Properties, Laws, Rules, Principles, and Theorems

Nonmenclature

The foundation of Algebra is built on the Properties of Real Numbers, Laws of Equality, and Pemdas. These statements represent what is allowed and what is not during computation of equations. The original purpose of these immutable truths was to formalize math so that everyone agreed on correct methods and procedure for achieving results. Without these foundations people would disagree and invent their own symbols, order, and produce different answers. This would make universal communication through math impossible. Just as the order of numbers is universally agreed (1,2,3, etc.) so to is what is legal during operations or not. For a modification of terms to occur it must be justified by the foundational properties which Malone Global University proudly presents.

Malone Math does away with naive explanantions about foundations of math and presents them in the purest form:
1. Properties of Real Numbers defines truth about numbers themselves.
2. Laws of Equality defines truth about equations.
3. Rules are procedural step by step explanation for computing numbers.
4. Principles are generalized single statements that capture the essence of a body of laws.
5. Theorems are algorithmic formulas tested to proved guarenteed results written using formal logic.
6. Formulas are generalized equations that solve infinite cases of a type.
7. Algorithms are equations that solve finite cases of a type.

Malone Math writes its own formulas generically called "algorithms". Technically speaking an alogrithm is an equation that uses symbols and steps in a specific order to derive computations and solve for an answer. Formulas themselves are algorithms but they go beyong special cases and solve for the general case, which means for all problems of that type not just some. Theorems are formulas that go above basic formula notation and use rigous logic to hard code what is stated by formulas but with a more professional tone using logical symbolism as would be seen in discrete mathematics courses.

In plain speak: Properties are about numbers. Laws are about equations, hence "equality". Rules are about procedures. Principles are about surmizing all the laws for a single branch of math. Theorems are formulas written in terse language using logical notation. Formulas are equations written in variables, that solve every case of a type, by replacing variables with actual values, to solve an equation. Algorithms are similar to formulas but do not solve every case of a type only narrowly chosen ones.

Therefore: Algorithms are custom made equations that deal with a small set of problems. This makes them highly specialized to a subset of problems. This allows optimization and faster results than standard formulas. Standard formulas are also algorithms but for every case and are not optimized for subsets. Theorems are also formulas but are written using rigous logic while formulas are written using naive logic. Rigor means "professional mathematician language", naive means "beginners intuition". Here "naive" does not mean wrong in the English sense, both naivete and rigor achieve the same answers, but rigor states conclusions in a smarter mathematically perfect tone while naive formulas are used to teach beginners. Beginners get the right answer but cannot articulate their process or results using the precise language of rigor. They are right in their answers but wrong for why the answer is what it is as they do not fully understand the hidden mechanics. This is the fundamental difference between naive beginner understanding and professional rigor as pure logic.

Properties of Real Numbers

1. Commutative Property
2. Associative Property
3. Substitutive Property
4. Distributive Property
5. Identity
6. Inverses
7. Closure

The Property Laws use letters like a or b or c to mean any three numbers that are not the same. Using letters in place of numbers is called using variables. Variables are very important in used in every algebra problem. In fact variables is what separates algebra from arithmetic. These laws are taken directly from the glossary at the beginning of the book "Malone Algebra".

Properties: Properties are rules that dictate how order of operations should behave and what is permissible when reordering terms to restructure an equation. Properties show an how the numbers behave and interact with each other on a foundational level as opposed to simply solving for an answer. Understanding and having innate sense of properties shows an advanced mastery of mathematics and goes beyond superficial plugging in of values to get a result. You know how the numbers behave conceptually rather then mechanically.

The Commutative Property states:

a + b = b + a and a ∙ b = b ∙ a

1. The commutative property of addition says you can move sums in any order.
2. The commutative property of multiplication says you can move factors in any order.

This states that the order of terms is irrelevant and moving them around does not change the answer for addition and multiplication. 1 + 2 is same as 2 + 1 , and 2 ∙ 3 is the same as 3 ∙ 2.

