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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.continue
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)
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.

(click for the list of pages in the lesson)

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`.

(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)