Is 0 (zero) a convention?

narrator

Hi, not sure if this is the right forum.. pls move if not.

Almost 30 years ago, I was studying engineering and my math tutor spoke about 0 (zero) being special. We talked about how nothing can be divided by zero, and sure enough, enter 1/0 into a calculator produces an error. (I vaguely recall learning this in high school).

Something occurred to me and I asked him about it. Here's the steps I put in front of him:

0/0 = 1
0/1 = 0
1/1 = 1
1/0= error

But 1/0 x 0/1 = 1

I asked, how can this be. He could not give me an answer. I've asked a few math teachers since then and none could give me an answer.

Is it simply a matter of convention, or is something else going on?

D H

0/0 is not one. It has no meaning. It is undefined.

micromass

In order to see why division by zero is undefined, you must consider the very definition of division. We say that m is divisible by n if and only if there is a unique integer k such that k*n=m. We then call k=m/n.

But if n=0 and m is nonzero, then this reduces to k*n=0=m. This can never be satisfied. So there is no such thing as division by zero.

What if m is zero? Certainly the equation holds then: if m=0 and n=0 then k*n=m reduces to 0=0. The problem is that k is not unique. So k=1,2,... all satisfies the equation. So we could potentially say that 0/0=1 or 0/0=2. Every possible value can be given to 0/0!! This is why we do not define 0/0.

As for your equation

[tex]\frac{1}{0}*\frac{0}{1}=1[/tex]

Well it just isn't true. 1/0 is not defined, so the left hand side is undefined.

Edgardo

Let us first consider what an inverse is. The inverse
is a number with a certain property, namely if you multiply
a number with its inverse you will get 1.

Example 1:
number=2
inverse=1/2
Proof: Multiplication yields
number*inverse = 2*(1/2) = 1

Example 2:
number=37
inverse=1/37
Proof: number*inverse = 37*(1/37) = 1

2 and 37 have inverses, which is equivalent to saying that 1/2 and 1/37 exist.

------


Now, consider the number 0.
Assumption: 0 has an inverse.
This assumption is equivalent to saying that 1/0 exists.

Then we have:

Example 3:
number=0
inverse=1/0
Proof: number*inverse = 0*(1/0) = 1

But this is a contradiction to the fact that 0 multiplied by
a number always equals 0. Therefore, our assumption that
the number 1/0 exists is wrong.

narrator

Thanks guys, that's actually helped and both explanations were easy to follow - the unique integer explanation and the inverse explanation make sense. As for 0/0, that explanation clarified my long held misconception, that anything divided by itself = 1, as taught in high school, because it needs to be qualified in either or both senses as you've explained, or more simply applied to numbers other than zero.


hmm.. Followup question.. Is ∞/∞ = 1 true?

D H

Undefined, for much the same reason. In one sense it can be any value, but in another sense saying that this has any specific value will lead to a contradiction.

narrator

Makes sense :)
thanks all