The notion of absolute value is used in the field of mathematics to name the value that has a number in addition to its sign. This means that the absolute value, also known as the modulus, is the numerical magnitude of the figure, regardless of whether its sign is positive or negative.
Consider the case of the absolute value 5 . This is the absolute value of +5 (positive 5) and -5 (negative 5). The absolute value, in short, is the same in the positive number and in the negative number: in this case, 5 . It should be noted that the absolute value is written between two parallel vertical bars; therefore, the correct notation is | 5 | .
The definition of the concept indicates that the absolute value is always equal to or greater than 0 and is never negative. From what has been said before, we may add that the absolute value of opposite numbers is the same; 8 and -8 therefore share the same absolute value: | 8 | . The absolute value can also be understood as the distance between the number and 0 . The number 563 and the number -563 are, on a number line, the same distance from 0 . This, therefore, is the absolute value of both: | 563 | . The distance between two real numbers, on the other hand, is the absolute value of their difference. Between 8 and 5 , for example, there is a distance of 3 . This difference has an absolute value of | 3 | . The concept of absolute value is present in several disciplines of mathematics, and the vector is one of them; more precisely, it is in the vector norm that we come across a similar definition. Before continuing, however, it is necessary to define Euclidean space, since these concepts are conjugated in this area. We understand by Euclidean space a kind of geometric space in which Euclid’s axioms are satisfied. An axiom is a proposition whose clarity is such that it does not require a proof to be admitted; specifically in the field of mathematics, this is the name given to the fundamental and improbable principles upon which theories are built. Euclid, meanwhile, was born in Greece around 325 BC. C., and his dedication to numbers made him worthy of the title of “Father of Geometry”. His most important work is a collection of thirteen books grouped under the title ”Elements“, where the aforementioned axioms (also known as Euclid’s postulates) are found, and we will see briefly below:
1) if we take any two points, it is possible to join them by means of a line; 2) it is possible to continuously extend all segments, regardless of direction; 3) circles can originate from any point, which will be taken as their center, and their radius can acquire any value; 4) any pair of right angles is congruent; 5) it is possible to draw a single line parallel to another from a point outside this one. Having exposed the bases of Euclidean spaces, we can say that the vectors in them can be represented in the form of segments that are oriented between any two points. If we take a vector, we can define its norm as the distance between two points, which serve as a limit; so much so, that in Euclidean space this norm corresponds to the modulus, that is, to the length of that vector. So as the absolute value, the module of a vector is always a positive number or zero, which represents a length, a distance. In this case, as in many others, associating this magnitude with a sign could lead to unnecessary complications. In the field of video game programming, on the other hand, the absolute value can appear on numerous occasions, according to the methodology of each developer. For example, when calculating the current speed of a person, we can ignore the direction in which it is moving and simply contemplate the segment that exists between 0 and the maximum speed, applying the corresponding acceleration; finally, just multiply the resulting value by the personaje direction vector to transfer it.