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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.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Introduction to Complex Numbers

Voice

Voice

Home

Complex number Representation

»  Abstraction of complex numbers
→  solution of algebraic equations in 2D plane
→  eg: x^3=1 has 3 solutions, (root(3)(1))_(1st), (root(3)(1))_(2nd), (root(3)(1))_(3rd)

»  complex numbers are numerical expressions
→  eg: 3+4 root(4)(-5) is one solution to ((x-3)/4)^4 = -5

»  solution to quadratic equation in the form a+ib
→  i=sqrt(-1)

Note: irrational numbers do not have a generic form. But, it is later proven that all complex numbers can be expressed in the form a+ib.

For now, only solution to quadratic equation is shown to be in the form a+ib.

### Representation of Complex Numbers (incomplete)

plain and simple summary

nub

plain and simple summary

nub

dummy

[Initial understanding] Solution to quadratic equation is in the form a+bi, where i=sqrt(-1).

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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In this page, as initial understanding of complex numbers, quadratic equations are examined. The solutions to quadratic equations is in the form a+ib, where i=sqrt(-1).

Keep tapping on the content to continue learning.
Starting on learning "Representation of complex numbers". ;; In this page, As initial understanding of complex numbers, quadratic equations are examined. The solutions to quadratic equations is in the form a plus i b , where i is square root of minus 1.

When using the real numbers, we come across problems that are mathematically modeled as x^2=-1. There is no solution to this in real number system. So, what is done?

• the equation has no solution
• number system is extended beyond real numbers

The answer is 'number system is extended beyond real numbers'. This number system that is over-and-above the real number system and is named as 'complex numbers'.

To include solutions to polynomials which number system is introduced?

Irrational numbers are represented with numerical expressions or symbols.
How a complex number is represented? In this topic an incomplete explanation to the form of representing complex number is discussed.

In the topic 'Generic form of Complex Numbers', the information discussed in here is further developed to complete the explanation.

In irrational number system, the solution to x^2=2 is given as +-sqrt(2).
Learning from that, what could be the solution to x^2=-1?

• +-sqrt(-1)
• 1
• -1

The answer is '+-sqrt(-1)'. The solution is represented as a numerical expression.

Note to students: In the attempt to develop knowledge in stages, the problems given in here are specific to quadratic equations. A more generic discussion will follow once first level of knowledge is acquired.

What is the solution to x^2 = -4?

• 2
• -2
• +-2sqrt(-1)
• +-2

The answer is '+-2sqrt(-1)'

What is the solution to (x+3)^2 = -4?

• -1
• 5
• -3+-2sqrt(-1)
• +-sqrt(-1)

The answer is '-3+-2sqrt(-1)'

It is noted that a quadratic equation of the form px^2+qx+r= 0 can be re-arranged to (x+q/(2p))^2 = -r/p + (q/(2p))^2.
For any equation in this form, we can arrive at a solution in the form a+b sqrt(-1).

sqrt(-1) is represented with a letter i
The solution to a quadratic equation is in the form a+bi.

Note : The said explanation covers only solutions to quadratic equations. Let us examine this representation in detail and then later generalize this for complex numbers.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Form of complex number : Solution to quadratic equation is a+bi where a, b in RR and i=sqrt(-1)

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

What are the solutions to the equation (x-1)^2 = -16?

• +-4
• +-4i
• 1+-4i
• +-1+-4i

The answer is '1+-4i'.

Progress

Progress

When using the real numbers, we come across problems that are mathematically modeled as x squared = minus 1. There is no solution to this in real number system. So, what is done?
equation;has;solution
the equation has no solution
number;system;extended;beyond;real
number system is extended beyond real numbers
The answer is 'number system is extended beyond real numbers'. This number system that is over-and-above the real number system and is named as 'complex numbers'.
To include solutions to polynomials which number system is introduced?
complex
Irrational numbers are represented with numerical expressions or symbols. ;; How a complex number is represented? In this topic an incomplete explanation to the form of representing complex number is discussed. ;; In the topic 'Generic form of Complex Numbers', the information discussed in here is further developed to complete the explanation.
In irrational number system, the solution to x squared =2 is given as plus or minus square root of 2. ;; Learning from that, what could be the solution to x squared = minus 1?
plus;minus;square;root
plus or minus square root minus 1
1
1
minus 1
minus 1
The answer is "plus or minus square root minus 1". The solution is represented as a numerical expression.
Note to students: In the attempt to develop knowledge in stages, the problems given in here are specific to quadratic equations. A more generic discussion will follow once first level of knowledge is acquired.
What is the solution to x squared = minus 4?
2
2
minus 2
minus 2
plus;minus;square;root;1
plus or minus 2 square root minus 1
plus;minus
plus or minus 2
The answer is "plus or minus 2 square root minus 1".
What is the solution to x plus 3 whole squared equals minus 4?
minus;1
minus 1
5
5
3;minus;square;root
minus 3 plus or minus square root minus 1
plus or minus square root
plus or minus square root minus 1
The answer is "minus 3 plus or minus square root minus 1".
It is noted that a quadratic equation of the form p x squared + q x + r= 0 can be re-arranged to x + q by 2p whole squared = minus r divided by p + q by 2p whole squared. ;; For any equation in this form, we can arrive at a solution in the form a+b square root of minus 1.
square root of minus 1 is represented with a letter i. ;The solution to a quadratic equation is in the form a plus b i. ;; Note : The said explanation covers only solutions to quadratic equations. Let us examine this representation in detail and then later generalize this for complex numbers.
Initial understanding : ;; Solution to quadratic equation is in the form a plus b i, where i equals square root of minus 1.
Form of complex number : Solution to quadratic equation is a plus b i where a, b in real numbers and i equals square root of minus 1.
What are the solutions to the equation?
minus 4
plus or minus 4
4i
plus or minus 4 i
one
one plus or minus 4 i
minus one
plus or minus one plus or minus 4 i
The answer is one plus or minus 4 i.

we are not perfect yet...