The Associative Property States:

(a + b) + c = a + (b + c) and (a ∙ b) ∙ c = a ∙ (b ∙ c).

1. The associative property of addition says you can add in any order.
2. The associative property of multiplication says you can multiply in any order.

For any numbers a, b, c (a + b) + c = a + (b + c) and (ab) c = a (bc). It says that if you change the grouping for pure + or pure ÷ the result is the same. If you mix addition and mulitplication operation in the same equation then order matters and PEMDAS states multiplication before addition, and to work left to right. For mixed equation like 1 + 2 ∙ 3 changing the grouping will change the result, so we say "muliplication is not associative with addition".

The Distributive Property states:

For any numbers a, b, c:

a (b + c) = ab + ac and (b + c) a = ba + ca
a (b – c) = ab – ac and (b – c) a = ba – ca

It says that a number times a quantity is equal to the sum of each term in the quantity being multiplied by that number. This allows you to write polynomial multiplication as a series.

The Substitutive Property states:

If a = b then a + c = b + c.

It is saying that if a variable equals another object (number, equation etc.) then you can erase the variable and replace it with that other object and vice versa. The example given above shows an application of the transitive law in reverse.The substitutive property states an equation first and then an equivalnce second. The transitive law reverses this order and states an equivalnece first and an equation second. In geometry this would be called positive versus contrapositive using an "if ... then" conditional statement.

Identity

Additive identity property: For any number a, a+0 = a. This states that adding or subtracting 0 does not change the value.

Multiplicative identity property: For any number a, a·1 = 1·a. This states that a number times 1 is itself.

Multiplicative property of zero: For any number a, a·0 = 0. This states number times zero is zero.

The identity property is calling the number's value it's "identity", similar to the way people have names as identity, numbers have values as identity.
For example this symbol " 5 " is called "five", that is it's name, it is the name of that number and so is called it's identity.

The three identity properties are saying that a numbers value (name/"identity") does not change during the given operations: 1. If you add zero to a number then number is unchanged. 2. If you multiply a number by 1 the number does not change. 3. Here they are referring to zero as the targeted number; If you muliply zero with another number zero is still zero. This last one flips the logic a little bit. Before they were targeting the variable "a" so this is sneaky: a + 0 = a, a is unchanged. a · 1 = a, a is unchanged, 0 · a = 0, 0 is unchanged. It is simpler to write it this way. These are different laws that have been rewritten as properties over hundreds of years so a bit pieced together logic wise. In plain speak we say , " anything times zero equals zero".

Inverses

Additive inverse: (a) + (-a) = 0 and (-a) + (a) = 0.

This says that two numbers of the same numerical value but opposite quality value sum to zero. This is called "canceling under addition" where two additive inverses sum to zero. "Numerical value" refers to quantity or "how big" the number is, "quality of value" refers to sign of the number such as whether negative or positive. "Numerical value" is stated in absolute value (always positive) and is another way of saying "quantity" or "quantity of value.

In an addition problem with many terms you can eliminate some work by finding additive inverses and canceling them. This removes them from the equation making it easier to solve. Since addivite inverses cancel to 0 and adding 0 does noty chnage value it is "eliminated" from computation thus saving work and moving one step closer to final answer.

Multiplicative inverse: x · ¹/_x = 1.

This says that the product of a number and it's multiplicative inverse (reciprocal) is equal to 1. The multiplicative inverse of a number is called it's "reciprocal". The reciprocal of a number or function is that number written as a fraction where the numerator and denominator are inverted. The product of two reciprocals is 1. Ex. ⅕∙5 = 1. We generalize this to any number can be written as a fraction over 1 such as x = ^x/₁, where x times it's reciprocal ¹/_x becomes ^x/₁ · ¹/_x = ^x/_x = 1. This is called "canceling under multiplication". Similar to additive inverses in an addition problem, in a multiplication problem two reciprocals cancel to 1 thus eliminating them since multiplying by 1 does not change the answer.

Closure

Laws of Equality

Properties or laws of equality: The following properties are true for any a, b, or c.

