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Properties, Laws, Rules, Principles, and Theorems

1. Properties of Real Numbers (fundamental behaviors of addition & multiplication)

Supporting Properties

  • Inverse Property
  • Zero Property of Multiplication
  • Closure Property

    • 2. Laws of Equality (how you can legally manipulate equations)

    Disclosure

    Different textbooks organize material differently or disagree on definitions of words, phrases, notation, and emphasis. As math evolves attempts are made to unify all branches into one, standardize all conventions into one, and reorder all guiding truths into a single structure. Before continuing I am morally obligated to tell you:

    1. Debate exists between educators on how math should be taught and distilled.
    2. Math, language, and physics are at odds with each other.
    3. Standard textbooks are written for use with a professor to give lectures and require important classroom discussion among peers.
    4. Standard textbooks emphasis self discovery and rigor which is required at the graduate level but inappropriate at the learning level. The authors forget they too were beginners once and required clear instruction over rigid formal logic.
    5. The balance between rigor and accessibility is delicate. Too much hand holding leads to rapid understanding in the beginning but confusion and disillusioned glass ceilings later.
    6. Easy courses lead to servers of the status quo (i.e. for dummies books, get a job etc.), Hard courses lead to designing the status quo. This is the difference between being perfunctory (as a "functionary") and being an arcitect of future civilizations (a founder, leader etc.)
    7. It is hard to teach "formal truth" upfront when a person has no exposure to advanced ways of thinking.
    8. Standard process is to give "naive" understanding first, teach computation, then once functionary status is achieved teach "formal logic".
    9. Formal logic is the truth, yet many professors lack intuition ("naiveite") and teach theoretical abstraction("logic") with no known real world application thus being "can't see the forest for the trees" or "head in the weeds".
    10. Dark economic forces suppress enlightenment to eliminate competition. "Bad school systems" create worker bees (functionaries), "good school systems" create manipulators of worker bees (drones/robots).

    The previous statements are well documented fact and NOT conjecture nor debated or contested. Simple crushing proofs for all statements can be provided on demand.

    Nomenclature

    In algebra, the properties (or laws) of operations are a handful of foundational rules that everything else builds on. Different textbooks group them slightly differently, but the core property laws usually include:

    The "Big Four" for addition and multiplication which are the Commutative Property, Associative Property, Distributive Property, and Identity Property. These four are what is usually meant by "The Properties of Algebra". The Closure Property is sometimes included, sometimes not. Here is a detailed breakdown:

    1. Properties (of operations)

    These describe how numbers behave when operations are applied. They’re called properties because they’re inherent features of the number system (like commutativity, associativity, distributivity, identity, inverses, closure). They tell you what’s true no matter what numbers you plug in.

    2. Laws of Equality

    These aren’t about the numbers themselves, but about what you’re allowed to do to both sides of an equation without changing it from being true. The main 4 are the Reflexive Law, Symmetric Law, Transitive Law, Substitution law, with Addition Law of Equality (addition principle), Multiplication Law of Equality (addition principle restated under multiplication) equally important but more supporting Laws. They get called laws rather than properties because they are rules of manipulation, like legal moves in algebra, instead of structural truths about the numbers themselves.

    3. Rules (of thumb)

    These are not formal "rules". Some people call properties and laws the "Rules of Algebra", Malone Math does not. In Malone Math rules are ordered steps to find totals or sub-totals. Rules focus not on numbers or equations but procedures. Such procedures can be called "algorithms", "routines", or even "tricks". These are not tricks merely codified logic for deriving answers or simplifying an expression.

    4. Principles

    "Principles" is an older term for the "laws of equality". Principles state foundational truth about algebraic manipulation of equations. A principle will try to generalize many laws into one "principle that rules them all". While laws themselves refer to individual operations or conditions, a principle will try to write one law to describe all conditions. Therefore a principle is a generalized rule for all laws of a certain type. As we master math branch by branch we will encounter laws titled "fundamental principles". Almost every branch of math has a fundamental principle. These laws act as rigorously defined foundational logic defines a single branch or topic of math.

