## Saturday, December 18, 2004

### 0/0 = ?

If a is NOT equal to zero, we say a/0 is undefined because it has no real solutions. No real number multiplied by 0 gives a non-zero product. Hence we disallow division by zero.

If a =0, then a/0 which is the same as 0/0 is also considered "meaningless" or "undefined", yet it seems to me that 0/0 has an infinite number of solutions, namely the set of real numbers. Any real number multiplied by 0 gives the product zero.

It's true enough that 0/0 doesn't indicate or demarcate any particular real number, but so what? Other operations can have more than one solution, (When X "squared" = 4, then X = +2 or -2), and that's considered perfectly acceptable. [Maybe mathematicians want to set things up so that "fundamental" operations (like +, multiplication, of course, is just repeated addition) have only one solution.] So, this operation, 0/0, has an infinite number of solutions. I guess because 0/0 has an infinite number of solutions it's characterized as "meaningless".

Maybe it's all just a matter of pragmatics in mathematics, insofar as since 0/0 is equal to any real number, it wouldn't do any interesting work.

Update: The above was just written off the top of my head. (Obviously.) As a commenter correctly points out, in some areas of mathematics, division by zero is accomplished. Albeit with different mathematical rules. I should have made it explicit in my posting above that I was referring to the rules in ordinary arithmetic and algebra. See http://en.wikipedia.org/wiki/Division_by_zero for more information. (Note: Wikipedia's reliability cannot be vouched for.)

Actually, this buttresses my point that pragmatics is involved with whether mathematics accepts division by zero. Different rules can be used to achieve different purposes.

Anonymous said...

This is complete rubbish. Mathematicians use zero divisors all the time; consider the fields of commutative algebra, algebraic geometry, algebraic number theory...

December 19, 2004 at 9:57 PM
Peter Wizenberg said...

Thanks for enlightening me.

December 19, 2004 at 9:59 PM
Peter Wizenberg said...

Division by zero outside of ordinary aritmetic and algebra is ONLY accomplished by modifying or changing some of the ordinary rules.

My posting was within the context of the normal rules of algebra and arithmetic. I regret that that was not made sufficiently clear.

December 19, 2004 at 10:09 PM