Disputing 1 / 0 = Infinity: Agree or Disagree?

[CRGreathouse]

If I take a real nonzero number a and divide it by the real number zero why should I expect the result to be a real number?

If you take the integer 2 and divide it by the integer 3, there is no integer that is the answer. But that doesn't mean that there's a unique answer outside the integers. In Q the answer is 2/3, but in Z/7Z the answer is 3.

That's why it's so important to know which system you're working in.

 

[Studiot]

But that doesn't mean that there's a unique answer outside the integers.

I agree, although adding uniqueness is another burden.

However that did not answer my question.

It doesn't mean there is or isn't; it just draws attention to a gap in the system of mathematics (numbers in this case) as do roots, surds and trancendental numbers, vector cross products, some Fourier analyses... the list goes on and on.

As a result of that attention it has been found possible and convenient to develop new mathematics to deal with some such gaps, but not all known gaps have been plugged this way.

1/0 is one such unplugged gap. So we say it is undefined.

 

[Hurkyl]

1/0 is one such unplugged gap. So we say it is undefined.

No, leaving 1/0 undefined is (these days anyways) a deliberate design decision.


Ponder this question: why would you want to divide 1/0?

Most of the other number systems that people use -- e.g. the real numbers, the extended real numbers, the projective complexes, modular arithmetic -- are used because they are good for some purpose. The projective complex line, for example, are especially well suited for studying rational functions of one variable.

OTOH, I believe wheels were defined specifically for the purpose of defining a good arithmetic system where +,-,*,/ are defined for any pair of numbers^*. But, AFAIK, nobody actually uses wheels, because they don't care if everything has a reciprocal, so there is no reason to put up with the extra complications involved with wheels.


*: Technically, / is a unary operator in a wheel: /x is the "reciprocal" of x, and 1/x is just 1 times the reciprocal of x.[/size]

 

[Studiot]

Ponder this question: why would you want to divide 1/0?

As a result of that attention it has been found possible and convenient...but not all...

No offence meant but that is what I said, right from the outset. It's the same thing from another point of view. 'We have chosen not to' is the opposite side of the same coin from 'we have chosen to'.

It's the same coin of reasons that we don't chose to define division as a fundamental operation between numbers and you used addition to discuss division in your post#29.

I was also trying to introduce a point of view that shows our current reticence in defining 1/0 is not just a whim but in keeping without current systems of mathematics in a wider sense to help make zeromodz more comfortable with this. Would you claim our system of maths is perfect?

 

[Mentallic]

Another way to show that 1/0 is not infinite, but actually undefined is as follows:

prove that if two complex numbers, z and w, are equal to each other, then their real parts and imaginary parts are equal.

i.e. z=w, let z=x+iy, w=a+ib

therefore, x+iy=a+ib
prove x=a, y=b

Doing a proof by contradiction: let's say that b$\neq$y. Then we have x-a=i(b-y) and then $$i=\frac{x-a}{b-y}$$ under the assumption that b$\neq$y. But the RHS is a real number, while the LHS is an imaginary number by definition, so obviously b=y (and thus a=x).

Notice that in this case, rather than having a dispute about 1/0=$\infty$ or not, we have 1/0 being imaginary. Clearly 1/0 is undefined in this number system.

 

[Hurkyl]

*sigh* By this point, I think any layperson who stumbles upon this thread is going to be more confused than helped, so I'm locking it.

If someone wants to start a thread on pedagogy, we can move over there and continue discussing that.


For the record, I'm not trying to be dogmatic here. I think much of the discussion has been reinforcing specific misunderstandings that people have about mathematics -- misunderstandings that the opening poster has even demonstrated!