Server Not Reachable. *This may be due to your internet connection or the nubtrek server is offline.*

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 *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.