Zero divided by zero must be equal to zero

Question

What is wrong with the following argument (if you don't involve ring theory)?

Proposition 1: $\frac{0}{0} = 0$

Proof: Suppose that $\frac{0}{0}$ is not equal to $0$

$\frac{0}{0}$ is not equal to $0 \Rightarrow \frac{0}{0} = x$ , some $x$ not equal to $0$ $\Rightarrow$ $2(\frac{0}{0}) = 2x$ $\Rightarrow$ $\frac{2\cdot 0}{0} = 2x$ $\Rightarrow$ $\frac{0}{0} = 2x$ $\Rightarrow$ $x = 2x$ $\Rightarrow$ $ x = 0$ $\Rightarrow$[because $x$ is not equal to $0$]$\Rightarrow$ contradiction

Therefore, it is not the case that $\frac{0}{0}$ is not equal to $0$

Therefore, $\frac{0}{0} = 0$.

Q.E.D.

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Update (2015-12-01) after your answers:

Proposition 2: $\frac{0}{0}$ is not a real number

Proof [Update (2015-12-07): Part 1 of this argument is not valid, as pointed out in the comments below]:

Suppose that $\frac{0}{0}= x$, where $x$ is a real number.

Then, either $x = 0$ or $x$ is not equal to $0$.

1) Suppose $x = 0$, that is $\frac{0}{0} = 0$

Then, $1 = 0 + 1 = \frac{0}{0} + \frac{1}{1} = \frac{0 \cdot 1}{0 \cdot 1} + \frac{1 \cdot 0}{1 \cdot 0} = \frac{0 \cdot 1 + 1 \cdot 0}{0 \cdot 1} = \frac{0 + 0}{0} = \frac{0}{0} = 0 $

Contradiction

Therefore, it is not the case that $x = 0$.

2) Suppose that $x$ is not equal to $0$.

$x = \frac{0}{0} \Rightarrow 2x = 2 \cdot \frac{0}{0} = \frac{2 \cdot 0}{0} = \frac{0}{0} = x \Rightarrow x = 0 \Rightarrow$ contradiction

Therefore, it is not the case that $x$ is a real number that is not equal to $0$.

Therefore, $\frac{0}{0}$ is not a real number.

Q.E.D.

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Update (2015-12-02)

If you accept the (almost) usual definition, that for all real numbers $a$, $b$ and $c$, we have $\frac{a}{b}=c$ iff $ a=cb $, then I think the following should be enough to exclude $\frac{0}{0}$ from the real numbers.

Proposition 3: $\frac{0}{0}$ is not a real number

Proof: Suppose that $\frac{0}{0} = x$, where $x$ is a real number.

$\frac{0}{0}=x \Leftrightarrow x \cdot 0 = 0 = (x + 1) \cdot 0 \Leftrightarrow \frac{0}{0}=x+1$

$ \therefore x = x + 1 \Leftrightarrow 0 = 1 \Leftrightarrow \bot$

Q.E.D.

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Update (2015-12-07):

How about the following improvement of Proposition 1 (it should be combined with a new definition of division and fraction, accounting for the $\frac{0}{0}$-case)?

Proposition 4: Suppose $\frac{0}{0}$ is defined, so that $\frac{0}{0} \in \mathbb{R}$, and that the rule $a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$ holds for all real numbers $a$, $b$ and $c$. Then, $\frac{0}{0} = 0$

Proof: Suppose that $\frac{0}{0}=x$, where $x \ne 0$.

$x = \frac{0}{0} \Rightarrow 2x = 2 \cdot \frac{0}{0} = \frac{2 \cdot 0}{0} = \frac{0}{0} = x \Rightarrow x = 0 \Rightarrow \bot$

$\therefore \frac{0}{0}=0$

Q.E.D.

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Suggested definition of division of real numbers:

If $b \ne 0$, then

$\frac{a}{b}=c$ iff $a=bc$

If $a=0$ and $b=0$, then

$\frac{a}{b}=0$

If $a \ne 0$ and $b=0$, then $\frac{a}{b}$ is undefined.

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A somewhat more minimalistic version:

Proposition 5. If $\frac{0}{0}$ is defined, so that $\frac{0}{0} \in \mathbb{R}$, then $\frac{0}{0}=0$.

Proof: Suppose $\frac{0}{0} \in \mathbb{R}$ and that $\frac{0}{0}=a \ne 0$.

$a = \frac{0}{0} = \frac{2 \cdot 0}{0} = 2a \Rightarrow a = 0 \Rightarrow \bot$

$\therefore \frac{0}{0}=0$

Q.E.D.

