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Thought-Process to Discover Knowledge
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: 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`.