What is 1 divided by 0?

This is part of a series on common misconceptions.

True or False?

$\frac10$ is undefined.

Why some people say it's true: Dividing by $0$ is not allowed.

Why some people say it's false: $\frac10 = \infty.$

Can you see which of these is the correct explanation?

The statement is $\color{blue}{\textbf{true}}$.

Proof:

If $\frac10 = r$ were a real number, then $r\cdot 0 = 1,$ but this is impossible for any $r.$ See division by zero for more details. $_\square$

There are some common responses to this logic, but they all have various flaws.

Rebuttal: In calculus, $\frac10$ equals $\infty.$

Reply: This statement is incorrect for two reasons. First, infinity is not a real number. The proof demonstrates that the quotient $\frac10$ is undefined over the real numbers.

It is true that, in some situations, the indeterminate form $\frac10$ can be interpreted as $\infty:$ for instance, when taking limits of a quotient of functions. But even this is not always true, as the following example shows:

Consider $\lim\limits_{x\to 0}\frac{1}{x}.$

Approaching from the right, $\lim\limits_{x \to 0^+} \frac{1}{x} = + \infty.$
Approaching from the left, $\lim\limits_{x \to 0^-} \frac{1}{x} = - \infty.$

In order for $\frac{1}{0}$ to be consistent, the limits from both directions should be equal, which is clearly not the case here.

Rebuttal: What about on the Riemann sphere?

Reply: For certain complex functions, it is convenient and consistent to extend their domain and range to ${\mathbb C} \cup \{\infty\}.$ This set has the geometric structure of a sphere, called the Riemann sphere. For instance, suppose $a,b,c,d$ are complex numbers such that $ad-bc\ne 0.$ Then the function $f(z) = \frac{az+b}{cz+d}$ can be extended by defining $f\left(-\frac dc\right) = \infty$ and $f(\infty) = \frac ac$ $\big($or $f(\infty) = \infty$ when $c=0\big).$ This makes $f$ a bijection on the Riemann sphere, with many nice properties.

So there are situations where $\frac10$ is defined, but they are defined in a tightly controlled way. It is still the case that $\frac10$ can never be a real (or complex) number, so—strictly speaking—it is undefined.

See the consequences of assuming that $\frac{1}{0}$ is defined for yourself in the following problem:

See Also

• List of Common Misconceptions

• Indeterminate Forms