# Something that has troubled me for a while now: absolute value

If the absolute value is usually defined as a term used in mathematics to indicate the distance of a point or number from the origin (zero point) of a number line or coordinate system. How can this be possible:

For x < 0, | x | = - x

• What about it strikes you as impossible? If you plug in a particular value for $x$ (for instance, $-1$), does it still seem wrong? – Milo Brandt Aug 24 '16 at 2:00
• It's simply a definition of a function. As Milo asks, what seems impossible about it? When $x<0$, the output of your function $|x|$ is given by $-x$. – Carser Aug 24 '16 at 2:01
• My suspicion is that you're under the impression that $-x$ is negative because it begins with a minus sign. But if $x<0$, then $-x$ is positive. $\qquad$ – Michael Hardy Aug 24 '16 at 2:19
• Note: -(-3)= 3. This is fine you can see the negative sign inside. But what if x =-3. Then -x = 3. Still okay. |x| =3. |-x| =3. But x <0 and -x >0. And -x =3. So |x| = 3 = -x. It looks wrong be if x is negative. Then -x is positive. – fleablood Aug 24 '16 at 8:00

Draw a number line. How far away is $-1$ from $0$? Well, it's exactly one unit away - that is, $\vert -1\vert=1$.

But $1=-(-1)$. Do you understand why?

In case you haven't seen this before, don't feel bad if it looks really strange. This is totally not obvious at first! Here's why it's true: the way we define negatives is $$\mbox{"-a is the thing you need to add to a to get 0."}$$ That is, the defining property of (say) $-2$ is that $2+(-2)=0$. Now, what do you add to $-1$ to get $0$? Well, the answer is just $1$! So $-(-1)=1$. This sort of reasoning by algebraic definitions can seem really weird at first, and I strongly suggest you talk to your teacher(s) about it until it makes sense. Right now it might seem a little random, but it's actually super important; and down the road, it will be one of the key ideas behind abstract algebra.

In general, if $x$ is negative, then $\vert x\vert=-x$ because - despite how it may seem! - $-x$ is the "positive" version of $x$!

(What's really going on here: $-x$ flips the sign of $x$. If $x$ is positive, $-x$ is negative, and vice versa.)

One way to define |x| is that it is the larger of x and -x, with the understanding that the larger of 0 and 0 is 0. So |-3| is the larger of -3 and -(-3) and the larger of these two numbers is -(-3) =3.

|x| is always positive. (Well,non-negative-- it could be zero.)

x might be positive or it might be negative. (Or it might be zero... let's just ignore the zero options for this answer...)

|x| is whatever it takes to express the "size" of x. If x is positive, the positive expression is "x". But if x is negative, x is not positive. Instead -x is positive.

So |x| is either, x or -x-- whichever one of those two is not negative.

If x < 0, then x is the one that is negative and -x is positive(!). And |x| is the positive one of x and -x. So which one is |x|?