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If $a^2 = b^2 $ and $a,b \gt 0$ then we can answer "Is $a > b$?"

I know that $a^2 = b^2 \Rightarrow |a| = |b|$ but still I don't understand why the second condition is absolutely necessary.

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If $|a|=|b|$ you don't know whether the signs are the same. Given, say, $|a|=|b|=2$, you could have a=-2, b=2 in which case b>a, or a=2, b=-2, so a>b, or they could have the same sign so a=b.

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From |a| = |b| it is impossible to answer to: Is a > b? For example, a might be -1 and b be 1, or viceversa. If you know both of them are positive, then you are done.

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The condition a,b>0 is not the only way to determine whether a>b, but you do need to know something about the signs since without specifying them you only have |a|=|b| and so, depending on which signs apply you could have any of a=b, a < b, or a > b

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HINT $\ $ For $\rm\ x = a/b\ $ it is $\rm\ x^2 = 1\:,\ x > 0\ \Rightarrow\ x = 1$

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