In mathematics, real numbers can be represented by points on a straight line called the number line, which includes a point called the origin, the direction of number growth, and a unit length. It is generally assumed that there is a one-to-one correspondence between real numbers and points on the number line, with the position of a point determining the size and order of the numbers. This essentially assumes that all real numbers have a definite position on the number line, and that there is a definite order between any two real numbers.This paper shows that there are real numbers with uncertain positions, and that all real numbers do not lie on the number line of the same dimension. The number line is composed of discrete points, which are "pure numbers"—that is, only pure numbers exist on the number line, while non-pure numbers exist in "empty space." Therefore, there is a logical contradiction between the continuity of real numbers and the real number line; the real number line is an incomplete and imperfect conception for representing real numbers. This paper gives the definition of a pure number and the relationship between its cardinality and the natural cardinality.These results verify the viewpoint of quantum theory in physics, namely that the straight line on the "macroscopic" number line is composed of "microscopic" discrete and discontinuous points.