algebra

Engineering Mathematics15 Aug 2026 · 8 min read

The Field Laws of Real Number Arithmetic

The commutative, associative, distributive, identity and inverse laws, what each one licenses, and how they justify every step of routine algebraic simplification.

Engineering Mathematics15 Aug 2026 · 6 min read

The Index Laws: a Complete Treatment

Why powers add on multiplication and multiply on nesting, how those two facts force the values of x0 and x-m, and why 00 is left undefined.

Engineering Mathematics14 Aug 2026 · 12 min read

Abstract Algebra Prerequisites: Number Theory, Set Theory and Linear Algebra

Abstract Algebra Prerequisites: Number Theory, Set Theory and Linear Algebra: core definitions, structural results and verification methods in abstract algebr

Adding and Subtracting Fractions with Common Denominators

Handbook guide to adding and subtracting fractions with common denominators, with rules, worked examples, verification, common errors and practical.

Engineering Mathematics14 Aug 2026 · 7 min read

Algebra as Coordinatisation: Objects, Operations and Mathematical Structure

An introduction to algebra as a method of coordinatising mathematical and physical objects with abstract quantities and operations, rather than treating algebra merely as symbol…

Engineering Mathematics14 Aug 2026 · 7 min read

Algebra Study Workflow: Diagnose, Practise and Verify

A structured handbook method for learning algebra through diagnosis, worked steps, deliberate practice, checking and targeted review.

Cancelling Factors in Algebraic Fractions

Handbook guide to cancelling factors in algebraic fractions, with rules, worked examples, verification, common errors and practical engineering applications.

Comparison and One-Variable Relationship Models

Handbook guide to comparison and one-variable relationship models, with rules, worked examples, verification, common errors and practical engineering.

Consecutive Integer and Number-Relationship Problems

Handbook guide to consecutive integer and number-relationship problems, with rules, worked examples, verification, common errors and practical engineering.

Decimal Arithmetic: Addition, Subtraction and Multiplication

Handbook guide to decimal arithmetic: addition, subtraction and multiplication, with rules, worked examples, verification, common errors and practical.

Decimal Fractions and Decimal Division

Handbook guide to decimal fractions and decimal division, with rules, worked examples, verification, common errors and practical engineering applications.

Decimals as Fractions and Place Value

Handbook guide to decimals as fractions and place value, with rules, worked examples, verification, common errors and practical engineering applications.

Distance–Rate–Time and Linear Geometry Applications

Handbook guide to distance–rate–time and linear geometry applications, with rules, worked examples, verification, common errors and practical engineering.

Distributive Property and Combining Like Terms

Handbook guide to distributive property and combining like terms, with rules, worked examples, verification, common errors and practical engineering.

Double Inequalities and Bounded Ranges

Handbook guide to double inequalities and bounded ranges, with rules, worked examples, verification, common errors and practical engineering applications.

Exponent Laws for Products, Quotients and Powers

Handbook guide to exponent laws for products, quotients and powers, with rules, worked examples, verification, common errors and practical engineering.

Engineering Mathematics14 Aug 2026 · 5 min read

Exponent Rules and Powers

Understand exponent notation and apply product, quotient and power rules accurately when simplifying algebraic expressions.

Factoring Quadratic Trinomials

Handbook guide to factoring quadratic trinomials, with rules, worked examples, verification, common errors and practical engineering applications.

Formula-Based Linear Modelling and Unit Relations

Handbook guide to formula-based linear modelling and unit relations, with rules, worked examples, verification, common errors and practical engineering.

Fraction Multiplication and Division

Handbook guide to fraction multiplication and division, with rules, worked examples, verification, common errors and practical engineering applications.

Greatest Common Factor and Factoring by Grouping

Handbook guide to greatest common factor and factoring by grouping, with rules, worked examples, verification, common errors and practical engineering.

Linear Equations with Fractions and Decimals

Handbook guide to linear equations with fractions and decimals, with rules, worked examples, verification, common errors and practical engineering.

