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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Welcome to *the only place where the trigonometric values on unit circle is properly connected to the trigonometric ratios of right-triangles*.

• The trigonometric ratios of right-triangles are defined as -- "The right-triangles having a given angle are similar. The ratio of sides for those right-triangles is a known constant".

• The trigonometric values on unit-circle are defined as -- "The representative similar triangle is taken in unit-circle with hypotenuse `1`. The trigonometric ratios become into the horizontal and vertical projections.".

The thought-process in the above is *revolutionary and aweinspiring*.

The trigonometric values for all quadrants and for compound angles are also covered. *(click for the list of lessons in this topic)*

Trigonometric Values for all angles in unit circle form

The *revolutionary and aweinspiring explanation of trigonometric values on unit circle* is provided.

• The right triangles with a given angle are similar. The ratio of sides of these similar triangles are defined as trigonometric ratios.

• One such similar triangle is the right-triangle in unit circle. This can be representative of all the similar triangles.

• Being in the unit circle the hypotenuse is `1` and the coordinate point `(x,y)` defines the trigonometric ratios.

In this, computing trigonometric values for an angle in any quadrant is explained.

Trigonometric Ratio for Angles in All Quadrants

Learn something *made astoundingly easy*: Representing trigonometric values for angles in all quadrants of coordinate plane to an equivalent trigonometric value in first quadrant is explained.

Trigonometric Identities for compound angles

In this page, expressing trigonometric values for sum or difference of two angles, in terms of the trigonometric functions of the two angles is explained.

This provides *very simple proofs* for trigonometric identities of compound angles, `sin(A+B)`, `cos(A+B)` etc.

Open your mind to something exciting -- relearn what you know already, the "whole numbers", in a *refreshing new form*.

Learn about

• *grouping*

• *regrouping* or carry over

• *de-grouping* or borrowing

• *First principles* of comparison, addition, subtraction, multiplication, and division

• *Procedural Simplifications* by place-value

• numerical expressions and *precedence order**(click for the list of lessons in this topic)*

Introduction to Whole Numbers

Relearn the basics to build the proper understanding to the

• digits, numeral, and numbers

• numbers in abstraction

• grouping 10 into 1 of higher place-value

• face and place-values

• approximation and estimation of numbers

The pages in this lesson are

Comparing Whole Numbers

Do you know the whole number is an *ordered sequence*? Based on the order of whole numbers, the following are defined:

• predecessor and successor,

• largest or smallest,

• ascending and descending orders.

Whole numbers addition and subtraction

Do you understand what is regrouping or carry over and what is de-grouping or borrow? Do you want to have a refreshing new perspective of addition and subtraction of whole numbers?

Addition is combining two numbers.

Subtraction is taking away part of a number.

Learn here how these translate into simplified procedures for large numbers

(1) addition by place-value with regrouping and

(2) subtraction by place-value with de-grouping.

Whole Numbers: Multiplication and Division

Do you want to have a refreshing new perspective on multiplication and division of whole numbers?

Multiplication is repetition of a quantity.

Division is splitting of a number into equal parts.

Learn here how these translate into simplified procedures for large numbers (1) multiplication by place-value and (2) division by place-value.

Numerical Expressions with Whole Numbers

Do you understand what a numerical expression is?

Do you want to know the precedence order to be followed in simplifying numerical expressions?

That is, do you want to understand how BODMAS or PEMDAS helps to have an uniform understanding of what a numerical expression evaluates to?

Learn here -- the fundamentals of precedence order of arithmetics in numerical expressions.

When dividing a whole number, dividend, by another whole number, divisor, the result is quotient and remainder. The remainder is `0`, or in other words, the divisor divides the dividend without a remainder. This basic property leads to understanding all the following

• odd and even numbers

• prime and composite numbers

• factors and multiples of a number

• LCM and HCF

• Divisibility tests

This lesson provides *breathtakingly simple and intuitive explanations* to the above topics. Especially, the divisibility tests are explained in a *simple-thought-process* to understand why the procedure works. *(click for the list of lessons in this topic)*

Classification of Numbers based on Remainder in Division: Odd-Even and Prime-Composite

This page introduces the following concepts in a simple thought process.

• The numbers that are divisible by `2` without a remainder, they are even. And the numbers that results in `1` as remainder, they are odd.

• Similarly prime numbers and composite numbers are defined.

Factors, Multiples, Prime Factorization

Factors, Multiples, and Prime factorization are made simple and clear.

Highest Common Factor

This lesson provides a brief overview of

• common factors

• highest common factor

• Simplified procedure to finding HCF

The above is explained in a simple-thought-process.

The pages in this lesson are

Least Common Multiple

This lesson provides a brief overview of

• common multiples

• least common multiple

• Simplified procedure to finding LCM

• Relationship between the numbers, HCF, and LCM

The above is explained in a simple-thought-process.

The pages in this lesson are

Basics of Divisibility Test

A number (dividend) is divisible by a divisor number if the remainder is `0`.

A procedure, to check if a given number is divisible by a divisor or not divisible by a divisor, is called divisibility test of the divisor. In this page, some basic divisibility tests for the following are introduced.

• product of multiple numbers

• sum of multiple numbers This forms the foundation to developing divisibility tests for numbers like `2, 3, 4, cdots`.

The pages in this lesson are

__Understanding Divisibility__ *redo *

Simple Divisibility Tests: 2, 10, 3, 4, 5, 11, 9, 6

In this page, a simple overview of the divisibility tests for `2, 10, 3, 4, 5, 11, 9, 6` are provided. The procedure is outlined with simple-reasoning, which helps students to understand the procedure.

Simplification of Divisibility Tests: 8, 12, 15

To develop divisibility tests for numbers like `8, 12, 15`, the following methods of simplification are analyzed and explained.

• Simplification of divisibility test by subtraction

• Simplification of divisibility test by division

• Simplification of divisibility test by factors

Using the above, Divisibility tests for `8, 12, 15` are explained.

