zero. For example $$\frac{1}{1} \;\;\text{and} \;\; \frac{1}{-1}$$ Then get closer to zero on either side. For example $$\frac{1}{.5} \;\;\text{and} \;\; \frac{1}{-.5}$$. Then even closer. $$\frac{1}{.1} \;\;\text{and} \;\; \frac{1}{-.1}$$. Do you see how the numbers are getting further and further apart? This is why we have no clue what dividing by zero might mean.

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Then that just proves that the "equalizing" methodology is flawed, if I have ten fingers, and divide my fingers by zero, I logically should still have ten fingers, as I've done nothing to them, to say that they magically got erased is incredibly stupid, and I don't care what "rule" says that's wrong, because it doesn't make sense any other way. –  MarcusJ Aug 13 '13 at 21:13
    
OK... If you say you have 10 fingers and divide your fingers by a rhinoceros, you logically should still have ten fingers. You've done nothing to them and they did not get magically erased! See, just like dividing by 0, dividing by a rhinoceros is undefined. It's not that you're doing nothing - we just having defined what it is your doing, so it's kind of like you're doing nothing at all (but not exactly). –  Imray Aug 13 '13 at 21:34

Who says X divided by zero does not equal X? Let's challenge the conventional thinking (wording) and agree that, yes, if you divide something by nothing, there is no division. Likewise, if you multiply something by nothing, there is no multiplication. Don't treat zero as though it were a number, as though it were a part of the continuity of infinitely shrinking numbers less than one. It's not. It's a limit those numbers will never reach. Zero is not another point on a line. It's a gap. It comes from the Sanskrit word meaning "empty."

Divide that apple on the table by ... nothing at all, and you've performed no operation. You're left with the apple, whole and undivided. Seems perfectly logical. You've done nothing to the apple. You've done something to the operation, which was to abandon it. So let's go with that and invite Yuan or another contributor to explain more convincingly why zero isn't the same as nothing.

That's going to be a hard sell, don't you think? Of course, I like to argue negative numbers don't exist because you can't have less than nothing; you can't take two apples away from one. But that's another topic.

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I'm sorry, but your analogy makes little sense. Multiplication, division and zero are well-defined mathematical concepts that you cannot simply discuss away or re-interpret as you like. –  Johannes Kloos Sep 10 '13 at 19:54

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