Answer 1

The error is in the very first implication. If $0/0$ is not equal to zero, there is no reason why $0/0$ must equal some $x$. There is no reason to believe that we can do this division and get a number.

Therefore you have a proof that $0/0$ cannot equal any nonzero $x$. Combine this with a proof that $0/0$ cannot equal zero, and you have proved that $0/0$ is not a number.

Answer 2

Your proof assumes that $0/0$ is a number since your argument involves arithmetic operations.

Now, if we assume the usual rules of arithmetic for the rationals, why can't we do this: $$ \frac00+\frac11=\frac{0\times1+1\times0}{1\times0}=\frac00. $$ Then $$ 1=\frac11=0. $$

Answer 3

In order to prove anything whatsoever about $\frac{0}{0}$, we need a definition of $\frac{a}{b}$ where $a$ and $b$ are, say, integers. The correct definition is $ \frac{a}{b} = ab^{-1}$, where the definition of $b^{-1}$ is the number (which is unique, when it exists, according to some basic properties of numbers) such that $bb^{-1} = 1$. But then one can see (again, assuming some basic properties of numbers) that $0^{-1}$ does not exist, so $\frac{0}{0}$ isn't even defined.

Answer 4

You can define $\frac{0}{0}$ to be anything you want; it can be $0$, or $1$, or $\pi$. But here's the catch: we want the choice of value for $\frac{0}{0}$ to be compatible with the usual laws of arithmetic.

From this standpoint, what you've really proved is that if $\frac{0}{0}$ is anything except $0$, then the law $2\frac{0}{0} = \frac{2 \cdot 0}{0}$ cannot hold. Therefore, the law: $a\frac{b}{c} = \frac{ab}{c}$ cannot hold, either. This suggests that simply defining $\frac{0}{0}=0$ might actually be a good idea. Unfortunately, this breaks another law of arithmetic, namely $\frac{a}{a} = 1.$

Since we cannot have the best of both worlds, perhaps it is best to simply leave $\frac{0}{0}$ undefined.

Actually, I think the best solution is to define division not of real numbers, but of affine subsets of $\mathbb{R}$. These are: the singleton subsets of $\mathbb{R}$, the empty subset, and $\mathbb{R}$ itself. So by passing to the affine subsets, we've effectively adjoined two new "points", namely $\emptyset$ and $\mathbb{R}$.

Now define that given affine subsets $Y$ and $X$ of $\mathbb{R}$, we have:

$$\frac{Y}{X} = \{r \in \mathbb{R} \mid Y \supseteq rX\}$$

You can check that the result of dividing one affine subset by another will itself always be affine.

Under these conventions, we have:

$$\frac{0}{0} = \mathbb{R}, \qquad \frac{1}{0} = \emptyset$$

This justifies the intuition that trying to divide $0$ by $0$ is somehow different from trying to divide a non-zero number by $0$.

Full disclosure: although passing to the affine subsets works algebraically, I'm not entirely sure how to put a topology or uniform structure or metric on the collection of affine subsets of $\mathbb{R}$. Until we can figure out how to do this, my proposed solution probably isn't that useful.

Answer 5

In my experience, when people have a problem with $\frac 0 0$ it generally means they have not developed a clear idea of the way mathematics forms models.

The character of mathematical systems

At first, every expression a pupil meets can be evaluated to a number they understand, but fairly soon $\frac 0 0$ and $\sqrt -1 $ turn up, yet they cling to the feeling that “if you can write it, it must mean something”. The point has to be made that we define what expressions mean in such a way as to help us solve problems, and sometimes these definitions just can’t be made to cover all cases without implying weird behaviour, and so we agree to consider some cases undefined. This makes maths sound a rather arbitrary process, but the constraint of solving problems actually makes it about the least contingent human activity there is. Of course, having made some definitions, we experiment with their implications, which gives rise to new problems, so pure mathematics feeds off itself as well as off applications. We discover that we can create things in an inevitable way.

Algorithmic v. solution seeking operations

We can distinguish algorithmic operations such as addition, multiplication and raising to powers, from solution-seeking operations such as subtraction, division and the taking of roots. Operations in the first class are defined by some terminating algorithm (such as repeated increment, addition or multiplication), often in the form of a recursive definition, which we show terminates. Operations in the second class are defined as solutions to equations, i.e. they are required to yield numbers which, when combined using previously defined operations, yield a certain result.

More generally, we may have conditions which are not written as equations, and we may be operating on other entities than ‘numbers’ (whatever they may be!¹).