    5. Theorems

    A theorem is an algorithm that produces guaranteed results, without saying anything about properties of numbers or laws of equality. In every branch of math there exists one or two main theorems. These theorems are the main takeaway from the branch to learn how to perform computations. Geometry has some 200 or more theorems! Keeping track of them all is a pain. The main ones are Pythagoras Theorem and Heron's Theorem. A theorem is a formal way of stating a general result, where formal means "proven".

    6. Formulas

    A formula is an equation that substitutes variables for numbers so that it can solve many equations. This is called "the general formula" or "general solution" to a problem. A "problem" in this sense refers not to a single math equation but entire class of problems that fit certain conditions that are classified by "type". The first formulas you will see will be for "area problems". This shows how the word "problem" is used with formulas. The format is "type-problem", so in this case "area" is the type of problem. Formulas are even more specific in that you have "the formula for the area of a rectangle" as a class of area problems. Naturally for every shape you have a different formula. When mathematicians say they "generalize solutions" they mean they are writing formulas. A general solution must solve all problems of a class (sub-type) to be called a formula. This means it must solve infinite equations within the class.

    7. Algorithms

    Algorithms are lesser form of formulas. They so do not solve every case so are not generalized solutions. They do solve some, many, or most of a sub class and only offer "special solutions" to a sub class of problems. Algorithms only solve finite problems so their best use is optimization for special of a very special type of equation within a sub class. This is useful if you are constantly working on a small set of problems with different values with many operators or steps. Specifically algorithm refers to "the number of steps taken, in what order, and what operation was performed at each step". The best example would be PEMDAS. PEMDAS tells what steps in what order, and what operation to perform, but since it does not have variables or numbers, only process, it is not a formula or solution in of itself. So algorithms serve two roles: Broad steps for completing computations, or specific formula like steps for solving a small set of problems within a sub class of a type that falls short of being a general case formula itself.

    6. Definitions

    1. Properties
      • About: Numbers (or objects) themselves.
      • Meaning: Intrinsic truths about how numbers behave under operations.
      • Example: a + b = b + a (commutative property).
    2. Laws
      • About: What you can do with equations, equalities, and equivalence statements.
      • Meaning: Formal rules that preserve truth when manipulating expressions.
      • Example: Law of Equality: If a = b, then a + c = b + c.
    3. Rules
      • About: Procedures.
      • Meaning: Step-by-step guidelines that tell you how to perform operations correctly.
      • Example: Order of operations (PEMDAS/BODMAS).
    4. Principles
      • About: Generalized truths that cover all laws within a branch of math.
      • Meaning: Broad foundations that unify multiple laws.
      • Example: Principle of Superposition in linear algebra — results add together if the system is linear.
    5. Theorems
      • About: Guaranteed results when conditions are met.
      • Meaning: Statements proven true within mathematics using definitions, axioms, and logic.
      • Example: Pythagorean Theorem (\(a² + b² = c²\) for right triangles).
    6. Formulas
      • About: General cases for all problems of a certain type.
      • Meaning: Ready-made expressions that apply universally in that problem class.
      • Example: Quadratic Formula: x = (\(−b ± √(b² − 4ac)) / 2a\).
    7. Algorithms
      • About: Targeted step-by-step processes.
      • Meaning: A subset of problem-solving methods, often optimized for efficiency, sometimes specialized for small sets of problems.
      • Example: Euclidean Algorithm for finding the greatest common divisor.

    In short:

    Takeaways:

    Summary:

    Modern education calls everything a property or law. Malone Math clearly delineates 5 types of classification based on what is being described: 1. Properties are of numbers; 2. Laws are of equations (balancing)[1]; 3. Rules are of computational steps (algorithms)[2] 4. Principles are "generalizations"[3] of "laws" that are singular per branch; 5. Theorems are "rules" that are rigorously defined using formal "logic"[4].

    foot notes: [1] "Balancing an equation" is foundational to solving equations in algebra and the main technique used. [2] Algorithms are defined as the "logic" used and the "steps" taken to complete a process. [3] Generalizations are defined in math as statements, called the "general case" that are true for all cases of a type not just some cases. The opposite of the general case is a "special" case that violates the law/rule/theory. [4] Logic is the symbols, operations, and steps used in a process, making it follow very much the definition or relation to an algorithm, however "logic" usually refers to "logical symbolism" and often refers to the symbols used in writing mathematical definitions using formulas called "formal logic".