A - B has, under our initial assumption, a precise value:
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
The two expressions we are dividing by zero are
+ 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, . 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.
As discovered in the solution to Fermat's
Really Last Theorem, here is a more promising stratagem for mathematical
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
+1 = -1
|No reason, just letting A = B = -1.
Oops, right side went imaginary.
By definition, i 2
So, that buck I owed you? Well, pay up.