Division by Zero

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
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