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I was reading 'The method of Coordinates - Gelfand' and in the section about the absolute value of a number, it is stated what follows :

if x > 0, then |x| = x,
if x < 0, then |x| = -x,
if x = 0, then |x| = 0

"Since the points a and -a are located at the same distance from the origin of coordinates, the numbers a and -a have the same absolute value: |x| = |-x|."

I don't understand the second statement : if x < 0, then |x| = -x. Why is the absolute value of a negative number negative if a distance between 2 points, I suppose, can't be negative ? Plus, the author contradicts himself by saying that |x| = |-x|, because if |x| = x and |x| = |-x|, then |-x| = x. Is it a typing error or am I missing something ?

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    $\begingroup$ if x < 0, then |x| = -x Replace $x$ with $-1$ and see what you get for $|-1|$. Hint: it's not negative. $\endgroup$ – dxiv Sep 9 '18 at 19:19
  • $\begingroup$ Because $-x> 0$ if $x < 0$ $\endgroup$ – ab123 Sep 9 '18 at 19:20
  • $\begingroup$ $-x$ isn’t necessarily negative, it’s just $-1\cdot x$. $\endgroup$ – Michael Hoppe Sep 9 '18 at 19:26
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    $\begingroup$ If x = -1, then |-1| = -(-1) = 1. So the author actually could not say that if x < 0, then |x| = x because that would mean, as x is a negative number, that it's modulus is a negative number (because x = -something), which is false. Now I understand, thanks for your help. $\endgroup$ – Cylian Barichard Sep 9 '18 at 19:54
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You can define the absolute value by

$$|x|=\max(x,-x)$$

if $x<0$, $-x$ is bigger than $x$.

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  • $\begingroup$ What is max(x, -x) ? Is it the biggest value between x and -x ? $\endgroup$ – Cylian Barichard Sep 9 '18 at 20:08
  • $\begingroup$ @CylianBarichard Yes. {}{}{}{} $\endgroup$ – hamam_Abdallah Sep 10 '18 at 15:48

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