Your thoughts on Division by 0?

[tition]

Also the definition of multiplication as a function R2->R is silly

No it's not: this is exactly how multiplication is defined.

Division is the inverse operation to multiplication, not the inverse function to multiplication.

Inverse is a word whose meaning depends on context: in this case it has two completely different meanings: welcome to mathematics.

[hamsterman]

Damn. But then, what is an inverse operation?

* is the inverse of / when ((x*y)/y) = x. However (y/(x*y)) = 1/x. That is, the inverse of a (not commutative and not unary) operation depends on whether you place its result as the first or second argument.

If (y/(x ? y)) = x (where ? is unknown operation), a ? b = b / a. So the second inverse of division is division?

[tition]

Damn. But then, what is an inverse operation?

Well, I think you explained it quite well in your preceding post: (multiplication by c) is the inverse of (division by c), provided that c is not 0.

Multiplication is a function taking two arguments. It's not injective ( (1/2,2)->1 and also (1/3,3)->1 ), so we better not speak of inverses*.

Multiplication by c is a function taking one argument. It is injective for c not equal to zero, so we speak of inverses.


*In general it is wise to avoid the term "inverse function". Let f(x):=x+1. Question: is the inverse f^{-1} (0) equal to -1, or is it equal to {-1} (the one element set containing -1)?

Answer: provided f is bijective, both are valid mathematical definitions. There is no established standard on which is better. What is meant by a particular author depends on preference.
Corollary: avoid using the word "inverse function" when you write mathematical texts. Similarly, avoid the use of f^{-1}(x). Under all circumstances you are better off saying
f(x):=x+1
g(y):=y-1
f(g(y))=y
g(f(x))=x
Oops I think I wrote a bunch of crap.

Wikipedia gives a definition of inverse function: f:X->Y has an inverse g if and only if (g f) is the identity on X and (f g) is the identity on Y.

So, the function (multiplication by c) is invertible if c is not zero.

The function multiplication (taking two arguments) is not invertible.


[Edit:] As far as the question,

what is an inverse operation?

- I don't think that "inverse operation" is a used mathematical term. Feel free to define it as you please :)

[zwA4jE8b]

I always thought that dividing by zero will equal infinity. Because it seems to me that you can put infinite nothing into something...

[Gaminic]

We know; the question is: can you put infinite nothing into nothing?

[ascii]

Yes.

for (int i = 0; i <= 0; i += 0)
        ;

Infinite loop :)

[atropos]

According to the axioms of set theory, for a ring to be a division ring, the set of numbers {xn}, where xn * a = 1 for some a(a is an element of the ring) must be the set of all non-zero units of the ring. [A ring is a set R that is closed under both addition and multiplication.]

The trivial set, {0}, satisfies the requirements to be called a ring, but doesn't meet the additional requirement to be classified as a division ring. That means no dividing by zero...


@Gaminic: An infinite amount of nothing is just nothing, and nothing can't go into nothing, by the above.

[Cheraphy]

I once thought that division by 0 should equal infinity. I guess I figured that since division by a fraction was the same as multiplying by the reciprocal, and the smaller the fraction the greater the quotient, thus since 0 is infinitely small, division by it should yield an infinitely great quotient.

Also:

Tition wrote:

Division is the inverse operation to multiplication, not the inverse function to multiplication.

Those would be one and the same if we were thinking declaratively ;D

[Moschops]

We know; the question is: can you put infinite nothing into nothing?

The informal but useful infinite hotel analogy indicates that you can :)