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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.
9th-10th Foundation

### Foundation of Algebra with Numerical Arithmetics

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)
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.

(click for the list of pages in the lesson)

The pages in this lesson are

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.

•  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.

(click for the list of pages in the lesson)

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.

(click for the list of pages in the lesson)

Foundation of Algebra (Summary)

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

(click for the list of pages in the lesson)

The pages in this lesson are