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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 (
), 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
-
where is the extended complex plane (complex plane extended with , 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.