if a^2 is even....prove that a is even...

Question

jiteshmeghwal9 · Answer

it's known to everyone that the square of a even number is even

shubhamsrg · Answer

if a = 2k
=>a^2 = 4k^2 = 2*something =>even
hence proved..

anonymous · Answer

u r showin proof to..if a is even...show that a^2 is even........:D

shubhamsrg · Answer

o.O ??

parthkohli · Answer

It's the same. lol

jiteshmeghwal9 · Answer

$\LARGE{a^2=even \space number}$
then the number whose square is $a^2$ is also even

parthkohli · Answer

If $a$ is even, then $a$ is in the form $2k$.
$$2k \times 2k = 4k^2 = 2(2k^2)$$

shubhamsrg · Answer

maybe your next ques will be:
if a is even prove that a+1 is odd!! 
or even
if a is even prove that 2a is even!!
;)

parthkohli · Answer

Haha, those are axioms! ^

shubhamsrg · Answer

in "or even" i didnt mean the mathematical meaning of "even" !! :P

anonymous · Answer

p->q is not the same as q->p....
i.e.,if A=40degrees,..then A is acute.....
where as if A is acute then S is 60 degrees.........????:DDDD

anonymous · Answer

sorry 40 degrees..

shubhamsrg · Answer

i thing B is 66.75 radians here ??

shubhamsrg · Answer

nevermind,,am outa here!! :D

parthkohli · Answer

Didn't I tell the proof?

jiteshmeghwal9 · Answer

haha ! some problem is with this site

jiteshmeghwal9 · Answer

http://answers.yahoo.com/question/index?qid=20071114004333AA1ql5F

jiteshmeghwal9 · Answer

well now look @ this site @anusha.p u will gt ur answer :)

parthkohli · Answer

Oh yes! I have this proof posted too as a tutorial!

parthkohli · Answer

http://openstudy.com/study#/updates/4fed7487e4b0bbec5cfccbfb

anonymous · Answer

@jiteshmeghwal9  i want it by direct proof....

jiteshmeghwal9 · Answer

yeah ! these site is providing diffrent types of answers

parthkohli · Answer

Wait, no.
Direct proofing is not possible in this case!

parthkohli · Answer

Okay, I lied. It actually is... but it's a long one. "To prove: For all whole numbers x, if x2 is even then x is even. A direct proof is difficult, so a proof by contrapositive is preferable." http://en.wikipedia.org/wiki/Proof_by_contrapositive

jiteshmeghwal9 · Answer

& u have gave u the proof using a axiom
that the square of a even number is even

parthkohli · Answer

If you are talking about the $a = 2k$ proof, then see above.

jiteshmeghwal9 · Answer

Given : $a^2$ is even 
then from above axiom the number whose square is $a^2$ is also even number

parthkohli · Answer

And then we continue with $a = 2k$.

jiteshmeghwal9 · Answer

too small proof @anusha.p :D

jiteshmeghwal9 · Answer

well i m also nt sure.
am i right @ParthKohli ???

parthkohli · Answer

Yes.

jiteshmeghwal9 · Answer

k!

anonymous · Answer

so..wat do u say finally???

parthkohli · Answer

In this case, if we are given a hypothesis that looks like "If $a^2$ is even, then $a$ is even", we may rearrange(only in this case) to make it look like "If $a$ is even, then $a^2$ s even too."

Presume that $a = 2k$ where $k$ is an integer.$$a^2 \implies (2k)^2 \implies 4k^2 = 2(2k^2) $$We know that this is an even number since 2 is a factor.

anonymous · Answer

can we rearrange......?
i mean is it allowed....if so..how??

parthkohli · Answer

It's just mind-work and logic. Think about it, and don't think about $p\Rightarrow q$ thing.

anonymous · Answer

but the actual thing here is the implication only....na

parthkohli · Answer

"If $a^2$ is even, then $a$ is even" is just a fancy way to say "If a is even, then $a^2$ is even too."

parthkohli · Answer

Think about it for a while.

theviper · Answer

$$\Large{a^2=a \times a}$$Since $a^2$is even number therefore a is also even because when we multiply anything with an even no. we get an even number ;)

anonymous · Answer

???

parthkohli · Answer

@TheViper That's a nice way, but you gotta use these statements as less as possible. 
"we multiply anything with an even no. we get an even number" is a theorem, and not an axiom. 

Rest is well :)

anonymous · Answer

http://uk.answers.yahoo.com/question/index?qid=20101012102515AAlORD8 check this out @parthkolhi

theviper · Answer

OK thanx @ParthKohli :)

anonymous · Answer

@ParthKohli

parthkohli · Answer

@anusha.p Haha, I did give a proof for the contrapositive.

parthkohli · Answer

Quote, unquote, "I want a direct proof."
I told you that we can't just directly prove, and so did Wikipedia. ;P

anonymous · Answer

so..what should i do.......this quest is given under direct proof secton.....:(

parthkohli · Answer

Just use my proof. Haha!

parthkohli · Answer

Okay, don't get the converse. 
You start with assuming that $a = 2k$.$$(2k)^2 =4k^2 = 2(2k)^2 $$