Division by zero

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Nonzero value divided by zero

There are those who will argue that any nonzero value divided by zero is [complex] infinity (zero divided by zero being undefined), thus zero to a negative power is [complex] infinity, on the evidence of such graphs as of the mapping of

1   x

from

ℝ

to

ℝ

:

As positive

x

gets very small and tends towards 0,

1 / x

does get very large and grows towards [complex] infinity from the positive direction (arbitrary large but finite positive values). On the other side of the vertical axis, as negative

x

gets very small and tends towards 0,

1 / x

grows towards [complex] infinity from the negative direction (arbitrary large but finite negative values). Note that there is only one complex infinity and it has no direction, whereas the potential infinities (arbitrary large but finite values) do have a direction (negative or positive in the case of the real line, any angle in the case of the complex plane). If we consider the mapping of

1 / z

from

ℂ

to

ℂ

, then as the norm

ρ

of

z = ρ e i θ

gets very small and tends towards 0 (thus

z

tending towards 0 from direction

θ

 ),

1 / z = ρ  − 1 e  − i θ

grows towards [complex] infinity from direction

− θ

. When we consider the mapping from

ℝ

to

ℝ

,

1 / x

grows towards [complex] infinity from the directions

θ = 0

or

θ = π

. When we consider

limρ→0 ρ  − 1 e  − i θ

, we are considering potentially infinite norms (arbitrarily large but finite norms) so there is a defined direction. When we consider the norm

ρ

of

z = ρ e i θ

to be absolute 0, then

θ

is undefined (directionless) and

1 / z

is [complex] infinity ($\textstyle {\tilde {\infty }}$), which is as directionless as 0 is. Complex infinity is the point at infinity corresponding to the North pole of the Riemann sphere.

The special case of zero divided by zero

Zero divided by zero is undefined since it can evaluate to any point of the extended complex plane. Assigning a specific value to 0 / 0 leads to contradictions.

If one supposes that 0 / 0 = 1, then

${\begin{array}{l}\displaystyle {{\frac {0}{0}}={\frac {0\,x}{0\,y}}={\frac {0}{0}}\cdot {\frac {x}{y}}={\frac {x}{y}}=z,\quad x,y\in \mathbb {C} ,\,z\in {\hat {\mathbb {C} }},}\end{array}}$

where $\textstyle {{\hat {\mathbb {C} }}=\mathbb {C} \cup \{{\tilde {\infty }}\}}$ is the extended complex plane (complex plane extended with $\textstyle {\tilde {\infty }}$, the complex infinity), also called the Riemann sphere.

External links

• Weisstein, Eric W., Complex Infinity, from MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W., C^*, from MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W., Extended Complex Plane, from MathWorld—A Wolfram Web Resource.
• Weisstein, Eric W., Riemann Sphere, from MathWorld—A Wolfram Web Resource.