Reducing number of Digits for Divisibility Test

To develop divisibility tests for numbers like `7` and `13`, "Simplification of divisibility test in number of digits" is analyzed and explained.

Using that, Divisibility tests for `8, 12, 15` are explained.

Integers take a new name "*directed-whole-numbers*". That is, received an amount of 3 and given an amount of 3 are two different numbers with +3 (aligned to the chosen direction) and -3 (opposed to the direction). Learn in this

• What are directed numbers?

• How to handle direction in numbers?

• Sign and Absolute values of directed numbers

• *First Principles* of Comparison, addition, subtraction, multiplication, and division : directed numbers.

• *Procedural Simplification by sign and place value*

• Numerical Expressions with Directed Numbers - Precedence order and Sequence *(click for the list of lessons in this topic)*

Introduction to Integers (directed whole numbers)

Get a refreshing new perspective into what integers are -- "directed whole numbers". For example, received 3 and given 3 are two different numbers -- +3 and -3 respectively. In this lesson, understand

• how directed numbers is specified using positive and negative numbers.

• handling of sign in numbers.

• ordinal property of negative and positive numbers.

• sign and absolute value of a given number.

Comparing Integers

Do you want to learn, Where the negative numbers fit in the "ordered sequence" 0, 1, 2, ...?

In this lesson, the *comparison in first principles* -- matching two quantities to find one as smaller, equal, or larger than the other -- is extended for directed numbers. Based on that, the place of negative numbers in ordered sequence is explained.

The comparison in first principles is extended to the simplified procedure *comparison by sign and place-value*.

Once the ordered sequence of positive and negative numbers is specified, the following are easily understood.

• predecessor and successor,

• largest or smallest,

• ascending and descending orders.

Integers: Addition & Subtraction

Do you want to understand how the negative numbers are handled in addition and subtraction?

In this lesson, learn the following for directed-whole-numbers.

• addition in first principles -- combining two quantities and measuring the combined

• subtraction in first principles -- taking away a part of a quantity and measuring the remaining quantity.

• Simplified procedure : Sign property of Addition

• Simplified procedure : Sign property of Subtraction.

Integer Multiplication & Division

Do you want to understand how the negative numbers are handled in multiplication and division?

In this lesson, learn the following for directed-whole-numbers.

• directed numbers multiplication in first principles -- repeatedly combining a quantity and measuring the combined and

• directed numbers division in first principles -- splitting a quantity into a number of parts and measuring one part

• simplified procedure : Sign Property of Multiplication

• simplified procedure : Sign Property of Division

Numerical Expressions with Integers

Do you want to understand how the negative numbers are handled in numerical expressions?

In this lesson, learn the following for directed-whole-numbers.

• precedence order : PEMDAS or BODMAS

• sequence of operations : left to right

Most importantly, learn handling of subtraction as inverse of addition.

Learn the *revolutionary new details uncovered about fractions*. Fractions are measure of "parts of a whole". Numerator is the count of the parts and Denominator represents the size of one part as *place-value* for the numerator. In this, learn the following.

• Fractions as numbers with specified place-values

• Fractions as directed-numbers (positive and negative)

• Fractions arithmetics : first principles

• fractions arithmetics : simplified procedures

Addition/Subtraction with LCM of denominators

Multiplication by multiplying numerators and denominators

Division as multiplication with multiplicative inverse *(click for the list of lessons in this topic)*

Fractions : Fundamentals

Fractions are numbers with a specified place value. In this page, the following are detailed

• Numerator is the count of measure of a quantity

• Denominator specifies the place-value of the numerator That is 3/5 means, 3 counts of parts, and each part having 1/5 place-value.

• Fractions are also used to specify a part of a group.

• Fractions are directed numbers having positive or negative signs

Types of fractions

Based on the properties, the fractions are classified as different types : like fractions, unlike fractions, proper fractions, improper fractions, mixed fractions, equivalent fractions.

Arithmetics with Fractions

Fractions are specified in different place values or denominators. This page explains arithmetics with fractions such as, comparison, addition, subtraction, multiplication, and division.

The arithmetic operations are first explained in *first-principles* for fractions - numbers with different place-values.

Then *simplified procedures* are explained.

Numerical Expressions with Fractions

Do you want to understand how the fractions are handled in numerical expressions?

In this lesson, learn the following for fractions.

• precedence order : PEMDAS or BODMAS

• sequence of operations : left to right

Most importantly, learn handling of *division as inverse of multiplication*.

Decimals are introduced as *standard form of fractions*.

Fractions are part-of-whole, with different place-values specified as denominators. The Decimals are standard form of fractions with standard form for

• place-values : `1//10`, `1//100`, etc.

• position of digits : decimal point and order of digits after decimal point.

This course on decimals is *revolutionary and amazingly simple*.

In this, learn the following.

• Decimals as standard form of fractions

• Decimals as directed-numbers (positive and negative)

• Decimal arithmetics : first principles

• Decimal arithmetics : simplified procedures *(click for the list of lessons in this topic)*

Fundamentals of Decimals

Decimals are introduced as "standard form of fractions". This introduction also provides a simple overview of expanded form of decimals and decimals in number-line.

To understand decimals and arithmetics with decimals, this foundation is very important.

The pages in this lesson are

__Standard form of Fractions: Decimals__ *redo *

Conversion of Decimals

This page gives a brief overview of converting fractions into decimals, and decimals into fractions.

At the end, an overview of numbers with digits in decimal place not repeating and not ending. These numbers are called irrational numbers.

This introduction on irrational numbers is *amazingly simple and revolutionary*. All students should go through this once to understand irrational numbers.

Decimal Arithmetics

This page explains arithmetics with decimals such as, comparison, addition, subtraction, multiplication, and division.

The arithmetic operations are first explained in *first-principles* for decimals.

Then *simplified procedures* are explained.

Numerical Expressions with Decimals

Do you want to understand how the decimals are handled in numerical expressions?

In this lesson, learn the following for decimals.

• precedence order : PEMDAS or BODMAS

• sequence of operations : left to right

Most importantly, learn modified precedence order *PEMA / BOMA*.

