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

Numbers and Arithmetic Operations

This page revises the numbers quickly. It is important to understand the following concepts from this lesson, Like
**Laws and Properties of Arithmetic : Numbers and Operations** :

• Ordinal property of numbers

• Comparison (greater, equal, or lesser)

• addition (combining two quantities)

• subtraction (inverse of addition)

• multiplication (repeated addition)

• division (inverse of multiplication)

• exponent (repeated multiplication)

• root (one inverse of exponent)

• logarithm (another inverse of exponent)

Arithmetic Operations And Precedence

This course is designed for students at 9th and 10th grade level. It assumes the following were introduced.

• Whole numbers, integers, fractions, decimal numbers.

• Addition, subtraction, multiplication, division, exponent, root, logarithm.

• Subtraction is inverse of addition

• Division is inverse of multiplication

• Root and Logarithm are inverses of Exponent

• introduction to variables in algebra

• Numerical expressions

• BODMAS or PEMDAS for precedence order

• BOMA or PEMA
The objective of this topic is to *formalize numerical arithmetic as applicable to algebra*.

It is very important to go through these to understand algebra.

Laws and Properties of Arithmetics : Comparison

This page introduces the Trichotomy property and Transitivity properties of numerical comparison. To understand in-equalities in algebra, these properties are used.

Laws and Properties of Arithmetics : Addition

In this lesson, the laws and properties of addition are revised. The properties are Closure, Commutative, Associative, Additive Inverse, Additive Identity.

The subtraction is handled as inverse of addition and in that case, the properties mentioned above are applicable.

eg: Subtraction is not commutative $a-b\ne b-a$

But Subtraction as inverse of addition $a-b=a+(-b)=-b+a$

It is very important to go through this once to understand algebra.

Laws and Properties of Arithmetics : Multiplication

In this lesson, the laws and properties of multiplication are revised. The properties are Closure, Commutative, Associative, Distributive, Multiplicative Identity, Multiplicative Inverse.

The division is handled as inverse of multiplication and in that case, the properties mentioned above are applicable.

eg: Division is not commutative $a\xf7b\ne b\xf7a$

But Division as inverse of multiplication $a\xf7b=a\times \left(\frac{1}{b}\right)=\frac{1}{b}\times a$

It is very important to go through this once to understand algebra.

Laws and Properties of Arithmetics : Exponents

This page introduces the the properties of exponents. To understand polynomials and equations in algebra, these properties are used.

Numerical Expressions

In this lesson, properties of expressions are explained.

• An expression is a statement of a value

• the value of an expression remains unchanged when the expressions are modified per PEMA / CADI

It is very important to go through this once to understand in-equalities in algebra.

Arithmetic Properties of Equality

In this lesson properties of equations are explained.

• equations are statement of equality between two expressions

• the statement of equality remains unchanged when the expressions are modified per PEMA / CADI

• the statement of equality remains unchanged for arithmetics between two equations

It is very important to go through this once to understand equalities in algebra.

Identities Explained with Numerical Arithmetics

In this lesson, identities are explained in general.

• identity is a statement of one expression modified as per PEMA / CADI properties to arrive at a different expression.

• These two expressions are identical.

• And, identities are studied because one form of identity can be modified into its equivalent form for some purpose.

It is very important to go through this once to understand identities in algebra.

Arithmetic Properties of Inequality

In this lesson properties of inequalities are explained.

• inequations are statement of comparison between two expressions

• statement of comparison remains unchanged when the expressions are modified per PEMA / CADI

• statement of comparison is modified when an in-equation is modified with another equation.

• statement of comparison remains unchanged as per the transitivity property of comparison

It is very important to go through this once to understand in-equalities in algebra.

Algebra : First Principles (Summary)

This topic provides a simple summary of foundation of algebra with some examples. Algebraic expressions are representation of quantities with variables, numbers, and arithmetic operations between them.

The expressions are modified as per the PEMA precedence and CADI Laws and properties of Arithmetics.