Secret Numbers

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.