Reflexive: a = a. This is similar to the identity of a number.
Symmetric: if a = b then b = a. This just restates substitutive property twice.
Specifically it says you can switch the order of lh and rh of an equation: 2+2 =4, 4 = 2+2.

1. Reflexive
2. Symmetrive
3. Transitive
4. Substitutive
5. Additive
6. Multiplicative

Reflexive

Symmetrive

Transitive

Substitutive

Additive

Multiplicative

Laws of Inference

1. Exponent Laws
2. Radical Laws
3. Logarithm laws
4. Reciprocal Law

Exponent Laws

The following rules are considered "laws" under the definition given in the chapter's nonmenclature section. Remember that laws state equality between two halves of an equation. Then these laws are not taking about the exponents as numbers but how to create equivalance statements. The goal is to take the left-hand side of the equation and rewrite it differently on the the right-hand side. The practical application is changing the expression from difficult to operate forms to easier ones. Once an expression has been converted to an easier for solving or canceling becomes straighforward.

1. Multiplication Law (same bases): \( x^ax^b = x^a+b \)
2. Division Law (same bases): \( x^a/x^b = x^a-b \)
3. Power Law: \( (x^a)^b = x^ab \)
4. Multiplication Law (same powers): \( x^ay^a = (xy)^a \)
5. Division Law (same powers): \( x^a/y^a = (x/y)^a \)
6. Negative exponents Law : \( x^0/x^b = x^0-b = x^-b = 1/x^b \)
7. Fractional exponents Law: \( x^1/n = n√x \)
8. Rational denominator law: \( 1/√x = x^0/x^-1 = (√x)^-1 = √x/x \)

1. Algebra: Variables, Functions, Graphing, the Quadratic Formula, Matrices, Series
2. Geometry: Unit Circle, properties of shapes and area formulas
3. Trigonometry: Trig Functions and Co-Functions (cosecant,secant,cotangent)
4. Calculus: Derivatives and Integrals

Radical Laws

Logarithm laws

Reciprocal Law

Rules of Computation

1. Division of zero
2. Negation
3. Negative Exponent rule
4. Negation Rules
5. Negating Negative Rule
6. Negative Factor Law
7. Two Negatives Rule
8. Negative Bases Rule
9. Adding Powers Rule
10. Subtracting Powers Rule
11. Multiplying Powers Rule
12. Do Not Multiply Different Bases By Adding Exponents Rule
13. Commutative Property Rules
14. The Associative Property Rules

Division of zero

Division of zero: Division by zero is undefined. It is undefined since dividing by zero does not give us a singular value but potentially infinite values. Whereas “definition of a number” means that a number is unique, “undefined” means such a unique number either does not exist or a domain error has occurred.

Negation

Negation: Negation means reversing then meaning of a a statenment, an operation, or taking the opposite of something. Negative statements negate positive ones. Likewise Subtraction negates or is the negation of addition. By multiplying by a -1 we can negate the sign of terms or parenthetical series. This is useful simplifying and reducing equations to make them cleaner looking or easier to solve.

Negative Exponent rule

Negation Rules

Negating Negative Rule

Negative Factor Law

Two Negatives Rule

Negative Bases Rule

Adding Powers Rule

Subtracting Powers Rule

Multiplying Powers Rule

Do Not Multiply Different Bases By Adding Exponents Rule

Commutative Property Rules

The Associative Property Rules

Principles of Math

1. Addition principle

Addition principle: If a = b then a+c = b+c. This says if we add something to one side of an equation we must must add same amount on the other. This is main concept of algebra. What follows is any operation done on one side must be done on the other. The addition principle is shorthand for all these other instances of balancing an equation.

This is an early priciple that is saying the same thing as the

Theorems

1. Fundamental Theorem of Arithmetic
2. Pythagoras theorem
3. Binomial theorem
4. Bayes Theorem
5. Stokes Theorem

Formulas

1. Linear Formulas
2. Quadratic Formula
3. Cubic Formula
4. Function formulas

Algorithms

1. Formula for Powers
2. PEMDAS RULES
3. Equation Rules