# Prove that 2x is always an even number.

How does one prove that for every value in $\Bbb N$ 2x = an even number?

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Define an even number –  Belgi Sep 14 '12 at 10:55

Isn't the definition of an even number all $n \in \mathbb{Z}$ such that n = 2k for some $k \in \mathbb{Z}$?

$2x$ with $x \in \mathbb{N}$ would always be an even number according to the definition above.

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Basically, if someone told you to prove the above statement, then you would simply write, "by definition" and maybe even invoke the statement of the definition. But even that would be excessive in my opinion. –  aviness Sep 14 '12 at 11:40

That's the definition of an even number. A natural number $y$ is said to be even if there is another natural number $x$ such that $y = 2x$

Another definition: A natural number $n$ is said to be even if its residue class in the quotient ring (field) $\Bbb{Z}/2\Bbb{Z}$ is not different from 0.

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My question was most aimed at the mathematical notation of the answer. Obviously, every integer doubled is an even number; is there however a way to write this in a mathematical notation? –  Gideon Potgieter Sep 14 '12 at 11:02
@GideonPotgieter Everything on this page is "mathematical notation" –  user39572 Sep 14 '12 at 11:10
What aviness and BenjaLim are saying is that a definition is not a theorem, so you don't have to prove it! A definition can't be false, it's just a way to call things: you only have to avoid definitions that lead to results which are in contradiction with already established ones. –  Andrea Orta Sep 14 '12 at 11:11
Thanks, got your point! –  Gideon Potgieter Sep 14 '12 at 11:52
Why say "class is not different from 0" vs. "class = 0" ? –  Bill Dubuque Sep 14 '12 at 13:34