Welcome to the *astonishingly clear introduction* to exponents, roots, and logarithm. Learn in this

• representation of exponents

• roots and logarithms are two inverses of exponents

• difference between common and natural logarithms

• arithmetics with exponents and logarithms

• Precedence -- PEMDAS or BODMAS (Exponent and Order)

• introduction to squares and square roots

• introduction to cubes and cube roots

All the above are given in a simple thought process. *(click for the list of lessons in this topic)*

Fundamentals of Exponents

Welcome to the astonishingly clear introduction to exponents, roots, and logarithm. Learn in this

• representation of exponents

• roots as one inverse of exponents -- finding base from given power and exponent result

• logarithms as the another inverse of exponents -- finding power from the given base and exponent result

This clarity is essential to understand the properties of roots and logarithms.

In this, common and natural logarithms are introduced. To explain natural logarithm, the number `e` is introduced in a *thought-provoking and revolutionary discourse*.

Exponents and Logarithm Arithmetics

Exponents are repeated multiplication. Root is an inverse of exponent to find the base. Roots are also considered as a form of exponent with fraction power. Logarithm is another inverse of exponent to find the power. This page explains arithmetics with exponents and logarithms -- providing a list of formulas as known results or properties.

At the end, the roots and logarithms that are not defined are explained. For example: `log_0 a` and `log_1 a` are not defined. `root(0)(a)` is not defined.

Numerical Expressions with Exponents

Do you want to understand how the exponents, roots, and logarithms are handled in numerical expressions?

In this lesson, learn the following.

• precedence order : PEMDAS or BODMAS

• sequence of operations : left to right

Most importantly, learn handling of *root and logarithm as inverse of exponent*.

Squares and Square Roots

Squares and Square roots are explained in a simple-thought-process. This also covers finding square root using prime-factorization and long division methods.

Cubes and Cube Roots

Cubes and cube roots are explained in a simple-thought-process. This also covers finding cube roots using prime-factorization.

Welcome to the *groundbreaking explanations* to topics in arithmetics.

• comparing 2 numbers as one is greater than other does not provide the information on the *relative magnitude of the numbers*. For example, 11 and 1000 are greater than 10, but 11 is close to 10 and 1000 is far greater than 10. This relative magnitude is specified with ratio, proportion, and percentage.

• *Direct and Inverse variation comes in a pair*. Learn the revolutionary insights into how a simple multiplication p*q=r leads to direct variation and inverse variation. p and r are in direct variation and p and q are in inverse variation.

• Simple and Compound Interest are simplified with couple of similar looking formulas. With this *novel explanation*, students are relieved of memorizing 10+ formulas.

• A special case of Direct and Inverse Variation pair is *rate * span = aggregate*. Speed-time-distance, time-work, and pipes-cisterns are explained in this unique lesson.

• Profit-loss, discount, and tax are simplified with one standardized formula for all. With this *extremely simple explanation*, students are relieved of memorizing 20+ formulas. *(click for the list of lessons in this topic)*

Comparing Quantities

In numerical arithmetics comparing two numbers is done by specifying one number as greater than, equal-to, or smaller than the other. In this, *how far a number is greater is not captured*. For example, 11 is greater than 10, and 1000 is also greater than 10. The relative magnitude of comparison is specified with ratio, fraction, proportion and percentage.

Unitary Method; Direct & Inverse Variations

Learn something exhilarating : the hidden secret in understanding direct and inverse variation is revealed.

In the variation problems, the underlying mathematical operation is multiplication. And

• multiplicand and product are in direct variation

• multiplier and product are in direct variation

• multiplicand and multiplier are in direct variation.

Direct variation and Inverse variation comes as a pair in the same formula. This is called as *Direct and Inverse Variation (DIV)* Pair.

Simple & Compound Interest

Learn something *made astoundingly easy* : Simple Interest and Compound Interest made easy. Students need not memorize 10+ formulas.

Simple Interest

Maturity Value in `T` time periods `A` = Principal `P + ` Interest `P R T `

`=P (1+R+R+R...T` times)

Compound Interest:

maturity value in `n` time periods `A = P (1+R)^n`

`=P(1+R)(1+R)(1+R)...n` times

These two equations look similar and has a simple explanation to the terms.

Rate and Span

Learn something unique and awesome : A special case of Direct and Inverse Variation pair is rate * span = aggregate.

The following are explained with the above rate-span equation.

• speed * time = distance,

• work rate * time = work done, and

• fill rate * time = filled amount in pipes-cisterns

Once the above is understood, it is extremely simple to solve problems in topics speed-distance-time, time-and-work, and pipes-and-cisterns.

Consumer Sales Business

Welcome to the *breathtakingly simple explanation* for the topics profit-loss, discount, and tax. Students need not memorize 20+ formulas anymore, just enjoy solving problems.

`text(Profit%) = 100 xx` `(text(SalePrice)-text(CostPrice))``//text(CostPrice)`

`text(Discount%) = 100 xx` `(text(MarkedPrice)-text(SalePrice))``//text(MarkedPrice)`

`text(Tax%) = 100 xx` `(text(BilledPrice)-text(SalePrice))``//text(SalePrice)`

All these three formulas are very similar and has a simple explanation to the terms in numerator and denominator.

Algebra is *introduced in the exactly relevant form* -- anchors the algebra in the numerical arithmetics.

• numbers are symbols representing value of quantities. The same can be represented with letters or symbols called variables

• numerical expressions are expressions with numbers and arithmetic operations between them. Algebraic expressions are similar.

• numerical equations are statement of equality between two numerical expressions. Algebraic equations are similar. The above is used to explain polynomials and algebraic identities.

A *revolutionary and ingenious formal foundation* (at 9th grade level) is given in the topic "Foundation of Algebra in Numerical Arithmetics". This course is a simplified version of that designed for 6th to 8th grades. *(click for the list of lessons in this topic)*

Algebra: Variables and Expressions

Algebra is *introduced in the exactly relevant form* -- anchors the algebra in the numerical arithmetics. It covers the following

• numbers and variables

• numerical expressions and algebraic expressions

• numerical equations and algebraic equations.

