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.

You can rewrite equations by changing the order in which the number is listed without changing result. The commutative property for addition and subtraction:

Division is not commutative as changing order effects result:

Of the six combinations we get three different answers. Because 1 cancels out when used as a denominator using a number other than 1 will give six different answers. Using 0 or 1 in proofs often is a bad way to check validity of an answer. Use numbers 2 and higher. You can commute exponential terms and radicals because they are simplified as constants, unless division operators come into conflict. If the division operator is present nothing can be moved to the left or right of it but anything on the left can be moved anywhere on the left and anything on the right can be moved on the right:

1. We have all operators present
2. The blue is left and the red is right of the obelus
3. Group using parentheses
4. Or write as a fraction

As a fraction everything left is the numerator and everything right is the denominator. It is easier to perform this calculation using parentheses but writing as a fraction shows you how division keeps things separate. The numerator (or LH) can be rewritten in different orders up to 3 ways :

2 ∙ √3 + 82 = √3 ∙ 2 + 82 = 82 + √3∙ 2

Because PEMA still applies we must do multiplication before the addition, as such there is no other way to re-order the terms. If we only had multiplication or addition only then we could have 6 re-orderings.

We have to find 82 first then multiply it with 2 then multiply it with √3. Notice that changing the order of the terms does nothing to change the order of operations.

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 th grouping will change the result, so we say "muliplication is not associative with addition".

The Associative Property with a focus on order of operations (pemdas) and reciprocity

The associative property says you can perform the operations in different orders and still get same answer.

We can group the first two numbers:

Or we can group the last two numbers

Whether we group the first two or last two numbers the answer is still the same. This will be true for any addition problem in that how you group terms does not change the result. Because of this addition is said be to “associative”.

Multiplication is also associative: 2∙3∙4 can be (2∙3)∙4 or 2∙(3∙4)

In general the associative property says that how you group addition or multiplication does not change result.
The reason being deals with PEMDAS and the order of operations.
For any operation and it's reciprocal operation math is associative.
Math is not associative when you have operations that are not reciprocal.

1. Addition is reciprocal with subtraction
2. Multiplication is reciprocal with division
3. Exponentiation is reciprocal with radicalization

PEMDAS says order matters, associativity appears to say order does not matter.
More correctly associativity should use the word “group or grouping” rather than the word “order”.
PEMDAS out ranks any property law, its order must be strictly adhered[1].
The only time PEMDAS appears to be broken is when performing a reciprocal operation[2].
We can't really talk about associative property without talking about PEMDAS and order of operations[3].

PEMDAS lists the order of operations, the discreet separate operations. Parentheses are always the first operation to do. What associativity does is not say order does not matter but that you can change how things are grouped. In other words parentheses always come first (according to PEMDAS) but you can change where the parentheses are ( according to associativity).

[1] Some people complain ignorantly take issue with PEMDAS by insisting that sometimes it is okay to break the order of PEMDAS like 2∙3÷4 where they state you can divide first instead of multiplying. Well duh. They are reciprocal operations and therefore both multiplication. [2] Using PEMA over PEMDAS is preferred by some teachers because it shuts up the people complaining about in the last statement. PEMA is superior but PEMDAS is what is easier to learn. If learning PEMA first it is easy to learn PEMDAS next but is not necessary, if learning PEMDAS first it is harder to learn PEMA but learning PEMA is necessary because it teaches reciprocal operations and does so in a concise manner.

1. Parentheses
2. . Exponents and radicals
3. Multiply and divide
4. 4. Add and subtract

PEMDAS uses six letters to state five operations and does not show reciprocity while PEMA uses four letters to state six operations and does show reciprocity. PEMA uses to less letters, includes radicals[4], and shows reciprocity. This makes PEMA better than PEMDAS for three reasons. Always do parentheses first (or any other grouping symbol). Consider a operation and its reciprocal “on the same level” and essentially to be the same operation therefore you can do them in alternate orders (not “any” order). Saying “any order” tends to confuse people and lose track of the correct order of operations. Without order of operations complex equations would have more than one answer, which is incorrect, and people would differ on the result leading to math being nonfunctional[5].

