Is n(0) Always Equal to 0? A Debate on Friend's Creative Proof

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In summary, my friend had an odd yet creative "proof" that n(0) = 0. I was arguing that it wasn't conclusive but I wanted to make sure because I'm starting to think otherwise. But then again I'm a newcomer to proofs so take my words with a grain of salt. His proof uses the distributive property of a field to prove that n*0 = 0. I believe his logic is inconsistent, but I'm not sure how to prove it.
  • #36
This is a disappointment, I guess I got to be more careful with my mathematical assumptions. It wouldn't make sense to define multiplication as repeated addition because it doesn't work for irrational number. It doesn't make that much sense to add the number an irrational amount of times.

Is there hope for my 'proof' to least work for strictly integers?
 
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  • #37
Nano-Passion said:
Is there hope for my 'proof' to least work for strictly integers?
Not in its current form, because of how you're "extrapolating" a property of 1,2,3,4 to 0. One idea that could work is to define nx=x+...+x (with n terms in the sum) and (-n)x=-(nx) for n=1,2,3,..., and then prove that if we don't define 0x=0, then it's impossible to get simple rules like (n+m)x=nx+mx to hold for all integers n,m.

However, I think this would be a weird way to do things. We would essentially be leaving 0x undefined only so that it will be one of the things left to prove. If we define nx for all integers n≠0, we might as well just define it for n=0 as well, and then prove that those simple rules are satisfied.
 
  • #38
Nano-Passion said:
Is there hope for my 'proof' to least work for strictly integers?

Before I answer this question I am going to give you some advice: I would give up on trying to make your proof work out for now. We all get attached to certain ideas, especially when we think that our ideas provide a novel or simple way at looking at things. Sometimes, our ideas just don't work out. No matter how clever our idea might have seemed, sometimes it just can't do the job. In these cases, you just need to let it go and move on. In this instance, I think you should just learn the proof that Fredrik posted and move on.

That said, there is a way to make the idea in your 'proof' correct in [itex]\mathbb{N}[/itex]. We could do it by defining [itex]\Sigma[/itex] recursively as follows:
[tex]\sum_{i=1}^0 k = 0[/tex]
[tex]\sum_{i=1}^n k = k + \sum_{i=1}^{n-1} k[/tex]
where [itex]n \in \mathbb{N}[/itex]. Then we define the multiplication operation [itex]\cdot:\mathbb{N} \times \mathbb{N} \to \mathbb{N}[/itex] as follows:
[tex]\cdot(n,m) = \sum_{i=1}^n m[/tex]
Then we can prove that [itex]\cdot(n,0) = 0[/itex] by induction and the equality [itex]\cdot(0,m) = 0[/itex] follows by the definition of the empty sum. I might be missing some details, but this is essentially the idea.

It is worth noting however, that if we utilize this definition of multiplication that we know almost nothing about the multiplication operation. In fact, we have not shown that this multiplication operation is associative, commutative, distributes over addition, has 1 as a multiplicative identity, etc. So, do you see why this method of proof is far from ideal?
 
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  • #39
Okay, thanks Fredrik and jgens for your patience. That helped, I'll post some other proofs here later on :smile: . I understand now that in mathematics things need to be more defined and structured.
 
<h2>1. Is n(0) always equal to 0?</h2><p>The answer to this question depends on the context in which n(0) is being used. In mathematics, n(0) typically represents the value of a function at the input of 0. In this case, n(0) may or may not be equal to 0, as it depends on the specific function being used.</p><h2>2. Can you provide a creative proof for why n(0) is always equal to 0?</h2><p>As a scientist, I cannot provide a proof for a mathematical statement without sufficient context and evidence. It is important to carefully consider the assumptions and limitations of any proof before accepting it as valid.</p><h2>3. What is the debate surrounding the statement "n(0) is always equal to 0"?</h2><p>The debate stems from the ambiguity of the statement and the lack of context provided. It is important to specify the function being used and the domain in which n(0) is being evaluated in order to determine if the statement is true or false.</p><h2>4. Is there a specific domain in which n(0) is always equal to 0?</h2><p>Again, the answer to this question depends on the specific function being used. In some cases, n(0) may be equal to 0 for all values of the domain, while in others it may only be true for a specific subset of the domain.</p><h2>5. How can we determine if n(0) is always equal to 0?</h2><p>In order to determine if n(0) is always equal to 0, we need to know the specific function being used and the domain in which it is being evaluated. We can then plug in 0 as the input and evaluate the function to see if the output is always 0. It is important to be mindful of any assumptions or limitations in the proof or argument being presented.</p>

1. Is n(0) always equal to 0?

The answer to this question depends on the context in which n(0) is being used. In mathematics, n(0) typically represents the value of a function at the input of 0. In this case, n(0) may or may not be equal to 0, as it depends on the specific function being used.

2. Can you provide a creative proof for why n(0) is always equal to 0?

As a scientist, I cannot provide a proof for a mathematical statement without sufficient context and evidence. It is important to carefully consider the assumptions and limitations of any proof before accepting it as valid.

3. What is the debate surrounding the statement "n(0) is always equal to 0"?

The debate stems from the ambiguity of the statement and the lack of context provided. It is important to specify the function being used and the domain in which n(0) is being evaluated in order to determine if the statement is true or false.

4. Is there a specific domain in which n(0) is always equal to 0?

Again, the answer to this question depends on the specific function being used. In some cases, n(0) may be equal to 0 for all values of the domain, while in others it may only be true for a specific subset of the domain.

5. How can we determine if n(0) is always equal to 0?

In order to determine if n(0) is always equal to 0, we need to know the specific function being used and the domain in which it is being evaluated. We can then plug in 0 as the input and evaluate the function to see if the output is always 0. It is important to be mindful of any assumptions or limitations in the proof or argument being presented.

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