RE: Dividing by zero
September 7, 2014 at 12:13 pm
(This post was last modified: September 7, 2014 at 12:18 pm by Alex K.)
Can't we simply say that dividing by zero is an abomination unto God, and leave it at that???
...damn, wrong forum.
One comment though (I hope noone made it yet):
What one can and cannot do is always a question of definitions.
One can for example extend the field of real numbers with "+" and "*" as operations by a quantity called infinity,
http://en.wikipedia.org/wiki/Alexandroff_extension
and then declare that by definition to be the value of x/0 (though what is 0/0 then...). The problem is that no matter how one defines x/0 in one's extension of the real numbers, one invariably loses cherished properties of the field of real numbers which one would like to preserve in order to do calculations with it - such as every number having a unique inverse with respect to "+" and "*".
So I'd rephrase the question slightly from
"why can't I divide by zero"
to
"Can't I consistently define the division by zero in some generalization of the field of real numbers without losing certain properties?".
The latter is a well-defined question which can be proven, and the answer is: no, you can't.
Residue theory was invented in the 19th century. It's also kindergarten stuff compared to the things invented in the 20th century. Algebraic topology anyone?
...damn, wrong forum.
One comment though (I hope noone made it yet):
What one can and cannot do is always a question of definitions.
One can for example extend the field of real numbers with "+" and "*" as operations by a quantity called infinity,
http://en.wikipedia.org/wiki/Alexandroff_extension
and then declare that by definition to be the value of x/0 (though what is 0/0 then...). The problem is that no matter how one defines x/0 in one's extension of the real numbers, one invariably loses cherished properties of the field of real numbers which one would like to preserve in order to do calculations with it - such as every number having a unique inverse with respect to "+" and "*".
So I'd rephrase the question slightly from
"why can't I divide by zero"
to
"Can't I consistently define the division by zero in some generalization of the field of real numbers without losing certain properties?".
The latter is a well-defined question which can be proven, and the answer is: no, you can't.
(October 1, 2013 at 12:20 pm)little_monkey Wrote: Taking limits is a concept invented in the 17th century. By now, it's really kindergarten stuff compared to complex integration and Residue theory.
Residue theory was invented in the 19th century. It's also kindergarten stuff compared to the things invented in the 20th century. Algebraic topology anyone?
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition
