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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

from

to

:

As positive

gets very small and tends towards 0,

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

gets very small and tends towards 0,

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

from

to

, then as the

norm of

gets very small and tends towards 0 (thus

tending towards 0 from direction

),

grows towards [complex] infinity from direction

. When we consider the mapping from

to

,

grows towards [complex] infinity from the directions

or

. When we consider

, we are considering

potentially infinite norms (

arbitrarily large but finite norms) so there is a defined direction. When we consider the norm

of

to be absolute 0, then

is undefined (directionless) and

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

, then

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

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. [http://mathworld.wolfram.com/ComplexInfinity.html]
- Weisstein, Eric W., C-*, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/C-Star.html]
- Weisstein, Eric W., Extended Complex Plane, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/ExtendedComplexPlane.html]
- Weisstein, Eric W., Riemann Sphere, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RiemannSphere.html]