Why is it necessary to use the absolute value of the difference when finding the distance on a number line, but not necessary when finding the differences of the coordinates in the coordinate plane? "

Remember that the distance on a number line is given by the formula: 
where 
is the distance between the points. 
is the second point in the number line.  
is the first point in the number line.
Now, suppose we are trying to find the distance between the points -1 and -5; our first point is -1, so , and our second point is , so . Suppose we are going to find the distance between the two point without using absolute value:




Look what we have here, a negative distance! Since distances cannot be negative, we must use absolute value to always get postie distances between two point on a umber line:




 

Now, the distance between two coordinates on a plane is given by the formula: 

is the distance between the two coordinates
are the coordinates of the first point
are the coordinates of the second point 
Notice that  and  are squared, so it doesn't matter if we get a negative distance because a negative number raised to an even power (like 2) is always positive; therefore we don't need absolute value in this case because we won't ever get a negative distance.