Double Integrity 

Copyright ©1999 by Paul Niquette. All rights reserved.

 
A = B
A2 = AB
A2 - B2 = AB - B2
(A + B)(A - B) = B(A - B)
No reason, just letting A = B.
Multiplying both sides by A.
Subtracting B2 from both sides.
Factoring both sides.
A + B = B
Dividing both sides by A - B
B + B = B
1 + 1 = 1
2 = 1
Substituting A = B
Dividing both sides by B.
Double Integrity

he expression A - B has, under our initial assumption, a precise value: zero. The step that says, "Dividing both sides by A - B" means "Dividing by zero." Just about everyone knows that dividing any finite number by zero results in an quotient of infinity -- a value sometimes described as "undefined" (although "increasing without bounds" may be more appropriate).  Division by zero is not, in any true mathematical sense, forbidden. It is a natural consequence of formulating any ratio in which the denominator can vanish to zero.
    An ideal vertical line has an infinite 'slope,' resulting from innocently dividing any part of its length by the corresponding horizontal distance, which is, of course, zero.
The situation in Double Integrity is a whole lot more interesting. Dating back to antiquity and some unknown originator, this paradox exemplifies mathematical monkey business at its best. 
Or worst.
The two expressions we are dividing by zero are (A + B)(A - B) and B(A-B). Each has a coefficient of A - B and therefore a precise value: zero.
    The sophisticated solver knows that dividing zero by zero results in a quotient of -- well, anything you want it to be. The "undefined" has become "definable at will" ("increasing -- or decreasing -- without limit").
It is easy to imagine practical applications: mathematically proving to a lender, for example, that you can repay your debt with half of what is owed or convincing a cashier to give you change for a dollar bill in the amount of $2.
 


Epilog
As discovered in the solution to Fermat's Really Last Theorem, here is a more promising stratagem for mathematical monkey business...
A = B
A2/2 = B2/2
( A2 )1/2= ( B1/2 )2
 No reason, just letting A = B.
Since 2/2 = 1, nothing changes.
Factoring each side differently.
 ( -12 )1/2= ( -11/2 )2
( +1 )1/2 = ( i )2
 +1 = -1
 No reason, just letting A = B = -1.
Oops, right side went imaginary.
By definition,  i 2 = -1.
So, that buck I owed you?  Well, pay up.


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