What is Algebra?

Nature of Algebra 

Algebra is a set of finitely many symbols with functions and operators acting on these symbols in such a fashion that all these functions and operators can be implemented in an electro-mechanicalelectromechanical device or a computer or computer program.

A human being or a machine, by applying mechanically oriented operations could simply operate on the symbols of an algebra without any requirements to understand what the symbols or the operations are supposed to mean!

The latter is called Abstraction.

Example: The Comma operator

a , b

separates the ‘a’ from ‘b’ so it is not confused with ‘ab’ or in general separates the Right Hand Side (rhs) from Left Hand Side (lhs).

Example: The Function Sin

Sin(a)

computes the well known trigonometric function Sinus.

Words

A Word in an algebra is a sequence of concatenated symbols and functions and operators.

Example:

c = a + b

symbols c, =, a, +, and b are concatenated or sequenced into a string of letters or a word as the final ensemble.

Valid Words

Not all sequences of symbols in an algebra are Valid, indeed there is another ‘mechanical’ process of an algebra which qualifies a word either being Valid or Invalid.

Example:

a = b + 1 a valid word in algebra of arithmetic

1 +++, b an invalid word in algebra of arithmetic

Repeatability 

The first motivation and the last goal of an algebra is one and only one namely Repeatability:

If a set of symbols and operations manipulated by a person living in the 17th century was again manipulated by a person today, both efforts yield the same results!

This is due to the fact that Algebra at its core construction is like unto an electro-mechanicalelectromechanical entity that requires no intelligence to operate! Even a machine can carry out the same operations!

Students

need

to

be

aware

that

being good at algebra or making proficient use of algebraic systems does not require any intelligence, rather requires skills and in particular skills in programming. A person being good at algebra is decided by their skills to manipulate symbols and with no reference to his/her level of intelligence.