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Secret Numbers

Hidden compartments in the number line
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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.

nineteenthly, Mar 02 2015

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       Not an idea [-]
Voice, Mar 02 2015
  

       So how's the herb garden coming along ?
FlyingToaster, Mar 02 2015
  

       The herb garden is now on the other side of the city and I'm weeding instead.   

       What counts as an idea then?
nineteenthly, Mar 02 2015
  

       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.
scad mientist, Mar 02 2015
  

       [scad] similar tricks have been done for DRM. No dongle and your 10-decimal-point-precision calculations are secretly being done to 3.
the porpoise, Mar 02 2015
  

       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?
Dub, Mar 02 2015
  

       "Let's imagine the Pentium FDIV bug exists, not in silico, but in vivo"   

       What [Voice] said.
lurch, Mar 02 2015
  

       Might be good for cryptography. Like imagine a number that's not quite pi, by being off one number in a random decimal place.
RayfordSteele, Mar 03 2015
  
      
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