Algebra

Elements

The fundamental element of algebra is the variable — an unknown quantity represented by a symbol. Variables stand in for numbers we have not yet determined; their types reflect their role in the expression.

Types of Variables

• Dependent / Independent — related by a function
• Constant — a fixed value
• Parameter — a variable defining a family of solutions
• Free / Bound — scope within expressions
• Dummy — placeholder in summation or integration

Forms (Expressions)

Expressions are the forms of algebra — structures built from variables and operations:

• Linear / Nonlinear — degree of the polynomial
• Binomial — two terms; Quadratic — degree 2
• Homogeneous / Heterogeneous — uniformity of degree
• Factorable — decomposable into simpler expressions

The equation is algebra’s central form: it asserts that two expressions are equal — $f(x) = g(x)$ — and the solution is the recovery of the variable that makes the assertion true.

Axiomatic Foundations

Algebra rests on field axioms — the minimal rules governing addition and multiplication. Every algebraic manipulation is a chain of inference from these axioms.

• Commutativity: $a + b = b + a$, $\; ab = ba$
• Associativity: $(a+b)+c = a+(b+c)$
• Distributivity: $a(b+c) = ab + ac$
• Identities: $a + 0 = a$, $\; a \cdot 1 = a$
• Inverses: $a + (-a) = 0$, $\; a \cdot a^{-1} = 1 \; (a \neq 0)$

Equality as Inference

Solving is deduction. Each step applies an inference rule — adding the same quantity to both sides, multiplying by the inverse — preserving the equality relation:

$\text{if } a = b, \text{ then } a + c = b + c$

The quadratic formula is a theorem derived from these axioms via completing the square:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Procedures

The algorithmic lens reduces algebra to finite effective procedures — sequences of steps that terminate with a solution.

Solving a Linear Equation

1. Expand all brackets; collect like terms on each side.
2. Move variable terms to one side, constants to the other.
3. Divide by the coefficient.

Cost: $O(n)$ steps for $n$ terms.

Factoring a Quadratic

Given $ax^2 + bx + c$, find two numbers with product $ac$ and sum $b$, then factor by grouping. When no integer factors exist, the quadratic formula is the fallback procedure.

Other Key Procedures

• Completing the square: $ax^2 + bx + c \;\to\; a(x - h)^2 + k$
• Polynomial long division: divide $p(x)$ by $d(x)$, yielding quotient and remainder
• Partial fraction decomposition: split rational expressions for integration

Each procedure takes an expression as input and yields an equivalent, simpler form as output.

Algebra as a System

Seen as a system, algebra has clear stocks and flows:

• Elements (stocks): variables (unknown), constants (known)
• Forms (structures): expressions and equations
• Process (flow): solving moves values from the unknown stock to the known stock
• Equilibrium: an equation is satisfied — the system has reached balance when both sides are equal under the assigned values

Feedback Loop

Substituting a candidate solution back into the equation is a balancing loop: if the equation holds, the process terminates; if not, the solver iterates.

Variables are the elements, expressions are the forms, equations are the equilibrium condition.

The system takes inputs (known values, constraints) and produces outputs (solutions or simplified forms). Its boundary is the set of operations permitted by the field axioms.

Algebra as a Tool

The engineering lens treats algebra as the symbolic machinery for modelling, controlling, and computing:

• Modelling: express a real-world relationship symbolically — $v = d/t$, $F = ma$ — before substituting numbers
• Simplification: reduce an expression to canonical form (factored, expanded, partial fractions) to minimize computation
• System solving: Gaussian elimination solves $n$ linear equations in $n$ unknowns in $O(n^3)$ — the workhorse of numerical engineering

Key Constraints

• Division by zero is undefined; solutions must be checked against domain restrictions.
• Polynomial equations of degree $\geq 5$ have no general closed-form solution (Abel–Ruffini) — numerical methods become necessary.

The objective is always to find the simplest equivalent form that makes the answer evident or the computation tractable.