<PRE><B>Question 650296</B>
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Forget you ever heard terms like &quot;invisible + sign&quot; or &quot;negative decimal sign&quot; (whatever &lt;b&gt;&lt;i&gt;that&lt;/i&gt;&lt;/b&gt; means).  And why would you replace an understood positive with a visible negative?  The absolute value function certainly doesn&#39;t tell you to do anything of the kind.


Ok, back to basics.  First let&#39;s talk about the reason we have an absolute value function in the first place.  If you are just dealing with regular numbers, you can tell how far away you are from zero &lt;b&gt;&lt;i&gt;and&lt;/i&gt;&lt;/b&gt; in which direction.  4 is on the right of zero by 4 units while -100,000 is way over in the next room on the left side of zero.  But there are situations in math where we don&#39;t actually care which side of zero we are on; we only care how far away we are.  Enter the absolute value function.  In conversational terms, the absolute value function leaves a number completely alone if it is already positive, or it strips away the negative sign if it is negative.  The bottom line:  the value of an absolute value function is &lt;b&gt;&lt;i&gt;always&lt;/i&gt;&lt;/b&gt; positive or zero.


Let&#39;s look at the formal mathematical definition and see if it makes sense compared to the conversational definition that I just shared.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ |x|\ =\ \left\{\ \ x\text{ if }x\ \geq\ 0\cr-x\text{ if }x\ &lt;\ \,0\right]


So what does that say?  Simply this:  If *[tex \LARGE x] is already positive, leave it alone -- *[tex \LARGE |x|\ =\ x] whenever *[tex \LARGE x] is positive.  On the other hand, if *[tex \LARGE x] is negative, return the &lt;b&gt;&lt;i&gt;opposite&lt;/i&gt;&lt;/b&gt; of *[tex \LARGE x] -- *[tex \LARGE |x|\ =\ -x] whenever *[tex \LARGE x] is negative.  The thing to remember is that if *[tex \LARGE x] is negative then *[tex \LARGE -x] is &lt;b&gt;&lt;i&gt;a positive number&lt;/i&gt;&lt;/b&gt;.  Again, same bottom line: no matter what value you submit to the function, the value of the function will either be positive or zero -- NEVER negative.


Back to your problem.  What sort of number is *[tex \LARGE \frac{1}{2}] -- positive or negative?  And then given the answer to that question, what is the value of *[tex \LARGE \left|\frac{1}{2}\right|]?


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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