What is 0 divided by 0?

This is part of a series on common misconceptions.

What is $\frac00?$

Why some people say it's 0: Zero divided by any number is 0.

Why some people say it's 1: A number divided by itself is 1.

Only one of these explanations is valid, and choosing the other explanations can lead to serious contradictions.

The expression is $\color{#D61F06}{\textbf{undefined}}$.

Here's why:

Remember that $\frac{a}{b}$ means "the number which when multiplied by $b$ gives $a.$" For example, the reason $\frac{1}{0}$ is undefined is because there is no number $x$ such that $0 \cdot x = 1.$

The situation with $\frac{0}{0}$ is strange, because every number $x$ satisfies $0 \cdot x = 0.$ Because there's no single choice of $x$ that works, there's no obvious way to define $\frac{0}{0}$, so by convention it is left undefined. $_\square$

Of course, there are many possible counterarguments to this. Here are a few common ones:

Rebuttal: Any number divided by itself is $1.$

Reply: This is true for any nonzero number, but dividing by $0$ is not allowed.

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Rebuttal: $0$ divided by any number is $0.$

Reply: This is true for any nonzero denominator, but dividing by $0$ is not allowed no matter what the numerator is.

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Rebuttal: Any number divided by $0$ is $\infty.$

Reply: Even for nonzero $y,$ writing $\frac{y}{0}=\infty$ is not entirely accurate: see 1/0 for a discussion. But this reasoning only makes sense for a nonzero numerator.

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Rebuttal: If we choose to set $\frac00=1,$ or $0,$ it is not inconsistent with other laws of arithmetic, and it makes one of the rules in the above rebuttals true in all cases.

Reply: This is a combination of the first two rebuttals, so here is a "big-picture" reply. Any specific choice of value for $\frac00$ will allow some function to be extended continuously. For instance, if we mandate $\frac00=1,$ then the function $f(x) = \frac xx$ becomes continuous at $x=0.$ If $\frac00=0,$ the function $f(x)=\frac0x$ becomes continuous at $x=0.$

But this is not satisfactory in all cases, and the arbitrariness of the choice will break other laws of arithmetic. For instance,

$$\begin{aligned} \frac00 + \frac11 &= \frac{0\cdot 1+1\cdot 0}{0\cdot 1} \\ &= \frac00, \end{aligned}$$

which doesn't make any sense for any (finite) choice of $\frac00.$

Introduction of terms like $\frac{0}{0}$ in otherwise sound arguments can break them down. See if you can spot the error in the problem below:

See Also

• List of Common Misconceptions

• Indeterminate Forms