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-- Gomathi Sankar

Whole Numbers

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

Learn about

• *grouping* to form place-value

• *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*

Divisibility in Whole Numbers

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.

Integers

Integers are "*directed-whole-numbers*". That is, "received an amount of 3" and "given an amount of 3" are two different numbers -- +3 (aligned to the chosen direction) and -3 (opposed to the direction) respectively.

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

Fractions

Learn the *ingenious details 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

Decimals

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

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.

• place-values standardized to $1/10$, $1/100$, etc.

• decimal point and position of digits after decimal point speficies the standarized place-value.

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

Exponents

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.

Commercial Arithmetics

Welcome to the topics in commercial 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 variations come 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.

Mensuration : Length, Area, and Volume

Welcome to the *refreshingly new views* 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 $1\times 1$ square

• Volume is Space-Span, measured in reference to span of $1\times 1\times 1$ cube

In this basic course, the following are covered.

• Perimeter and area of simple 2D shapes

• surface area and volume of simple 3D shapes

Construction / Practical Geometry (basics)

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.

Algebra : Foundation with Numerical Arithmetics

Welcome to the *ingenious* course on 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 numerical 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

Trigonometry (Introduction)

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 $\mathrm{sin}$, $\mathrm{cos}$, $\mathrm{tan}$ etc.

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

• trigonometric ratios for standard angles

• trigonometric identities based on Pythagoras Theorem

Statistics & Probability

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*. The details are *ingenious and provided nowhere else*.

• how continuous data can be represented as grouped data and analyzed

Construction / Practical Geometry (High)

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.

Complex Numbers

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: $2+\sqrt[5]{3}$

» 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: $2+\sqrt[2]{2}-3\times \sqrt[3]{5}$

» Similar to irrational numbers, the direct extension is to express complex numbers as numerical expressions. eg: ${x}^{3}=1$ has $3$ solutions, $\left(\sqrt[3]{1}\right)}_{1st$, $\left(\sqrt[3]{1}\right)}_{2nd$, $\left(\sqrt[3]{1}\right)}_{3rd$

» But, Any complex number is given in a standard form $a+ib$, because of Euler Formula $r{e}^{i\theta}$$=r(\mathrm{cos}\theta +i\mathrm{sin}\theta )$

How so? Go through the first few lessons to get the *astoundingly new perspective* of complex numbers.

Mensuration : Length, Area, and Volume

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 derivations of the formulas of mensuration.
* This requires some review and rework. Though all relevant information is documented, it is not sufficiently explained and properly connected. If people read, then I'll put my effort in fixing that. *

Trigonometry (advanced)

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 details explained are *ingenious and found nowhere else*.

The trigonometric values for all quadrants and for compound angles are also covered.

Vector Algebra

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*.

Calculus - Limit

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

• $\frac{0}{0}$ is called as indeterminate value -- meaning a function evaluating to $\frac{0}{0}$ can take any value, it could be $0$, or $1$, or $7$, or $\infty$, or undefined.

• other forms of indeterminate values are: $\frac{\infty}{\infty}$, $\infty -\infty$, $0}^{0$, $0\times \infty$, or $\infty}^{0$

Rigorous arithmetic calculations may result in $\frac{0}{0}$, but the expression may take some other value. The objective of limits is to find that value. The details explained are *ingenious and found nowhere else*.

Once that is explained, the topics in limits are covered.

Differential Calculus

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

The details explained are *ingenious and found nowhere else*.

Integral Calculus

Integration takes a new name "*continuous aggregate*". That is, a quantity, given as a function, is aggregated continuously over an interval.The details explained are *ingenious and provided nowhere else*.

Learn in this:

• First Principles of Integration : *Continuous aggregate*

• 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.