New numbers to make insoluble equations soluble

Sometimes we find that our condition can be satisfied by members of the set defined so far. For example, given numbers $a$ and $b$, we want to solve $ a + n =b $ for $n$, and call that $ b - a $; then, since $ 1 + 2 = 3 $, we say $ 3 - 1 = 2 $. At other times, we find that there is no such solution, e.g. there is no natural (counting) number $n$ such that $ 3 + n = 2 $ — this is when we ask ourselves if we can define $ 2 - 3 $. Thus we move from natural to integers.

Given a type of equation we are trying to solve, we hope we can define the new ‘numbers’ in a useful way:

• They should include the old numbers, or something very like them.
• We want to be able to apply the existing operations to them.
• We want them to obey they same rules that the old numbers did.

As goblin’s answer said,

You can define $\frac 0 0$ to be anything you want; it can be 0, or 1, or π. But here's the catch: we want the choice of value for $ \frac 0 0 $ to be compatible with the usual laws of arithmetic.

Actually, though you can define it as you want, it may make you rather unpopular if you insist on something really quirky, so perhaps it is better to say “one could” etc.

Two ways of formalising a new number system

Typically we find that obeying the existing rules means that several expressions, such as $ 2 - 3 $ and $ 3 - 4 $, should have the same value. One way we deal with that is to define equivalence classes of pairs of numbers (if the operation is binary) that should yield the same value; we then show that we can define the various operations on the pairs so that they respect the classes and work as in the original situation.

Another way to deal with it is to introduce one or two new symbols and a syntax for combining them: we might then define integers as formal expressions of the form $+n$ or $-n$, and show that these formal expressions behave as desired. Similarly, introducing $i$ lets us define complex numbers.

These are two important ways to build a formal ‘working model’ of a number system, but there are many others, such as Dedekind cuts, von Neumann’s construction of $\mathbb N$ as ‘free sets’ and Conway’s construction of Surreal numbers.

The error in the question

The problem with $\frac 0 0$ in this question arises partly from failing to recognise that the formal expressions we introduce have only those properties we (consistently) define them to have and anything that follows from those. A bigger problem, though, is not realising that the formal expressions do not automatically mean anything, until we have shown that they reproduce the old numbers as a subset. Once we realise that they do not automatically mean anything, it is easier to accept that our definition may exclude expressions like $\frac 0 0$ (or call them ‘undefined’) in order to ensure desirable behaviour of the new number system.

Another error in the question as first formulated was to assume that the new expressions behave like the numbers they were constructed from, rather than considering that as something to be investigated.

The solution set approach

Goblin’s answer, (after making the essential point that we decide how to define expressions) went on to suggest an alternative definition of division in terms of the solution set of $a \times n = b$. This suggests you consider $ \frac 0 0 $ as $ \mathbb R $ (why not $ \mathbb Q $ or $ \mathbb C $ … or even $ \mathbb Z $ or $ \mathbb N $ ?), but it does not help you deal with $n^2 = -1$, which needs the approach described above.

Notes

¹ As Dedekind asked, Was sind und was sollen die Zahlen? “What are numbers and what should they be?” or just possibly “… what is the point of them”!

Answer 6

How do you define a fraction?

The definition I know is that for any number $a \neq 0$ we say the symbol $\frac{1}{a}$ satisfies $a \cdot \frac{1}{a} = 1$. One can show that this is well defined for any $a \neq 0$ and unique. Further, the symbol $\frac{b}{a}$ for any numbers $b,a$ with $a\neq 0$ is defined as $\frac{b}{a}:=b \cdot \frac{1}{a}$.

If $\frac{1}{0}$ would exists, then clearly by definition we would have $$\frac{0}{0}=0 \cdot \frac{1}{0} = 1.$$ However, the symbol $\frac{1}{a}$ is not definied for $a=0$, because there exists no number $b$ with $0 \cdot b = 1$, so there is no value to which we can assign the symbol $\frac{1}{0}$.

Example of the above definition:

We can compute $\frac{1}{2}$, because we know

$$ 2 \cdot \frac{1}{2} = \frac{1}{2} +\frac{1}{2} =1$$ and since

$$ 2 \cdot 0.5 = 0.5 + 0.5 = 1 $$ it follows that $$\frac{1}{2} = 0.5$$ holds.

Answer 7

If $\frac00 = 0$ then $0\times0=0$ would work as a proof. The problem is that also $3\times0=0$ and $7\times0$ and so on would all work.