Linear Inequalities, Number Lines and Interval Notation

Handbook guide to linear inequalities, number lines and interval notation, with rules, worked examples, verification, common errors and practical engineering.

Mixed Numbers, Improper Fractions and Compound Fractions

Handbook guide to mixed numbers, improper fractions and compound fractions, with rules, worked examples, verification, common errors and practical.

Mixture and Concentration Problems

Handbook guide to mixture and concentration problems, with rules, worked examples, verification, common errors and practical engineering applications.

Modelling Unknown Quantities with One Variable

Handbook guide to modelling unknown quantities with one variable, with rules, worked examples, verification, common errors and practical engineering.

Multiplying and Dividing Negative Numbers

Handbook guide to multiplying and dividing negative numbers, with rules, worked examples, verification, common errors and practical engineering applications.

Multiplying Binomials and the FOIL Pattern

Handbook guide to multiplying binomials and the foil pattern, with rules, worked examples, verification, common errors and practical engineering applications.

Operations with Algebraic Fractions

Handbook guide to operations with algebraic fractions, with rules, worked examples, verification, common errors and practical engineering applications.

Percent Increase, Decrease, Markups, Discounts and Reverse Percent

Handbook guide to percent increase, decrease, markups, discounts and reverse percent, with rules, worked examples, verification, common errors and practical.

Percents: Part, Whole and Rate

Handbook guide to percents: part, whole and rate, with rules, worked examples, verification, common errors and practical engineering applications.

Prime Factorisation, Divisibility, GCD and LCM

Handbook guide to prime factorisation, divisibility, gcd and lcm, with rules, worked examples, verification, common errors and practical engineering.

Quadratic Applications: Number, Revenue and Work Models

Handbook guide to quadratic applications: number, revenue and work models, with rules, worked examples, verification, common errors and practical engineering.

Quadratic Applications: Projectile and Geometry Models

Handbook guide to quadratic applications: projectile and geometry models, with rules, worked examples, verification, common errors and practical engineering.

Quadratic Distance, Rate and Motion Problems

Handbook guide to quadratic distance, rate and motion problems, with rules, worked examples, verification, common errors and practical engineering.

Quadratic Equations by Factoring

Handbook guide to quadratic equations by factoring, with rules, worked examples, verification, common errors and practical engineering applications.

Quadratic Equations by Square Roots and the Quadratic Formula

Handbook guide to quadratic equations by square roots and the quadratic formula, with rules, worked examples, verification, common errors and practical.

Engineering Mathematics14 Aug 2026 · 5 min read

Quadratic Formula: Solving General Quadratic Equations

Use the quadratic formula for ax²+bx+c=0, identify coefficients correctly, simplify the discriminant and interpret resulting roots.

Quadratic-Type Factoring and Rational Expression Reduction

Handbook guide to quadratic-type factoring and rational expression reduction, with rules, worked examples, verification, common errors and practical.

Radicals and Root Properties

Handbook guide to radicals and root properties, with rules, worked examples, verification, common errors and practical engineering applications.

Rational Equations That Lead to Quadratics

Handbook guide to rational equations that lead to quadratics, with rules, worked examples, verification, common errors and practical engineering applications.

Rational Equations That Reduce to Linear Equations

Handbook guide to rational equations that reduce to linear equations, with rules, worked examples, verification, common errors and practical engineering.

Rearranging Formulas and Solving for a Variable

Handbook guide to rearranging formulas and solving for a variable, with rules, worked examples, verification, common errors and practical engineering.

Reducing Fractions to Lowest Terms

Handbook guide to reducing fractions to lowest terms, with rules, worked examples, verification, common errors and practical engineering applications.

Signed Numbers: Addition and Subtraction

Handbook guide to signed numbers: addition and subtraction, with rules, worked examples, verification, common errors and practical engineering applications.

Simple Interest and Investment Allocation Problems

Handbook guide to simple interest and investment allocation problems, with rules, worked examples, verification, common errors and practical engineering.

Simplifying Monomials and Algebraic Powers

Handbook guide to simplifying monomials and algebraic powers, with rules, worked examples, verification, common errors and practical engineering applications.