This course is designed for 6th to 8th grades. Other advanced readers are directed to the *revolutionary and ingenious formal foundation* in the topic "Foundation of Algebra in Numerical Arithmetics".

Polynomials - Basics

Welcome to the simplified and refreshingly new explanation to what are polynomials. Algebraic expressions are expressions with constants, variables and arithmetic operations between them. *Polynomials are simplified form of algebraic expressions*. To understand how this simplified form is useful, please read the lesson.

The classification of polynomials based on the following are also discussed.

• number of variables (polynomial of n variables)

• number of terms (monomials, binomials, polynomials of n terms)

• power of variables (linear, quadratic, cubic, polynomials of nth power)

Polynomial Arithmetics

Like numbers or numerical expressions, polynomials can be added, subtracted, multiplied or divided. This page provides a *simple-though-process* to understand these.

The pages in this lesson are

__Addition and Subtraction of Polynomials__ *redo *

Basic Algebraic Identities

Identities are statements equating two expressions. In this, standard algebraic identities are introduced and explained in a simple-thought-process.

Welcome to the *refreshingly new explanation* to calculating perimeter, area, and volume of 2D and 3D shapes.

• Length is Distance-Span, measured in reference span of to `1` unit long line

• Area is Surface-Span, measured in reference to span of `1xx1` square

• Volume is Space-Span, measured in reference to span of `1xx1xx1` cube

In this basic course, the following are covered.

• Perimeter and area of simple 2D shapes

• surface area and volume of simple 3D shapes *(click for the list of lessons in this topic)*

Introduction to Measurements

Welcome to the *refreshingly new explanation* to calculating perimeter, area, and volume of 2D and 3D shapes. The following are covered in this

• absolute and derived standard in measurements

• length (distance-span), area (surface-span), and volume (space-span).

The accuracy in measurement and measurement units are also covered.

Mensuration: Two Dimensional Shapes

The topics provide a simple overview of formulas for perimeter and area of

• Polygon

• Rectangle

• Triangle

• Circle

• Quadrilaterals

Clear reasoning for the formulas are provided that helps students to understand and retain.

Mensuration: Three Dimensional Shapes

The topics provide a simple overview of formulas for surface area and volume of of

• cube

• cuboid

• cylinder

Welcome to the only place where practical geometry is explained in an *ingenious and simplified form*.

The geometrical instruments are introduced as *four fundamental elements of practical geometry*

• collinear points (straight-line using a ruler)

• equidistant points (arch using a compass)

• equiangular points (angle using a protractor)

• parallel points (parallel using set-squares)

Based on the four fundamental elements, the topics in practical geometry are explained. *(click for the list of lessons in this topic)*

4 Fundamentals Elements of Practical Geometry

Welcome to the *ingenious lessons that redefine how students learn*. The geometrical instruments are introduced as *four fundamental elements of practical geometry*

• collinear points (straight-line using a ruler)

• equidistant points (arch using a compass)

• equiangular points (angle using a protractor)

• parallel points (parallel using set-squares)

These four fundamental elements are explained in the following lessons.

Basic Shapes: Lines, Circles, Angles

The topics in this cover the *simple-thought-process* to constructing the following

• line segment

• circle

• copying a line-segment

• copying an angle

Secondary Elements of Practical Geometry

The topics in this cover a *simple-thought-process* to constructing the following

• perpendicular bisector

• perpendicular line on a point

• perpendicular line to a point

• bisector of an angle

The reasoning on how each of the procedures work is provided in a simple thought-process, which makes it easy for students to retain knowledge and work out the procedure.

Standard angles Using Campus

A *simple-thought-process* to constructing standard angles (`60^@`, `30^@`, `15^@`, `120^@`, `90^@`) using a compass are provided.

The reasoning on how each of the procedures work is provided, which makes it easy for students to retain knowledge and work out the procedure.

Construction of Triangles (Fundamental Shape of Polygons)

This provides a simple thought process to construction of triangles using given 3 parameters.

• side-side-side

• side-angle-side

• angle-side-angle

• rightangle-hypotenuse-side

• side-angle-latitude

This also analyses why the following do not work

• angle-angle-angle (only 2 parameters)

• side-side-angle (two possible triangles)

• side-side-altitude (two possible triangles)

Construction of Quadrilaterals

This provides a simple thought process to construction of quadrilaterals of various forms.

• Irregular Quadrilateral

• Parallelogram

• Rhombus

• Rectangle

• Square

• Trapezium

• Kite

This ingenious lesson provides a method to approach the contruction : Consider these as combination of two triangles of forms (sss, sas, asa, rhs, sal) and use the methods studies earlier.

Welcome to the *revolutionary and ingenious* formal foundation of algebra with Numerical Arithmetics.

Algebra is based on the following basics of numerical arithmetics.

• PEMA Precedence Order (Parenthesis, Exponent, Multiplication, and Addition)

Subtraction is inverse of Addition

Division is inverse of Multiplication

Root and Logarithm are two inverses of Exponent

• CADI Properties of Addition and Multiplication (Closure, Commutative, Associative, Distributive, Identity, Inverse).

• Numerical Expressions are statement of a value

• Value of a Numerical Expression does not change when modified per PEMA / CADI

• Equations are statements of equality of two expressions

• And statement of equality does not change ...(explained in the lesson)

• And so for In-equations *(click for the list of lessons in this topic)*

Numerical Arithmetics: Revision for Algebra

Formal foundation in Algebra starts with numerical arithmetics. This part reviews and establishes the following important points

• Subtraction is handled as inverse of addition

• Division is handled as inverse of multiplication

• Root and Logarithms are handled as inverses of exponent

• Numerical arithmetic precedence order is Parenthesis, Exponent, Multiplication and Addition, in that order.

The above is a *fresh new look* at what you would know already, and that is organized in a smart way to use in Algebra.

