# Division by Zero

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## Mistake

Let $x$ and $y$ be numbers such that $y = 0$.

The quantity $\dfrac x y$ is **undefined**.

It is a common mistake to forget this fact when evaluating formulas.

## Also see

- L'Hôpital's Rule: while $\dfrac 0 0$ is undefined, the limit of a rational function whose denominator approaches $0$ is not necessarily undefined.

## Historical Note

Brahmagupta stated that:

*positive or negative divided by cipher is a fraction with that for denominator*.

This was then referred to as **quantity with zero as denominator**.

Mahaviracharya writes in *Ganita Sara Samgraha* of c. $850$ CE:

*A number multiplied by zero is zero and that number remains unchanged which is divided by, added to or diminished by zero.*

which is of course seriously questionable.

## Sources

- 1967: Michael Spivak:
*Calculus*... (previous) ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers: $(\text P 8)$ - 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Equations - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.3$: Arithmetic

- For a video presentation of the contents of this page, visit the Khan Academy.