Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm having a really hard time learning proofs, and cannot pick up on why this is actually proved. This isn't a homework problem, it's straight from the solutions manual (it just doesn't tell me why the proof is actually solved).

The Proof:

If $x$ is an even integer, then $x^2$ is even.

Proof. Suppose $x$ is even. Thus $x=2a$ for some $a\in\mathbb Z$.
Consequently $x^2=(2a)^2=4a^2=2(2a^2)$.
Therefore $x^2=2b$, where $b$ is the integer $2a^2$.
Thus $x^2$ is even by definition of an even number.

I'm confused on the third line. I understand that $x^2=2b$, but why do we need to know "where $b=2a^2$, and why do we need to know it? How did we even get that value?

Shoot, I think I just got it as I typed it out... What it's really saying is that $x^2=2(2a^2)$, and since $x^2=(2a)^2$ factors down to $2(2a^2)$, $b$ is a valid even integer $2a^2$ when it's value is for '$b$' in $x^2=2b$.

share|improve this question
    
I think you miss opined the proof or it is wrong. $4a^2=2(2a^2)$ is the correct step. –  Thomas Andrews Jan 10 '13 at 5:43
    
you're right, updated. –  user56763 Jan 10 '13 at 5:55
    
related: math.stackexchange.com/questions/260240 –  A.Schulz Jan 10 '13 at 6:42
    
Many times you'll find in such "collections of examples" steps that are obvious (and that you'd probably just omit), for completeness and clarity. For homework/exams you'll have to find out what level of detail is appropiate (you´ll have read of the 100+ page proof that 2 + 2 = 4...) –  vonbrand Jan 23 '13 at 2:15

1 Answer 1

up vote 3 down vote accepted

To show $x^2$ even, you need to exhibit some integer $b$ with $x^2=2b$. The preceding calculation shows that the (obviously integer) expression $2a^2$ can be chosen as $b$.

share|improve this answer
    
cool, weird how writing it out a few times helps me get it. I think I need to start with something very basic on these proofs and think more about the math as individual steps. Thanks! –  user56763 Jan 10 '13 at 5:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.