When a point 'a' on a plane crosses the x-axis, its value is considered to be 0. This is because the x-axis is defined as the horizontal axis in a Cartesian coordinate system, where the value of the x-coordinate is zero at any point on this axis. The x-axis extends infinitely in both the positive and negative directions along the horizontal plane, and any point on this axis has an x-coordinate of zero. Therefore, when 'a' crosses the x-axis, it aligns with the x-axis, making its x-coordinate equal to zero.

The concept of the x-axis being associated with zero is rooted in the Cartesian coordinate system, which was developed by the French philosopher and mathematician René Descartes in the 17th century. In this system, each point on the plane is represented by an ordered pair of numbers (x, y), where x is the horizontal distance from the origin (0,0) and y is the vertical distance. When a point crosses the x-axis, its y-coordinate is zero, as it lies directly on the horizontal line that defines the x-axis. This is consistent with the definition of the x-axis as the set of all points with an x-coordinate of zero.

The importance of this concept is evident in various fields, such as physics, engineering, and computer graphics. For instance, in physics, the x-axis is often used to represent displacement or position in one dimension. When an object is at rest or moves along the x-axis, its displacement is zero, which is represented by a point on the x-axis with an x-coordinate of zero. Similarly, in computer graphics, the x-axis is used to define the horizontal position of objects within a 2D or 3D space.

Data supporting the concept of the x-axis being associated with zero can be found in the historical development of the Cartesian coordinate system. Descartes' work, "La Géométrie," published in 1637, is considered a foundational text in the development of modern mathematics. In this work, Descartes introduces the Cartesian coordinate system and defines the x-axis as the horizontal axis with an origin at the point (0,0). This definition has been widely accepted and used in mathematics and its applications ever since.

In conclusion, when a point 'a' on a plane crosses the x-axis, its value is 0 due to the definition of the x-axis as the horizontal axis in the Cartesian coordinate system, where the x-coordinate of any point on this axis is zero. This concept is fundamental to various fields and is supported by historical data from the development of the Cartesian coordinate system.