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.

the part after "ergo" hasn't been proven yet. all that has been proven is:

n*0 + n*0 = n*0

however, this in fact does imply that n*0 = 0:

subtract n*0 from both sides, and we get:

n*0 = 0.

(doing this uses implicitly that addition is cancellative: that a+b = c+b implies a = c. this is always the case if every number (or whatever we are dealing with) has an additive inverse, but is also true for just the non-negative integers).

this same "proof" holds for more general things than just numbers (integers). for example, if "n" represents a real number, and "0" is a 0-vector in R^{k}, we get that multiplying the 0-vector by any scalar is also the 0-vector.

But you didn't have "n(1)+n(0)" equal to anything to start with! You simply added an equal sign after; it's not really consistent.

If you did want a short demonstration that a*0 = 0 for all a, then consider this:
[tex]
\begin{align}
Let \ a, b, c \ \epsilon \ \mathbb{R},\ and\ consider \ the \ postulate: \\
a\cdot (b+c) = a\cdot b + a\cdot c \\
\ It \ follows \ that: \\
a\cdot (0+0) = a\cdot 0 + a\cdot 0 = a\cdot 0 \\
\ We \ can \ then \ subtract \ a\cdot 0 \ from \ both \ sides \ to \ obtain: \\
a\cdot 0 = 0 \\
\end{align}
[/tex]
As required.

Is this kind of what you wanted to see?

(I can't seem to align my TeX to the left. Help would be appreciated :) )

I just wanted to see if there is a loop hole in his proof. Something in your proof was confusing. At one point you have

[tex]a(0+0) = a(0) + a(0)[/tex]
[tex]a(0)+a(0) = a(0)[/tex]
How did you conclude that?? It looks that your using the property that your trying to prove, which isn't allowed.

Also, I have a proof of my own for this which I thought was pretty cool. I'll post it up here after this is taken care of.

It is generally known because you have been taught it in since you were a little kid. But in mathematics everything has to be proven, which is why its so darn self-consistent and powerful.

I had that a(b+c) = ab + ac. This is a common postulate in some books. I take it for granted most of the time. Surely that's not what's being proved here. What we're trying to prove is that:
a*0 = 0 for all a.

Using b=c=0 in my second line of manipulations, a*(0+0)= a*0 + a*0.
This is close to what your friend had, except that there were two steps missing. My approach is perfectly valid and in fact is also shown in some textbooks.

Here's another way you can look at this (From a less rigorous perspective perhaps):

[tex]
\begin{align}
n\cdot x = x\cdot (n+1)-x \\
\ for \ some \ x,n \ \epsilon \ \mathbb{R} \\
\ Let \ n=0, \ then: \\
\ 0\cdot x = x\cdot (1)-x = 0 \\
As \ required. \blacksquare \\
\end{align}
[/tex]

Which is actually kind of a silly way to put it. If you expand my first line you get : nx = xn.
Of course if you let n=0, then 0*x = x*0. If I was grading an assignment I'm not sure I'd mark my above "proof" correct.

Whoops, I don't know how I missed that. Anyhow, I guess that justifies my friend's proof. I was just wondering if his had any loopholes or not. At any rate, here is my proof, which I thought was rather interesting -- though I am biased. =p

This is one case where it becomes extremely important to make it explicit what you are assuming in your proof. If you begin with just the axioms for the real numbers or the rational numbers (or any ring for that matter), then you don't have a proof there.

However, if you are talking about constructing R from N (or something along these lines), where multiplication is defined in terms of repeated addition, then you can turn what you have written up into a rigorous proof using the notion of the empty sum.

I am guessing you are just assuming the field axioms for R,Q so you would actually need to utilize your friend's method for proving this result.

Not a common postulate, it is called an axiom.
In mathematics a number of axioms are asserted and from there everything is proven.
This is in all books.

Another axiom that is being used is a+0=0+a=a, meaning in particular that 0+0=0.

n(0)=0 is not an axiom, so that indeed needs to be proven.
DivisionByZro's first proof is correct.

The align environment is fine, but you need to use & symbols to indicate where the lines are to be aligned with each other. In this case, you should probably just start each line with &. You also need to use the \text command if you don't want everything you write to be interpreted as variables.