Originally
posted by
Rockman:
The action of taking a square root is NOT well-defined, thus it cannot be used in proofs.
Its like saying that 47=95 because 0=0 and 47*0=95*0
It just makes no sense.
actually no, your example is something different.
sqrt is a specific power (1/2) and it is very well and precisely definsed in mathematics as are all powers and you can definately use them in proofs if you apply them correctly and completely(which is not what I did)
Certain operations alter the domain of possible solutions and hence once you've done that you get a solution set for the new equation that won't work for the old one.
for example if you try to solve
x^2 = 1
x = 1, x = -1
but if you square both sides
x^4 = 1
to which the solutions are:
x = 1, x= -1, x=i, x= -i
but those aren't the solutions to the original equation but you have "created solutions" by raising both sides to a new power.
you can also get situations like this (without complex numbers)
by solving equations in the form (sqrt(ax^2+bx+c))= dx+e
(x is the variable, a,b,c,d,e are real numbers)
or even sqrt(ax+b) = cx
So to summarize the issue isn't with the definition of the functions, it's that we aren't being rigerous enough in the illustrated solutions and hence we end up with the wrong answer.
Division by 0 is undefined for the reason you stated.
as is lim as x->0 of sin(1/x)
:P