How should we do this problem: 2∙3÷4

There are two operation “∙” and “÷”, whichever one we perform first we get the same result. A dumb way to say this is “because ∙ and ÷ are the same operation. Close but inaccurate. This is the same inaccuracy as saying taking a square root is the same as exponents[6] in that is shows a lack of fundamental understanding. A more accurate response is that ∙ and ÷ are reciprocal and alternate orders of performing them are possible because reciprocity allows it.

A closer look:

There are different ways to perform this but all follow the same order of operations whether you call it PEMDAS BODMAS, BIDMAS, PEMA or whatever. The point is the result is the same.

Calling multiplication the same as division is an inaccuracy because it isn't the “same” it is the reciprocal and reciprocal means the “opposite”. Same does not mean opposite. Reciprocal operations are antonyms NOT synonyms. Likewise exponents are NOT the same as radicals. Defining them as such is a gross inaccuracy.

[4] PEMDAS and BODMAS/BIDMAS does not explicitly state radicals but only exponents/indices. They only list five operations and group exponents and radicals under a singular title. This lacks consistency. Using the same logic they should also group multiply with divide and add with subtract but don't. PEMA does and therefore is more logical and concise and more instructive.

[5] Calling five or more different things merely “power” or “powers” is confusing and lacks specificy. You might as well just say “number” as you are not differentiating one part of an equation or another. This leads to much pointing to an object to illustrate it rather than simply referring to it by a well defined name and being completely understood.

Ideally all equations should have well defined labels for each part of the equation. Consider that in a binary operation there are three numbers, two numbers are combined using an operator and a third number left as the result. Even in long equations with many terms only two numbers can be operated at a time. Of the six operations the four basic operations are well understood and defined, it is only when reaching higher math such as powering and rooting there is less definition and a greater lack of clarity as to what is what. At any time up to five things can be referred to as a “power” under PEMDAS. This is sloppy and does not provide enough clarity. Math should be well defined by clarifying terms with unique designations as to eliminate confusion when referring to a specific term. The following diagram lists the six operations with an unambiguous naming convention.

The picture does not completely give details how to pronounce but is drawn from that perspective, further explanation is required namely with radicals. Also a fraction is considered division here but special attention is given to renaming the dividend and divisor as numerator and denominator respectively. There may be more ways to write these operations but this is as far as I got. Not everyone is going to talk like an egghead all the time but if they do this is the proposed method to do it. Anything else is considered slang or inaccuracies for casual discussion (which is okay).

The number left of the operator (or top in the case of fractions) is considered the first term and is in blue.
The number right of the operator (or bottom in the case of fractions) is considered the second term and is in red.
The number right of the equals sign is considered the result and is in green.

All math consists of a binary operation in which three numbers are involved; the first number to the left of the operator, the second number to the right of the operator, the third number right of the equals/equality sign as the answer. Most only refer to the left hand or right hand of the equation this takes things farther.

This is all for now. This shows how the associative property fits with pemdas and reciprocity to form a more complete picture rather than teaching individually and leaving students to connect the dots themselves. The hope is that students connect the dots in a triangle and understand these concepts and use them together as a working unit rather than individual concepts that are scattered across math.

The Substitutive Property states:

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

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

To remove parenthetical terms you must use multiplication by the factor outside the parentheses, in this case “a”. If there is no factor then multiply by “1” to cancel the parentheses. If there is a minus sign in front of the parenthetical then you must multiply every term inside by –1. This inverts all signs of the terms (sign flipping) changing them to there additive inverses: 5(x+2) = 5x+10 –5(x+2) = –5x – 2 –(5x+3x–4) becomes –5x–3x+4.

To distribute means to multiply every term inside parentheses by the term outside of it. The reciprocal operation of distribution is factoring, where we divde every term by a factor. The factoring property states (ab + ac) = a(b + c), which is distriubtion in reverse. Factoring says that the sum of two products, "(ab + ac)", with a like factor, " a ", can be written as that factor multiplied by the "summed quantity" of the other two factors. Anything inside paranthese is called a quantity. A summed quantity means two numbers are being added in the quantity. So the factor "a" is being multipled with the quantity (b + c). The first factor is "a" and the second factor is "(b + c)", where (b + c) is a sum of b and c so it is called a summed quantity. Factoring is the reverse process of distribution. You divide every temr inside the paranthese by a factor that both sums share. Both "ab" and "ac" share " a " so we divide both sums (also called "terms") by "a". Divding both terms by " a" is called "factoring out" the a leaving b and c inside the paranetheses.

