The absolute value of #x# is always greater than or equal to zero: #|x|\ge 0#.

For example, the distance from both #4# and #-4# to zero is #4#. After all, #|4| = 4# and #|- 4| =-(-4)= 4#.

The equality #|x|=\sqrt{x^2}# follows from a distinction in two cases:

• If #x\ge0#, then #|x|=x=\sqrt{x^2}#.
• If #x\lt0#, then it follows from the previous case and the equality #(-x)^2=x^2# that #|x|=-x=\sqrt{(-x)^2}=\sqrt{x^2}#.

These two interpretations are related: in the flat plane, where we calculate with two coordinates #\rv{x,y}#, the distance from the origin #\rv{0,0}# is given by #\sqrt{x^2+y^2}#. Our number line is the special case when #y=0#. The distance to the origin is obtained by enterering #y=0# in this formula: #\sqrt{x^2}=|x|#.

Because distance is not dependent on the choice of the origin on the number line, the mutual distance between two numbers is #a# and #b# on the number line is same as the distance between them after shifting the two numbers each by (the same) random number. For that randum number we choose #-b#. In that case, #a# shifts to #a-b# and #b# to #b-b=0#. Because according the above the distance of #a-b# to #0# is #|ab|#, follows that the distance from #a# to #b# is equal to #|a-b|#.