Numerical Arithmetics: Laws and Properties for Algebra

Formal foundation in Algebra is established with laws and properties of Numerical Arithmetics.

• Comparison : Trichotomy and transitivity properties.

• Addition : Closure, commutative, Associative, Additive Identity, Additive Inverse properties

• Subtraction : Handled as inverse of addition, and holds properties of addition. eg: Commutative property `a-b = a+(-b) = -b+a (!=b-a)`

• Multiplication : Closure, commutative, Associative, Distributive over addition, Multiplicative Identity, Multiplicative Inverse properties

• Division : Handled as inverse of multiplication, and holds properties of multiplication. eg: `a-:b(c+d) ``= a xx (1/b)xx(c+d) ``= axx (c/b + d/b)` `!= a-:(bc+bd)`

• Exponents : Addition, multiplication, division properties of exponents.

The above is *exemplary and ingenious foundation* in learning algebra. For example, `x+(y+x)` is simplified using the commutative law `= x+(x+y)` and the associative law `= 2x+y`.

Numerical expressions, equations, identities, and in-equations for Algebra

Formal foundation in Algebra is established with the following.

• Numerical Expressions are statement of a value

• The value of a Numerical Expression does not change when modified per PEMA / CADI

• Equations are statements of equality of two expressions

• Statement of equality is maintained when expressions are modified

• Statement of equality is maintained for arithmetic operations between multiple equations (eg: addition of two equations)

• Identities are statement of equality of an expression to another as per PEMA / CADI

• In-equations are statements of comparison of two expressions

• Statement of comparison is maintained when an in-equation is modified per PEMA / CADI

• Statement of comparison is maintained when an in-equation is modified with another equation

• Statement of comparison is maintained under transitivity property of in-equations

The above is *exemplary and ingenious foundation* in learning algebra.

Foundation of Algebra (Summary)

This topic provides a simple summary of foundation of algebra with some examples.

Welcome to *the only place where the essence of trigonometry is explained*.

• a right-triangle is specified by 2 independent parameters.

• That means, if an angle and the length of a side is given, then one should be able to calculate the length of the other two sides.

• What property one can use to calculate the above? For a given angle, the ratio of sides is a constant.

Thus, the ratio of sides comes into existence as `sin`, `cos`, `tan` etc.
The thought-process is *revolutionary and aweinspiring*.

Beyond the definitions of trigonometric ratios, the following are covered.

• trigonometric ratios for standard angles

• trigonometric identities based on Pythagoras Theorem *(click for the list of lessons in this topic)*

Basics of Trigonometry

In this lesson, first the basics required to understand trigonometry are revised. Then, the *revolutionary and aweinspiring explanation of trigonometric ratios* is provided.

• A triangle has `6` or `7` parameters (`3` sides, `3` angles, sometimes height as the `7`th parameter)

• It has `3` independent parameters, meaning the other parameters can be calculated from the given `3`.

• A right triangle has `2` independent parameters, as one angle is already given as `90^@`.

• Given an angle and a side (which are the `2` independent parameters), how to compute the other two sides?

• The answer is that, given an angle, the ratio of two sides is a known constant. So, the known constant and the given side are used to compute the other sides.

This leads to defining the trigonometric ratios (ratio of sides as known constant) `sin`, `cos`, `tan`.

Trigonometric Ratios for Standard Angles

Learn something *made astoundingly easy*: Trigonometric ratios for standard angles, derived from

• equilateral triangles `30^@` and `60^@`,

• isosceles right-triangles `45^@`, and

• a triangle with one-side zero `90^@` and `0^@`.

Once the basis of standard angles is understood as the different triangles given above, the values of trigonometric ratios are very very easy to quickly calculate. *No need to blindly memorize*.

Trigonometric Identities and Complementary Angles

In this topic, some basic patterns and relations are explained in very simple thought process. The Pythagorean trigonometric identities and trigonometric values for complementary angles are explained.

Welcome to the *astoundingly clear introduction* to statistics and probability. Learn in this

• how data can be organized in tally and table form

• how data collected in statistics can be used to *predict future* occurrences

• how the predicting the outcome can be captured as a *probability in random experiments*.

• how continuous data can be represented as grouped data and analyzed *(click for the list of lessons in this topic)*

Statistics : Fundamentals of Data

Do you want to understand handling collection of numbers? This topic introduces the following in simple terms.

• data

• tally representation

• table representation

• pictograph representation

• bar-graph representation.

Statistics : Analysis of Data

Learn simple explanations with examples for

• cumulative frequency

• mean, median, mode of data

• reading bar-graphs

• bargraphs with two sets of data

• pie-charts.

Basics of Probability

Probability is made *breathtakingly simple* : read the *game-changing explanation* to

• using statistics to *predict future*

• how the *prediction becomes into probability* for random experiments.

Statistics: Grouped Data

Learn simple explanations with examples for

• grouped data

• finding probability in grouped data

• class parameters of grouped data

• finding, mean, mode, and median of grouped data

Welcome to the lessons that give *firm foundation* in measuring perimeter, area, and volume.

To measure a length, area, or volume one of the following methods is used.

• Measurement by Superimposition, Calculation and Equivalence

• Equivalence by infinitesimal pieces, and Cavalieri's Principle

Once the above is understood, the formula for perimeter, area, and volume of 2D and 3D shapes are provided.

Start learning the *exceptional discourse* that derive the formulas of mensuration. *(click for the list of lessons in this topic)*

Measurement Methods

Welcome to the lessons that provide *firm foundation*' in mensuration topics : perimeter, area, and volume. To measure a length, area, or volume one of the following methods is used.

• Measurement by Superimposition

• Measurement by Calculation

• Measurement by Equivalence

Three methods are used to find equivalence.

• Equivalence by infinitesimal pieces

• Equivalence by Cavalieri's Principle (2D)

• Equivalence by Cavalieri's Principle (3D)

Mensuration: 2D Shapes

The topics provide a simple overview of formulas for perimeter and area of

• polygon

• square and rectangle

• triangle

• circle

• quadrilateral

The derivation of these formulas are outlined for students to better understand and retain.