Distribution: A factor multiplied by a binomial or other series can be written as the sum of its products.

Distribution multiplies a to both numbers inside the () then writes answer as a sum of products.

For binomial and trinomial multiplication:

Distribution on top versus PEMDAS on bottom:

Top uses Pemdas and bottom uses distribution property:

Factoring: Dividing a product by one of it's factors to get the other factor.

Factoring is the oppposite of distribution. Distribution multiplies a factor with a binomial factoe to create a product. We sometimes call distribution "binomial multiplication". Factoring divides a "binomial product" by one of its known factors to find the other factor. This is covered in detail in multiplying polynomials.

Identity

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

Multiplicative identity property: For any number a, a·1 = 1·a. 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. What the three identity properties are saying is 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

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.
Transitive: if a = b and b = c then a = c. Two numbers are equal if they both equal a third number.
Substitutive: if a = b then a may be replaced by b. This allows known values to replace given variables.

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: \( 1/√x = x^0/x^-1 = (√x)^-1 = √x/x \)

What does it mean to have a power of – 1? To drop 1 power from exponent 0 using division.
What does it mean to have a power of 0? To drop 1 power from exponent 1 using division.

When you power you multiply by a factor of the same value as the base. The exponent is how many factors you are multiplying. If x^3 then x∙x∙x, if x^2 then x∙x, if x^5 ÷ x then x^5-1 = x^4. When dividing by the same base you subtract powers, this is called dropping powers or dropping a power. In x^5 ÷x we drop 1 power of x, in x^5 ÷ x^2 we drop 2 powers. In geometry raising or lowering powers is called losing or gaining dimensionality.

10^3 = 1000 10∙10∙10 2^3 = 8 2∙2∙2 x^3 = x∙x∙x
10^2 =100 10∙10 2^2 = 4 2∙2 x^2 = x∙x
10^1 = 10 10 2^1 = 2 2 x^1 = x
10^0 = 1 10/10 2^0 = 1 2/2 x0 = x/x = 1
10^-1 = 1/10 10/10∙10 or 10/10^2 2-1 = 1/2 2/2∙2 x^-1 = x/x^2 = 1/x
10^-2 = 1/100 10/10∙10∙10 or 10/10^3 2-2 = 1/4 2/2∙2∙2 x^-2 = x/x^3 = 1/x^2
10^-3 = 1/1000 10/10∙10∙10∙10 or 10/10^4 2-3 = 1/8 2/2∙2∙2∙2 x^-3 = x/x^4 = 1/x^3

This proves the law of the zero-ith power and the law of negative exponents for all x. It also gives a better intuition and pattern for why instead of only stating it as a fact and moving on. Positive and negative powers are reciprocals of each other. At its core exponentiation is simply multiplication of a repeated factor called the base or root. To justify x^0 = 1 first consider when a product equals 1. Only through reciprocity and the subsequent division does a product equal 1. A number times its reciprocal equals 1 as a product, stated simply as “a number divided by itself equals 1”. The only time a product equals 1 is when you perform a reciprocal operation, and every reciprocal operation simplifies to a number being divided by itself.

reciprocity = number divided by itself = 1
x∙1/x = x/x = 1
10∙1/10 = 10/10 = 1
7∙1/7 = 7/7 = 1

(+) exponents are multiplicative and (–) exponents are divisive

x^2 = x∙x 2 groups of x. In geometry this is a perfect square.
x^1 = x 1 group of x. In geometry this is a line.
x^0 = 1 1 group divided by 1 group is x/x.
x-1 = x/x^2 = 1/x 1 group divided by two groups
x-2 = x/x^3 = 1/x^2 pattern continues...

