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 May 8th, 2018, 12:37 AM #1 Newbie   Joined: May 2018 From: India Posts: 1 Thanks: 0 Each even number is equal to zero Dear Friends I have come to know an interesting fact that every even number is equal to zero and every odd number is equal to 1; what's your view? Last edited by skipjack; May 8th, 2018 at 05:06 AM.
May 8th, 2018, 01:34 AM   #2
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Quote:
 Originally Posted by asitsikary Dear Friends I have come to know an interesting fact that every even number is equal to zero and every odd number is equal to 1; what's your view?
Okay, I'll bite. This is true if we are working in $\displaystyle \mathbb{Z} _2$ (or anything isomorphic to it.) But something tells me you are talking about something different.

Please tell us why you are saying that.

-Dan

Last edited by skipjack; May 8th, 2018 at 05:06 AM.

 May 8th, 2018, 03:22 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,344 Thanks: 2466 Math Focus: Mainly analysis and algebra I think that it's completely uninteresting until you explain yourself properly.
 May 8th, 2018, 04:27 AM #4 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,921 Thanks: 883 Hmmm.....I wonder if any "integration" is involved....
 May 8th, 2018, 05:18 AM #5 Senior Member     Joined: Feb 2010 Posts: 683 Thanks: 129 I wonder how old he is? Zero or one?
 May 8th, 2018, 05:19 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,344 Thanks: 2466 Math Focus: Mainly analysis and algebra Goodness knows. But if $2a=2a+2$, then $a=a+1$ if dividing by $2$ makes any sense. So $0=1$. Thanks from topsquark
May 8th, 2018, 07:29 PM   #7
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Quote:
 Originally Posted by v8archie Goodness knows. But if $2a=2a+2$, then $a=a+1$ if dividing by $2$ makes any sense. So $0=1$.
To elaborate on this, what it essentially proves is that any field satisfying the OP's claims must have characteristic 2. This is another way of saying that division by 2 must not make any sense since it leads to a contradiction as you point out.

While I know you are aware of the distinction, I want to point out that the usual computation doesn't assume anything about invertibility.
The argument goes like:

If $2a = 2a + 2$, then $2a = 2(a+1)$. Since it holds for any $a$, it must hold in the case that $a$ is a unit and it follows immediately from uniqueness of inverses that $2a = 0 = 2(a+1)$. In a field, every element is invertible, hence the field has characteristic 2. The argument does not require any mention of inverting 2.

In fact, everything in this argument goes through equally well for arbitrary local rings. In this case, the characteristic is replaced by the Jacobson radical, but otherwise, there is no reason to mention inverting elements or requiring more than just the ring structure.

 May 8th, 2018, 07:48 PM #8 Math Team     Joined: May 2013 From: The Astral plane Posts: 1,855 Thanks: 750 Math Focus: Wibbly wobbly timey-wimey stuff. Good comments, but I just wonder when we'll hear from asitsikary. Unless, of course, he's a troll. -Dan
May 8th, 2018, 10:25 PM   #9
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Quote:
 Originally Posted by topsquark Unless, of course, he's a troll.
Or a spammer (There's a writing that something was edited by skipjack, probably a link).

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