Mensuration: 3D Shapes

The topics provide a simple overview of formulas for perimeter and area of

• cube

• cuboid

• cylinder

• prism

• pyramid

• cone

• sphere

The derivation of these formulas are outlined for students to better understand and retain.

Mensuration with combination of shapes and Solids

The topics provide a simple overview of formulas for part shapes listed below.

• area of segments and sectors of circle

• surface area and volume of frustum

The derivation of these formulas are outlined for students to better understand and retain.

Welcome to the simple thought process to construction problems in practical geometry.

In the lesson on basics, the four fundamental elements of practical geometry is explained. And, the construction of various shapes using the fundamental elements is explained.

That is extended to the following.

• Construction of triangles using secondary information (sum of sides, difference of sides, and perimeter of sides).

• Scaling of lines, triangles, and other shapes.

• Consturction of elements of circle (tangents and chords)

The reasoning on how each of the procedures work is provided, which makes it easy for students to retain knowledge and work out the procedure. *(click for the list of lessons in this topic)*

Construction of Triangles with Secondary Information

In this page, a short overview of constructing triangles using secondary parameters is provided. The primary parameters are the 3 sides and 3 angles. The secondary parameters are sum of two sides, difference of two sides, and perimeter. Construction of triangles using the secondary parameters is studied.

It covers the following.

• side-angle-sum of sides

• side-angle-difference between sides

• angle-angle-perimeter

The reasoning on how each of the procedures work is provided, which makes it easy for students to retain knowledge and work out the procedure.

The pages in this lesson are

__Construction of Triangles With Secondary Information__ *redo *

__Construction of Triangles using Angle, Side, Sum of 2 Sides__ *redo *

__Construction of Triangles using Angle, Side, difference between 2 Sides__ *redo *

__Construction of Triangles using Angle, Side, Negative difference between 2 Sides__ *redo *

__Construction of Triangles using Angle, Angle, Perimeter__ *redo *

Scaling of Lines and Shapes

In this page, a short overview of scaling various geometrical shapes is provided. It covers the following.

• Scaling a line segment

• Scaling a triangle

• Scaling a polygon.

The reasoning on how each of the procedures work is provided, which makes it easy for students to retain knowledge and work out the procedure.

Construction of Chord, and Tangents to Circle

In this page, a short overview of constructing tangents and chords to a circle is provided. It covers the following.

• Construction of a tangent on a point on the circle

• Construction of tangents from a point outside the circle

• Construction of chords of a given length

• Construction of chords at a distance from the center

The reasoning on how each of the procedures work is provided, which makes it easy for students to retain knowledge and work out the procedure.

Welcome to *the only place where the essence of "limit of a function" is explained*.

• `0/0` is called as indeterminate value -- meaning a function evaluating to `0/0` can take any value, it could be `0`, or `1`, or `7`, or `oo`, or undefined.

• other forms of indeterminate values are: `oo/oo`, `oo-oo`, `0^0`, `0xx oo`, or `oo^0`

Rigorous arithmetic calculations may result in `0/0`, but the expression may take some other value. The objective of limits is to find that value. The details explained are *revolutionary and provided nowhere else*.

Once that is explained, the topics in limits are covered. *(click for the list of lessons in this topic)*

Basics: Limit of a function

*One and only place where the essence of limit(calculus) is explained*.

• Indeterminate Value

→ `0/0`

→ represented by an expression

→ other forms: `oo/oo`, `oo-oo`, `0^0`, `0xx oo`, or `oo^0`

• The following can be true

→ `0/0 = 0`

→ `0/0 = 1`

→ `0/0 = oo`

→ `0/0 = 6` or `8` or `-3`

Understanding the above is essential to understanding limits(calculus). The topic involves figuring out the expected value of a function when it evaluates to `0/0`.

Understanding limits with Graphs

Welcome to the *astoundingly clear and simple lesson on understanding limits*. The geometrical meaning of left-hand-limit and right-hand-limit are explained with graph of a function.

The function is considered as two constituent functions of numerator and denominator and using the graphs of these functions, the limit is explained.

Based on the understanding of numerator and denominator, the L'Hospital's Rule is explained.

Calculating Limits

Examining a function at an input value is made *simple and clear*. Based on the information, how to determine if the function is defined, continuous, or not defined. The following are covered.

• examining a function at an input value

• limit of a continuous function

• limit of a piecewise function

• limit of functions with abrupt change

• limit of functions that are not defined

Algebra of Limits

In this page, Algebra of limits is detailed in a *coherent and simple form* -- the following are covered.

• understanding algebra of limits,

• Limit distributes over Addition and Subtraction when the value is not `oo-oo`.,

• Limit distributes over multiplication, when the value is not `oo xx 0`,

• Limit distributes over division when the value is not `0 -: 0` or `oo -:oo`,

• Limit distributes over exponent, when the value is not `oo^0` or `0^0`

• Limit with a variable can be substituted when value is not any of the forms of `0/0`

Limit of Algebraic Expressions

Standard results for limits of function involving polynomials and evaluating to `0/0`, `oo/oo`, or `oo - oo` are examined and explained with examples.

• limit of polynomials

• limit of functions evaluating to `0/0`

• limit of functions evaluating to `oo/oo`

• limit of binomials

• limit of some standard trigonometric, logarithmic, and exponent functions

The differentiation or derivative calculus is explained in *astonishingly simple and clear thought process*. The differentiation is covered in the following topics.

• application scenario of differentiation

• first principles of differentiation

• graphical meaning of derivatives

• differentiability of a function

• algebra of derivatives

• standard results in derivatives *(click for the list of lessons in this topic)*

Introduction to Differential Calculus

Some quantities are related such that one is rate of change of another. Finding the rate of change of a quantity, as a function of variable, is the derivative or differentiation. In this, the following are explained in simple and clear thought-process.

• application scenario of differentiation

• first principles of differentiation

• graphical meaning of derivatives

• differentiability of a function

Algebra of Differentiation

In this page, algebra of derivatives is detailed in a *coherent and simple form* -- The following are covered.

