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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 **nub****trek**.

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. continue

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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.

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

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. continue

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. continue

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. continue

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)`

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

*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

You are learning the free content, however do shake hands with a coffee to show appreciation.

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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)`.

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?

- Practice Saying the Answer

The answer is 'Complex Numbers'

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`'.

*your progress details*

Progress

*About you*

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

The answer is 'Complex Numbers'

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.