Somewhere along the number line between 3 and 4, there is a short
interval which, when considered in base 7 to 49 decimal places,
comprises a "trap door" which falls through to a different number
system hidden from the general public. The interval is precisely 7^-49
long.

In almost all circumstances,
arithmetic and algebra behave normally.
However, if the result of an operation returns a result in the secret
region (it also applies to complex numbers etc in a somewhat different
way I haven't thought about), the result is mapped onto a number in this
region in a rigorously defined manner. From that point onwards, most
arithmetic or algebraic operations will return results in this region,
which uses its own numerical notation and vocabulary. Furthermore,
hidden within this system itself are a number of further "wormholes"
which access further similar number systems hidden within them and
nested within each other. There are also secret escapes back into the
real, complex and hypercomplex systems of numbers.

Among mathematicians and possibly numerologists, there exists a
Freemasonry alike hierarchical cabal which only allows access to this
special system when they notice someone perform an operation with a
result in the range. This shadowy organisation has a number of
degrees to which one becomes admitted when one finds the other
secret trapdoors. The precise number of degrees in the hierarchy is
knowable but only expressed in terms of the ultimate secret number
system. Lower degrees cannot even comprehend the complexity of the
system used in the higher echelons.

It could be implemented in individual pieces of
software or hardware. For example, getting such a
result in a certain spreadsheet program could
cause it to enter a special mode and send a
notification to the organization headquarters.
Similar logic could even be built into the floating
point processor on the CPU. That would apply this
to all running software but would make contacting
the secret organization to induct the new member
a bit more complicated.

It would be a real bummer if the trapdoor is hit
while doing mission critical calculations for a
manned space flight or something.

Are you suggesting a numerical "Black Hole" number? One which, if encountered in-passing during a calculation which just happens to accidentally come within its clutches, causes the rest of the equation to blip out of existence? The ultimate proof that Turing was wrong, perhaps?