• understanding algebra of derivatives.

• Derivatives of addition, subtraction, multiplication, and division of functions

• Derivatives of function of functions, and parametric forms of functions.

Standard Results in Derivatives

In this page, standard results in derivative, that are repeatedly used, are explained and proven. This provides a *simple and coherent* study of the standard results.

• Derivatives of Algebraic Expressions

• Derivatives of Trigonometric Functions

• Derivatives of Inverse Trigonometric Functions

• Derivatives of Logarithms and Exponents

Integration takes a new name "*continuous aggregate of change*". That is, change, given as a function, is aggregated over an interval. Learn in this:

• First Principles of Integration : *Continuous Aggregate of Change*

• Graphical Meaning of Integration : *Area under a curve*

• Definition of *definite and indefinite integrals*

• Fundamental theorem of Calculus : Integration as *anti-derivatives*

Apart from these, the algebra of integrals and various results of integration of standard functions are explained. *(click for the list of lessons in this topic)*

Basics of Integration

Welcome to the *groundbreaking and unique* lesson on "*continuous aggregate of change*". Learn in this:

• First Principles of Integration : Change given by a function aggregates over an interval of the variable, given as *Continuous Aggregate of Change*.

• Graphical Meaning of Integration : If the function is considered as a curve, then the *area under a curve* is the integration.

• *Definite integral* as the integration over interval between two values of the variable.

• *Indefinite integral* as the integration over interval from 0 to a varying value for the variable.

• Fundamental theorem of Calculus : Connecting the "continuous aggregate of change" (integration) as the inverse of "instantaneous rate of change" (differentiation), in both definite and indefinite forms. Thus, integration is *anti-derivative*.

This is the only place one gets to learn the essence of integration.

Algebra of Integrals

In this page, Algebra of integrals is detailed in a *coherent and simple form* -- Integration of the following are covered.

• addition of functions,

• subtraction of functions,

• product of functions,

• division of functions,

• function-of-function,

• parametric form of functions.

Apart from the above, standard results -- integration of some standard functions are explained.

Various Forms and Results of Integrals

In this page, several methods are explained to work out integrals of standard forms of functions. This provides a *coherent thought-process to approach integration problems*. The following are covered.

• understanding the complexity and methods

• Integration by Substitution

• Integration using identities

• Integration by Parts

• Integration by Partial Fraction

• Integration by combination of methods

Welcome to the *novel approach to understanding complex numbers: it completely changes the way complex numbers are thought-about and learned*.

• Irrational numbers are numerical expressions, eg: `2+root(5)(3)`

» Complex numbers are numerical expressions too.

• Irrational numbers do not have a standard form. They are expressed as numerical expressions, with some having many terms. eg: `2+ root(2)(2)-3xx root(3)(5)`

» Similar to irrational numbers, the direct extension is to express complex numbers as numerical expressions. eg: `x^3=1` has `3` solutions, `(root(3)(1))_(1st)`, `(root(3)(1))_(2nd)`, `(root(3)(1))_(3rd)`

» But, Any complex number is given in a standard form `a+ib`, because of Euler Formula `re^(i theta)``= r(cos theta + i sin theta)`

How so? Go through the first few lessons to get the *astoundingly new perspective* of complex numbers. *(click for the list of lessons in this topic)*

Introduction to Complex Numbers

This lesson provides the *astoundingly new perspective* of complex numbers.

Irrational numbers were introduced as solution to algebraic equations similar to `x^2=2`. Similarly, complex numbers are introduced as *solutions to algebraic equations* similar to `(x-a)^2=-b`.

At the end of this topic, the *generic form of complex numbers* is proven to be `a+ib`.

Complex Plane and Polar Form

Complex number `a+i b` is equivalently an ordered pair `(a,b)` which can be abstracted to represent a 2D plane. This is named after the mathematician JR Argand as *Argand Plane* or *complex plane* with real and imaginary axes.

Complex numbers, having abstracted to a complex plane, are represented in polar form.

Learn these in a *simple-thought-process*.

Algebra of Complex Numbers

Algebra of complex numbers details out the operations addition, subtraction, multiplication , division, conjugate, and exponent.

The algebra of complex numbers is quite easy to understand in abstraction. This topic goes beyond and explains the application context.

Learn these in a *simple-thought-process*.

The pages in this lesson are

__Complex Number Arithmetic__ *redo *

__Complex Number: Modeling sine waves__ *redo *

__Understanding Complex Arithmetic__ *redo *

__Addition of two Complex numbers__ *redo *

__Subtraction of Complex Numbers__ *redo *

__Multiplication of two Complex numbers__ *redo *

__Conjugate of a complex Number__ *redo *

__Division of complex numbers__ *redo *

__Exponent of a complex Number__ *redo *

Properties of Complex Number Arithmetic

Complex number system is an extension of Real number system with inclusion of a new number `i=sqrt(-1)`. The complex arithmetic has the * properties mostly identical* to real arithmetic. The additional features are related to modulus, argument, and conjugate.

Learn these in a *simple-thought-process*.

The pages in this lesson are

__Understanding Properties of Complex Arithmetic__ *redo *

__Addition : Commutative Law __ *redo *

__Addition : Associative Law__ *redo *

__Multiplication : Closure Law__ *redo *

__Multiplication : Commutative Law __ *redo *

__Multiplication : Associative Law__ *redo *

__Modulus in multiplication__ *redo *

__Argument in multiplication__ *redo *

Welcome to *the only place where the essence of vectors is explained*.

• A vector in first-principles is a quantity with spatial-direction specified. Example: `2m` north-east.

• A vector in component form is linear-combination of unit vectors of independent directions. eg: `2i+2j`

• A vector in 3D co-ordinate system is a ray initiating from the origin.

Vector addition, dot product, and cross product are explained in

» first principle,

» geometrical meaning, and

» component form.

The details explained here are *revolutionary and astonishingly simple to understand*. *(click for the list of lessons in this topic)*

Introduction to Vector Quantities

Welcome to *the only place where the essence of vectors is explained*.

