Is 0 (zero) a convention?

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  • #1
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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?
 
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  • #2
D H
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0/0 is not one. It has no meaning. It is undefined.
 
  • #3
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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.
 
  • #4
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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.
 
  • #5
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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.
:smile:

hmm.. Followup question.. Is ∞/∞ = 1 true?
 
  • #6
D H
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Is ∞/∞ = 1 true?
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.
 
  • #7
222
11
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.
Makes sense :)
thanks all
 

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