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Friend's proof that n(0) = 0 
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#1
Dec1011, 04:39 PM

P: 1,306

I'm sorry if this is in the wrong place..
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. 


#2
Dec1011, 05:12 PM

P: 252

What exactly is "n(0)"? What type of function is this? I'm not familiar with the notation.
In any case, I can, for the moment, only speculate that you can't really split up "n(0)" into "n(0) = n(0) + n(0)". 


#3
Dec1011, 05:38 PM

HW Helper
P: 6,187

Hi NanoPassion!
Just guessing here... Are we talking about a group homomorphism? Or about the distributive property of a ring or field? Or... 


#4
Dec1011, 05:41 PM

Sci Advisor
P: 6,056

Friend's proof that n(0) = 0
My guess is simply multiplication. n(0) means nx0, where x is multiply.



#5
Dec1011, 05:47 PM

P: 1,306

[tex]n(0)[/tex] which is the same as [tex]n(0+0)[/tex] [tex]n(0)+n(0)[/tex] This kind of seems logical to me, kind of like saying [tex]n(1)[/tex] [tex]n(1+0)[/tex] [tex]n(1)+n(0)[/tex] Wait wait.. that would imply.. [tex]n(1)=n(0)[/tex] Which can't be true. So therefore his logic is inconsistent, correct? My apology, I meant simple multiplication. Some number n times 0 > n(0). 


#6
Dec1011, 05:54 PM

HW Helper
P: 6,187

Ah, okay, that would be the distributive property of a field.
(Math mumbo jumbo for the same thing ;) But isn't it already generally known that a number times zero is zero? Why proof it? (Unless you want to proof it for a ring or a field.) 


#7
Dec1011, 06:13 PM

Sci Advisor
P: 906

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 nonnegative 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 0vector in R^{k}, we get that multiplying the 0vector by any scalar is also the 0vector. 


#8
Dec1011, 06:22 PM

P: 252

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 :) ) 


#10
Dec1011, 06:36 PM

P: 252




#11
Dec1011, 07:48 PM

P: 1,306

[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. 


#12
Dec1011, 07:49 PM

P: 1,306




#13
Dec1011, 08:52 PM

P: 252

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. 


#14
Dec1011, 08:59 PM

P: 252

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. 


#15
Dec1011, 08:59 PM

P: 1,306

[tex]a(0)+a(0) = a(0)[/tex] 


#16
Dec1011, 09:03 PM

P: 252

"No my concern was this part actually:
[tex] a(0)+a(0) = a(0) [/tex] " If b=c=0, then b+c=0, as you had in your opening post. The lefthand side of the fourth line of the proof is: [tex] a\cdot (0+0) = a\cdot 0 + a\cdot 0 = a\cdot 0 [/tex] Which, if you remove the middle part, says: [tex] a\cdot (0+0) = a\cdot 0 [/tex] There is no mistake here. 


#17
Dec1011, 11:36 PM

P: 1,306




#18
Dec1111, 12:28 AM

P: 1,622

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. 


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