So $\frac00=0,\space\frac00=1,\space\frac00=3,\space\frac00=$ just about anything.

There are too many possible results, we want such operations to return defined results (one and only one result), otherwise the operation itself is not actually defined. In fact, dividing any number by $0$ is 'undefined', 'illegal'.

When dividing a non-zero integer by zero, we have the same problem. only, in that case, we simply have no-results.

ie: $\frac60 =$ number which multiplied by $0$ equals $6$.

We can say $\frac60=\infty$ only to cut short on a theorem, stating that the limit of $\frac mn$ with m integer and $n$ going to $0$ is infinite. It is obviously a notation-artifact. Otherwise we should say that $\infty\times0$ equals $6$, and $7$, and $8$, and $9$...

Answer 8

The first fault is in the very statement. You assume that there is a $0/0$ in the first place, you have to have that before you even assume that it's not equal to zero.

Then even if there were a $0/0$ you're assuming that there is a $2{0\over0}$ and then that it is equal to ${2\cdot 0\over0}={0\over0}$.

Finally you assume that if $2x=x$ you must have that $x=0$, even if you've got this far there's no reason to think that $x$ is a number and must obey the rules for numbers.

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Your updated proposition is true, but your proof has some issues.

In the first case you use the identity ${a\over b}+{c\over d} = {ad+bc\over bd}$ which is only shown to be true if $bd\ne0$.

In the second case you use the identity $a{b\over c} = {ab\over c}$ which is only shown to be true if $c\ne 0$.

That is you use twice identities where the prerequisite is not fulfilled.

Answer 9

The multiplicative semigroup of a field is not a group, but only an inverse semigroup. In the context of semigroups, we call $y$ an inverse element of $x$, if $x=xyx$ and $y=yxy$. An inverse semigroup is a semigroup where each element has an unique inverse element, such that an inverse operation $^{-1}$ can be defined via $x=xyx \land y=yxy \Leftrightarrow y=x^{-1}$. The unique multiplicative inverse element of $0$ in this context is $0$, so $\frac{0}{0}=0\cdot 0^{-1}=0\cdot 0=0$. But note that we also have $\frac{1}{0}=0$.

Note however that the rules for dealing with fractions get more complicated, because the sum of two symbolic fractions is not necessarily a pure symbolic fraction: $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}+0_d\frac{a}{b}+0_b\frac{c}{d}$. Here $0_x:=1-\frac{x}{x}$ is idempotent, because $\frac{x}{x}$ is idempotent, i.e. $0_x^2=1-2\frac{x}{x}+\frac{x^2}{x^2}=1-\frac{x}{x}=0_x$. The details of this standard representation are are worked out in a paper by Bergstra et al.

Inverse semigroups are the appropriate generalization of groups if we look at partial bijections instead of total bijections. In the context of rings, a ring whose multiplicative semigroup is an inverse semigroup is called a strongly regular ring. The category of strongly regular rings is related to the category of skew fields in a similar way as the integers are related to the prime numbers (and the same applies to the relation of the category of regular commutative rings to the category of fields). The name meadow has also been proposed for such rings. I have collected some scattered informations on those subjects in a blog post.

If we add the universal formula $a = 0 \lor (ab = ac \Rightarrow b = c)$ characterizing integral domains to the equations characterizing regular commutative rings, then we end up with a finite set of universal formulas characterizing the category of fields, as I learned from a paper by Hiroakira Ono. From this we can learn that the class of fields is closed under isomorphism, substructures and ultraproducts, see theorem 2.20 in chapter V "Connections with Model Theory" of Burris and Sankappanavar's A Course in Universal Algebra.

Now goblin decided to give an answer with a slightly different spin, but goblin has played with meadows himself in the past. He probably noticed that inversion defined like this is not a continuous function, and so kept on searching. I decided to mention this, not in order to avoid partial functions, but in order to point out that partial functions (and especially partial bijections) do have nice properties on their own, which are different from the properties of total functions.

It is important to keep in mind that this definition of $\frac{x}{0}$ leads to discontinuous operations, so it cannot be used to evaluate limits. And for meromorphic functions, using this definition would even destroy part of the inherent structure, but declaring division by zero as undefined will do so too.

Answer 10

Using the definition of a Ring, it is pretty simple to show that $$\forall a, 0\times a=a\times 0=0$$ Thus, If we were to define division (only in the case where $c=ab=ba$): $$\forall a, a=\frac{0}{0}$$ Which can be true if and only if your ring is the Zero ring $$\{0\}$$ Therefore, when $\frac{0}{0}$ exists, it is actually $0$ (but there aren't any other numbers in the ring)

Answer 11

Actually there are solid arguments why $0/0$ can be defined as $0$.

