0 isn't a real number.
Distance: the extent or amount of space between two things, points or lines.
0 is not a amount as it is not a number and is the "Distance" were to be 0 then it wouldn't be any leanth away.
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The distance travelled by a particle cannot be zero when displacement is not zero because unlike distance which is a scalar, displacement is a vector quantity implying that it has both direction and magnitude.
If the distance is measured from a point other than the initial position of the object.
It cannot be zero.
Absolute value
Numbers the same distance from zero (linearly) have the same "absolute value" whether positive or negative.