Prove equality (real numbers)

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October 17th 2011, 10:55 AM
Fabio010

Prove equality (real numbers)

a is a real number.

Demonstrate the following equality.

$\surd(a^2) = |a|$

I tried in this way but i think it is wrong.

$\surd(a^2) = a~~~(a\geq0)$

$a\geq0~~so~~a = |a|$

Is it correct? :)
October 17th 2011, 01:45 PM
emakarov

Re: Prove equality (real numbers)

You considered only the case $a\ge 0$ . Also, since the problem is so basic, the proof has to be really detailed, and I am not sure you sufficiently explained why $\sqrt{a^2}=a$ when $a\ge 0$ . So, what happen when $a < 0$ ? Recall that by definition $\sqrt{x}$ is a nonnegative number $y$ such that $y^2=x$ .
October 18th 2011, 10:58 AM
Fabio010

Re: Prove equality (real numbers)

Quote:

Originally Posted by emakarov

You considered only the case $a\ge 0$ . Also, since the problem is so basic, the proof has to be really detailed, and I am not sure you sufficiently explained why $\sqrt{a^2}=a$ when $a\ge 0$ . So, what happen when $a < 0$ ? Recall that by definition $\sqrt{x}$ is a nonnegative number $y$ such that $y^2=x$ .

Thats the point the exercise is so basic. But hard to do.
October 18th 2011, 11:07 AM
Plato

Re: Prove equality (real numbers)

Quote:

Originally Posted by Fabio010

Thats the point the exercise is so basic. But hard to do.

Any proof of this depends on what definitions you have and what other properties of real numbers you know.
If could be this simple:
1) if $a\ge 0$ then $\sqrt{a^2}=a=|a|$
2) if $a< 0$ then $\sqrt{a^2}=-a=|a|$ .
October 18th 2011, 11:42 AM
Fabio010

Re: Prove equality (real numbers)

Quote:

Originally Posted by Plato

Any proof of this depends on what definitions you have and what other properties of real numbers you know.
If could be this simple:
1) if $a\ge 0$ then $\sqrt{a^2}=a=|a|$
2) if $a< 0$ then $\sqrt{a^2}=-a=|a|$ .

I found some properties in internet , and i tried:

$\sqrt{a^2}=b~~b\geq0~~so~~b^2=a^2~so~(b-a)(b+a)=0$

$b = a,~in~this~case~a\geq0$
$b = -a, in~this~case~a\leq0$

so $\sqrt{a^2}=b=|a|$

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