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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 *novel approach to understanding complex numbers: it completely changes the way complex numbers are thought-about and learned*.

• Irrational numbers are numerical expressions, eg: `2+root(5)(3)`

» Complex numbers are numerical expressions too.

• Irrational numbers do not have a standard form. They are expressed as numerical expressions, with some having many terms. eg: `2+ root(2)(2)-3xx root(3)(5)`

» Similar to irrational numbers, the direct extension is to express complex numbers as numerical expressions. eg: `x^3=1` has `3` solutions, `(root(3)(1))_(1st)`, `(root(3)(1))_(2nd)`, `(root(3)(1))_(3rd)`

» But, Any complex number is given in a standard form `a+ib`, because of Euler Formula `re^(i theta)``= r(cos theta + i sin theta)`

How so? Go through the first few lessons to get the *astoundingly new perspective* of complex numbers. *(click for the list of lessons in this topic)*

Introduction to Complex Numbers

This lesson provides the *astoundingly new perspective* of complex numbers.

Irrational numbers were introduced as solution to algebraic equations similar to `x^2=2`. Similarly, complex numbers are introduced as *solutions to algebraic equations* similar to `(x-a)^2=-b`.

At the end of this topic, the *generic form of complex numbers* is proven to be `a+ib`.

Complex Plane and Polar Form

Complex number `a+i b` is equivalently an ordered pair `(a,b)` which can be abstracted to represent a 2D plane. This is named after the mathematician JR Argand as *Argand Plane* or *complex plane* with real and imaginary axes.

Complex numbers, having abstracted to a complex plane, are represented in polar form.

Learn these in a *simple-thought-process*.

Algebra of Complex Numbers

Algebra of complex numbers details out the operations addition, subtraction, multiplication , division, conjugate, and exponent.

The algebra of complex numbers is quite easy to understand in abstraction. This topic goes beyond and explains the application context.

Learn these in a *simple-thought-process*.

The pages in this lesson are

__Complex Number Arithmetic__ *redo *

__Complex Number: Modeling sine waves__ *redo *

__Understanding Complex Arithmetic__ *redo *

__Addition of two Complex numbers__ *redo *

__Subtraction of Complex Numbers__ *redo *

__Multiplication of two Complex numbers__ *redo *

__Conjugate of a complex Number__ *redo *

__Division of complex numbers__ *redo *

__Exponent of a complex Number__ *redo *

Properties of Complex Number Arithmetic

Complex number system is an extension of Real number system with inclusion of a new number `i=sqrt(-1)`. The complex arithmetic has the * properties mostly identical* to real arithmetic. The additional features are related to modulus, argument, and conjugate.

Learn these in a *simple-thought-process*.

The pages in this lesson are

__Understanding Properties of Complex Arithmetic__ *redo *

__Addition : Commutative Law __ *redo *

__Addition : Associative Law__ *redo *

__Multiplication : Closure Law__ *redo *

__Multiplication : Commutative Law __ *redo *

__Multiplication : Associative Law__ *redo *

__Modulus in multiplication__ *redo *

__Argument in multiplication__ *redo *