Here is a graphic of the function $f(x,y)=x/y$:

The function is odd against both $x$ and $y$ variables.

Along the axis $x=0$ it is constant zero.

Along the axis $y=0$ it is unsigned infinity, but its Cauchy principal value is constant zero.

Along the diagonal $y=x$ it is constant 1.

Along the diagonal $y=-x$ it is constant -1.

As all odd functions its behavior around zero does not have any preference to positive or negative values. As all odd analytic functions, this function in zero(if defined) can either be zero or unsigned infinity. But unsigned infinity is not a part of affinely extended real line or complex plane, which is usually used in analysis (it is part of projectively extended real line/complex plane). As such, the only variant that remains is $0$.

Note also that we can define a Cauchy principal value of a two-variable function in a point $(x_0,y_0)$ as a limit of the following integral:

$$\lim_{r\to 0}\frac 1{2\pi}\int_0^{2\pi}f(x_0+ r \sin (t),y_0+r \cos(t)) dt$$

This integrates the values of a function over a circle around a point. For analytic functions as long as the radius of the circle r comes to 0, the value of the integral comes to the value of the function in $(x_0,y_0)$. For our function, in $x_0=0, y_0=0$ the value of the integral is constant zero (because the function is odd) and does not depend on r. Consequently, the limit is also zero.

We can extend our analysis to complex numbers, but we again will get the same result, $0$.

Answer 12

The answer to this might be better understood in layman's terms. Going back to grade-school level word puzzles, $\frac{x}{0}$ means:

If you have $x$ items, and you distribute them equally among $0$ people, how many items does each have?

The answer is not a number—it is "I can't answer that" because it rests on an invalid assumption, exposed by noticing that each doesn't refer to any people. The correct response is "each who?" or "there are no people!"

Just because a math problem is expressed using normal numbers and operators, taking the form of other valid math problems to which an answer is possible, doesn't mean that it must have an answer nor that the answer is a number. Division by 0 really means "don't divide."

What's the result of a division where you don't divide? Whatever the answer is, it's $NaN$—not a number.

Using a logic analogy (since math is ultimately just a specialized branch of logic), if someone asks you a question that normally has a true/false answer, such as "Have you stopped beating your wife?" but the premise is wrong because you have never done so, then neither true nor false is a correct answer. It is either the case that the answer is outside the domain of Boolean, or there is no answer at all. One cannot do further Boolean or true-false logic on such a non-Boolean. Attempting to do so is a category mistake.

Similarly, doing any number math with the result of the requested operation $\frac{x}{0}$ is also a category mistake. Proceeding as though it must equal some number $n$ is to act like "What does the color blue taste like?" is a sensible question. But blue doesn't taste like anything, because it isn't a flavor.

• If you don't have a temperature, what is your temperature?
• If you punch no one in the nose, how mad will they all be?
• If you forgot to buy a coffee cup, does your empty cup have 0 ml caffeinated or 0 ml decaffeinated coffee in it?
• Of all the times you've successfully committed suicide, which one hurt the most?
• Were all the parents you never had blue-eyed or green-eyed?
• How many ccs does a little white lie occupy?

Answer 13

The problem is that you assume $\frac00$ is = some $x$ belong to the real number group.$\frac00$ is just not defined for you to assume it to be $=x$.

Basically its one of the seven indeterminate forms of mathematics.Look here for the complete list-https://en.wikipedia.org/wiki/Indeterminate_form#List_of_indeterminate_forms

Answer 14

Try to cancel 0/0 with different numbers each time you will get different solutions for 0/0 . Suppose 1×0/1×0 you get solution 1. Suppose 2×0/0×1 you get solution 2 . Suppose n×0/0×1 you get solution n. ∵0/0 has infinitely many solutions 0/0 is not defined.

Answer 15

0/0 is indeterminate. Which means it has infinite number of values.

(0)(1) = 0     =>     0/0 = 1
(0)(2) = 0     =>     0/0 = 2
(0)(4) = 0     =>     0/0 = 3 .....

From above it can be shown that 0/0 has infinite number of values which make it meaningless. Hence the term.

Answer 16

Well, then $n \cdot 0=0\; \forall \mathbb{R} \implies \frac{0}{0}=n=\frac{1}{n}$ so, $\frac{0}{0}$ is pretty much any real number. This is definitely not the case. This is why it is undefined.