Integers are "directed-whole-numbers", with positive numbers aligned to a chosen direction, and negative numbers opposed to the chosen direction.

Vectors are "spatial-directed-quantities". The spatial directions are

• x-axis (eg: right and left for a person)

• y-axis (eg: forward and backward for a person) and

• z-axis (eg: up and down for a person)

This page introduces the concept of vectors and the representation in component form in a simple thought-process.

Properties of Vectors

Welcome to the *astoundingly clear and simple lesson* on basic properties of vectors.

This topic discusses magnitude of a vector and the properties of that. This also discusses the different types of vectors like null, proper, collinear, coplanar, etc.

The pages in this lesson are

Vectors and Coordinate Geometry

Welcome to the *astoundingly clear and simple lesson* on vectors and coordinate geometry.

Coordinate Geometry is the system of geometry where the position of points on the 3D coordinates are described as ordered pairs of numbers corresponding to the three axes. In the mathematical representation of vectors, the components along 3D coordinates are described individually. In this topic, you will learn how these two are related.

Direction - Unique Feature of Vectors

Welcome to *the only place where the essence of vector arithmetics is explained*.

Vectors are quantities with spatial-direction specified.

If two vectors interact, then usually, one vector is split into two components

• component along the direction of the other vector

• component perpendicular to the direction of the other vector.

In vector arithmetics these two components behave differently.

• Vector addition: the component along the direction adds in magnitude.

• Vector dot-product: the component along the direction takes part in the product and the component in perpendicular is lost.

• Vector cross-product: the component in perpendicular to the direction takes part and the component along the direction is lost.

This explanation is *coherent and simple* for students to connect and remember.

Vector Addition

Welcome to the *only place where essence of vector addition* is explained.

• vector addition in first-principles: components aligned to each other add.

• component form of vector addition: individual components add independently.

• triangular law: sequential addition of vectors

• parallelogram law: continuous addition of vectors

The details in these pages provide powerful and clear insights.

The pages in this lesson are

__Vector Addition - First principles__ *redo *

Multiplication of Vectors by Scalar

Welcome to the *only place where essence of scalar multiplication of vectors* is explained. The following are covered.

• Scalar multiplication of vector

• Standard unit vectors

• Representing a vector as a linear combination of multiple vectors

• Revisiting component form of vectors as linear combination of standard unit vectors

The details in these pages provide powerful and clear insights.

Vector Dot Product

Welcome to the *only place where essence of vector dot product* is explained. The following are covered.

• Multiplication of two vectors : either component in parallel take part or the component in perpendicular take part

• Vector dot product in first principles

• Projection form of vector dot product

• Component form of vector dot product

The details in these pages provide powerful and clear insights.

Vector Cross Product

Welcome to the *only place where essence of vector cross product* is explained. The following are covered.

• Multiplication of two vectors : either component in parallel take part or the component in perpendicular take part

• Vector cross product in first principles

• Area of Parallelogram form of vector cross product

• Component form of vector cross product

The details in these pages provide powerful and clear insights.

The topics provide a simple overview of properties of vector arithmetics.

• Vector Addition

• Vector Multiplication by a scalar

• Vector Dot Product

• Vector Cross Product

The details in these pages provide clear insights. *(click for the list of lessons in this topic)*

Properties of Vector Addition

The topics provide a simple overview of Properties of Vector addition. The properties of vector addition can be easily understood with the properties of real number addition. The following are covered.

• Closure property of Vector Addition

• Commutative property of Vector Addition

• Associative property of Vector Addition

• Additive Identity

• Additive Inverse

• Magnitude of Sum of Vectors

The details in these pages provide clear insights.

Properties of Vector Multiplication by Scalar

The topics provide a simple overview of Properties of Scalar multiplication of Vectors is provided. The following are covered.

• Order of Scalar Multiplication

• Distributive over Addition

• Multiplication by Multiple Scalars

• Magnitude of Scalar multiple of vector

• Unit vector along a vector

The details in these pages provide clear insights.

Properties of Dot Product

The topics provide a simple overview of Properties of Vector Dot product. The following are covered.

• not closed

• commutative

• product by a negative

• product by a scalar multiple

• product with a null vector

• product of orthogonal vectors

• product of collinear vectors

• distributive over vector addition

• bilinear property

• not associative

• no cancellation in an equation

• no multiplicative identity

• no multiplicative inverse

• magnitude of dot product

The details in these pages provide clear insights.

The pages in this lesson are

__Understanding Properties of Dot Product__ *redo *

__Dot Product: Commutative Property__ *redo *

__Dot product of a negative__ *redo *

__Dot product of a scalar multiple__ *redo *

__Dot product with a null vector__ *redo *

__Dot product of Orthogonal Vectors__ *redo *

__Dot product of Collinear Vectors__ *redo *

__Distributive Property over Vector Addition__ *redo *

__When products of two vectors are equal__ *redo *

__Dot Product : No Multiplicative Identity__ *redo *

Properties of Cross Product

The topics provide a simple overview of Properties of Vector Cross product. The following are covered.

• closed

• commutative

• product by a negative

• product by a scalar multiple

• product with a null vector

• product of orthogonal vectors

• product of collinear vectors

• distributive over vector addition

• bilinear property

• not associative

• no cancellation in an equation

• no multiplicative identity

• no multiplicative inverse

• magnitude of cross product

The details in these pages provide clear insights.

The pages in this lesson are

__Understanding Properties of Cross Product__ *redo *

__Cross Product : Closure Property__ *redo *

__Cross Product : Not Commutative__ *redo *

__cross product of a negative__ *redo *

__Cross product of a Scalar Multiple__ *redo *

__Cross product with a null vector__ *redo *

__Cross Product of Orthogonal Vectors__ *redo *

__Cross Product of Collinear Vectors__ *redo *

__Distributive Property over Vector Addition__ *redo *

__When products of two vectors are equal__ *redo *

__Cross Product : No Multiplicative Identity__ *redo *