Question 254575
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If *[tex \Large a] is a negative real number, then *[tex \Large -a] must be a positive real number.


Let's look at the definition of absolute value:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ |x|\ =\ \left\{\ \ x\text{ if }x\ \geq\ 0\cr-x\text{ if }x\ <\ \,0\right]


When you read this definition, read *[tex \LARGE -x] as "the opposite" of *[tex \LARGE x] rather than "negative" *[tex \LARGE x] or "minus" *[tex \LARGE x].   I find it much less confusing to think of it in this way.


Part b is *[tex \LARGE |-a|].  In part a of the problem we established that *[tex \Large -a] must be a positive real number.  Looking at the definition of absolute value -- if the thing inside of the absolute value bars is positive, then the value of the absolute value function is identical to the value of the thing inside.  Since *[tex \Large -a] is positive, the absolute value of *[tex \Large -a] must be *[tex \Large -a] which is positive.


Part c.  First determine *[tex \LARGE |a|].  We know *[tex \LARGE a] to be negative, therefore, using the definition, the absolute value of *[tex \LARGE a] must be the opposite of *[tex \LARGE a] or *[tex \LARGE -a].  From part a of the problem we know that *[tex \LARGE -a] is a positive number.  Hence *[tex \LARGE |a|] is positive.  But part c asks for *[tex \LARGE -|a|].  This is the opposite of a positive number, i.e. a negative number.


Part d.  We know from part a that *[tex \LARGE -a] is a positive number.  Part d asks for the opposite of a positive number, to wit, a negative number.


Part e.  We know from part b that *[tex \LARGE |-a|] is a positive number.  Just like in part d, the opposite of a positive number is a negative number.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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