Fractional exponents are simply a notation convention for writing radicals without √ symbol.
It tells the person what operation to be performed and in what order.
If we want to take a first, second, third, fourth, or higher root we use a “1 over” fractional exponent.

x^1/1 = ^1√x = x a first root is simply the number itself
x^1/2 = ^2√x = √x a second root is a square root
x^1/3 = ^3√x a third root is a cubic root
x^1/4 = ^4√x a fourth root is a hyperbolic root

Higher roots have Latin names or simply use ordinal names. Names are used ubiquitously and this causes confusion over when and what someone is referring. Base and root mean the same thing but root can mean exponent if doing a radical operation. Power can refer to the exponent, the product or the entire term.

4^2 = 16 is of the form: Base^exponent = Product where the exponent is written in the index area in superscript.
Here we would call the base the root, but base is clearer. It is being multiplied so is getting bigger.

^2√16 = 4 is the form: ^exponent√Product = Base where the exponent is written in the index area of the radical sign and the product is called the radicand but product is clearer. A product is the result of multiplication. Here the product is being divided so it is in red meaning it is getting smaller.

Here we would call the exponent the “second root” or “square root”. Square roots omit the number two, higher roots than 2 write the number of the root. First roots are not really a thing. Calling the exponent the “root” is normal but calling it the exponent might be clearer. The base is actually the root of the product so if saying root you can refer to either the exponent or base. The “index” can refer to the area an exponent is or usually the exponent itself. Saying exponent is clearer. I would always use the labels as written because however odd sounding to some people there is no ambiguity.

There are other eccentric vocabulary and naming conventions but they all stem from lack of clarity leading to more and more made up words that serve to cause more confusion than eliminate it.

x^a/b : is of the form: Baseexponent/root , it is this form that contains fractional exponents that we will work on. 16^1/2 : means to take the square root of 16, √16 = 4 √16 : the radical sign omitted the exponent 2. When there is no exponent take a square root. 8^1/3 : means to take the cubic root of 8. ∛8 : the radical sign has an exponent of 3 in the index so we take the third root, ∛8=2. The exponents ½ and ⅓ are fractions. The top number is the power of the base number, and the bottom number is the power of the radical operation. In ½ the 2 means to take a square root and in ⅓ the 3 means to take a third root. In “ Baseexponent/root ” we have a base, raised to a power, and then are told to take a root of that “Baseexponent ”.

We can take as high or low a root as we want. For fractional exponents there is an algorithm where one part of the fraction powers the base and the other part powers the rooting process. The numerator is the exponent of the product while the denominator is the exponent of the radical operation, it is the number of roots to take.

^41/2 the base “4” has a exponent of 1. The “2” is the number of roots to take. We raise 4 to a power of 1 and then take the second root of it: 4^1/2 = 2√41 = √4 = 2. When square rooting we do not write 2 in the index of the radical. When product or other number has a power of 1 we do not write 1 in the index of the number.

2^4 = 2∙2∙2∙2 = 16 2 to the fourth power is 16. There are four factors of the base “2”.
√16 = 4 The square root of 16 is 4. Here we are calling 4 a “root” of 16.
^4√16 = 2 The fourth root of 16 is 2. 2 is a base or root of 16. It means the same thing.

Since 2∙2∙2∙2 = 16 there are four roots of base 2. The base is the factor and four factors means four roots means taking a fourth root will give 2 as an answer. Fractional exponents that do not have 1 as a numerator require calculating the product of the radicand and then taking a root.

4^3/2 = 8 Because we are taking the square root of 4^3.
4^3/2 = 2√43 = √64 = 8
4^3/2 = 8 The square root of four cubed is 8

Most times fractional exponents can lead to irrational roots/bases just like regular square roots can. The difference is that it is easier to find rational roots of products of integer exponents.

13^2/3 We need to first square 13 and then take its 3rd root:
13^2/3 = ^3√132 = ^3√169 ≈ 5.529 The number is irrational so use an approximation sign.

The highlight of arithmetic is learning the six operations and being able to solve any problem that uses +–∙÷^√ . This is the end of arithmetic. Next subjects have main topics with focused highlights:

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