We define Algebra to be that first course you take that is higher than “General Math”.
Algebra is the study of math that uses variables to replace numbers. If someone writes c + 2 = 5 you stare at it and then realize that c=3. In this example, the symbol ‘c’ is a variable.
Algebra reviews the different types of numbers: Integers, Rational Numbers, Irrational Numbers, Complex Numbers.
Algebra includes the study of the work that can be done with the Equality Sign, =, with a focus on three properties known as the Axioms of Equality: Reflexivity of Equality, Symmetry of Equality and Transitivity of Equality.
Algebra looks at the basic properties of two math operations, Addition and Multiplication
Algebra includes the study of Identities and Identity Elements (‘0’ for Addition and ‘1’ for Multiplication):
- a+0=a
- a*1=a
Cancellation is one of the tools of Algebra used to simplify equations. When using Algebra to derive a law in Physics, it is common for it to get more complicated in the beginning steps, and then in the last few steps, cancellations start occurring, bringing it down to something simple.
Algebra includes the investigation of Equations, also known as Equalities:
- a = a
- a+b = b+a
- a*b = b*a
- a + (-b) = a – b
Algebra includes the study of Binomial Multiplication, with multiplications such as the following:
- (x+a)(x-a)
- (x+a)(x+a)
- (x-a)(x-a)
Algebra will introduce the student to graphs and the equation y=mx+b where m is the slope and b is the y-intercept.
Algebra includes Exponents and the multiplication of exponents.
Work using the Polynomial, 
Appendix A
High School educators in the mid to late 20th century split Algebra into two courses, Algebra I and Algebra II, and they complicated things further by putting Algebra I into the first year and Algebra II into the third year, with Geometry, offered the second year, seeming to be a sort of “intermission”, except that Geometry stole the most important part of Trigonometry (scheduled to be taught with Algebra II in the year–kind of like how
When we explain Algebra here, should we stop at the content offered for that first year Algebra Course, or should we continue and on and provide content offered in that third year course called “Algebra II and Trigonometry”?
Appendix B — The Higher Meaning
What do they mean when they say “algebra” but they are referring to something much higher than the class offered in high school or middle school?
- An Algebra over a Field is a Vector Space equipped with a Bilinear Product.
- Al Algebra is an Algebraic structure, which consists of a Set, together with Operations of Multiplication, Addition and Scalar Multiplication by elements of the underlying Field, and satisfies the axioms implied by Vector Space and Bilinear.
…
(we love this next quote): “Algebra” is one of the most overloaded terms in mathematics. Hell, even for “algebras over a field”, people will ask you if you mean unital and associative algebras or not.
Appendix C – a bit of a problem with the nomenclature
It also seems we need to specify if we mean for the word Algebra to be the study of something (like Biology or Chemistry). Some authors use the word algebra as a shorthand for algebraic structure. To us, that seems kind of like calling a bear a biology.
Appendix D – a note of caution about Bilinear Products
When you first see “Bilinear Product” it seems reasonable that you might expect it to take its place in a set whose other elements include “Dot Product”, “Cross Product”, “Scalar Product”, “Inner Product”, “Outer Product”, and several more.
The name “Bilinear Product” made it to the definition used in the Wikipedia page “Algebra over a field”. If you mouseover the “bilinear product”, you discover that it is not a link to a single page, but links to two separate pages for “bilinear” and “product.” The word product is interesting, for being explained as the result of multiplying two things that can be multiplied, and we might joke that “multiplication” suffers a “used in too many ways” almost as bad as “algebra”. With this in mind, we will be more forgiving if some authors don’t write “bilinear product” and instead write something more nebulous–we normally hesitate to accept this, because it means the better students might spend a lot of time trying reach the point where they feel they safely understand everything that the nebulous part of a sentence was trying to say.
Rob Sterling 