The Number Line is an extremely useful human invention. What is not so obvious to many is that its essence expresses Pj Problems.
Consider the illustration of the number line. The matter being grouped are coffee beans. The mapping of the coffee beans into the number line is possible because the number line itself is partitioned into unique spaces where unique numbers reside. The partitioning of the number line is made possible by unit movements along the number line away from a central space called zero (0) which signifies nothing (e.g. no coffee beans). The other spaces are then given unique identities ( 1, 2, 3, ...).
The containership and identity functionality of the number line is further extended by defining the basic arithmetic operations of addition, subtraction, multiplication and division and applying them as forces of the number line. These basic arithmetic operations operate on any two or more numbers to get another number.
On the number line are numerous groups: even numbers, odd numbers, prime numbers, etc. There are infinitely many ways these numbers interact.
The number line is adaptive and so can be changed to represent various number systems. The binary number system which is dominant in computer science was extracted from the basic decimal number system which can be extended to form the hexadecimal number system, or any number system.
The number line is made stable by basic binding rules. For example division by zero is not allowed; the √-1 is not allowed in the real number plane but allowed in the complex number plane.
The matter mapped into the partitions of the number line; the partitions of the number line; the unit movements that established the partitions; the unique identities of the numbers in the partitions; the arithmetic operations; the adaptiveness of the number line; the groupings of the number line; the stability of the number line and the creation of the number line are all expressions of Pj Problems. Consequently, the number line is an expression of Pj Problems