Arithmetic can be done without rings: arithmetics can be done without the use of numbers, e.g., decimals (xn – xs) and ratios (a+b/x). However, in practice, most problems involving numbers still require rings. For instance, determining the value of the hypotenuse of a triangle, the difference between two right angles, and the roots of a polynomial are all examples of arithmetical problems in which the use of numbers plays a key role. A ring, therefore, is often necessary to help calculate some of the more complex arithmetic operations.

Rings play an important role in number theory. The prime number ring, which contains the prime number i through z, is one such example. In mathematics, rings are geometrical structures which generalize algebraic fields: addition, multiplication, and division of numbers are not commutative or monotonic, and multiplication and addition of polynomials are not commutative. Simply put, a ring is an interlocking set equipped with a pair of binary operations equivalent to the natural logarithm of an underlying natural number, satisfying certain properties analogous to those of multiplication and division of real numbers.

Although there are no standard definitions for rings in mathematics, they can be visualized as a finite geometric structure consisting of closed curves on which numbers, say angles and poles, can be measured. The equator, whose graph represents the symmetrical angles formed by the poles of the rings, represents the second order of the ring structure. The equator is symmetrical about the central point of the ring, while the central point lies at the midpoint of the ring, representing the first order of the ring.

In the case of a circular ring, the ring particles have zero mass, while their equinoxes are not infinitely hot nor infinitely cold. The spokes of the ring, however, have a constant temperature, which changes as the ring moves on its axis. When the equinoxes near each other become very hot, the rings close together, while the equinoxes far from each other become very cold. The equator is symmetrical about its central point because the equinoxes are perfect cubes.

In astronomy, rings are called either satellite moons or exosphere satellites. Satellite moons are around nearly every celestial body in the solar system except the asteroid belt. Exosphere satellites, by contrast, are only found around our own moon.

The moon is almost perfectly spherical, so it can be used to calculate the gravitational pull that the earth has on itself. It also has rings, but they are very irregular and much smaller than the main ring particles. Thus, astronomers can measure the moon’s position with great precision and determine the position of the rings by taking a series of satellite-based maps, each showing the positions of the rings for a year or more since Apollo 14 made the first map. These maps show the position of many other satellites and the distribution of gravity within the disk of the moon. Since the moon does not have a large concentration of small, icy debris that could collect and become a moon rock, we can conclude that the distribution of moon rock throughout the solar system is quite random.