Finite sequence in the context of Vector (physics and mathematics)

A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: $${\displaystyle r=b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots }$$Here . is the decimal separator, k is a nonnegative integer, and ${\displaystyle b_{0},\cdots ,b_{k},a_{1},a_{2},\cdots }$ are digits, which are symbols representing integers in the range 0, ..., 9.

Commonly, ${\displaystyle b_{k}\neq 0}$ if ${\displaystyle k\geq 1.}$ The sequence of the ${\displaystyle a_{i}}$—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all ${\displaystyle a_{i}}$ are 0, the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number.

In mathematics and physics, a vector is a physical quantity that cannot be expressed by a single number (a scalar). The term may also be used to refer to elements of some vector spaces, and in some contexts, is used for tuples, which are finite sequences (of numbers or other objects) of a fixed length.

Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.

In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set A is usually denoted A. The free semigroup on A is the subsemigroup of A containing all elements except the empty string. It is usually denoted A.

More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set.