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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

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Representations of Complex Numbers

» coordinate form `a+ib`

» polar form `r(cos theta + i sin theta)`*abstracted from `a+ib`*

→ `r=sqrt(a^2+b^2)`

→ `cos theta = a/r`

→ `sin theta = b/r`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

A complex number in the form `a+ib ` is equivalently given in the form `r (cos theta + i sin theta)`

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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Complex number in the form `a+ib` can be equivalently represented in the form `r(cos theta + i sin theta)`. This representation, has an advantage to understanding the generic form of complex numbers.

Starting on learning "Equivalent Representation of complex numbers". ;; Complex number in the form a plus i b can be equivalently represented in the form r cos theta plus i sine theta. This representation, has an advantage to understanding the generic form of complex numbers.

Which of the following equals `2`?

- `4-2`
- `1+1`
- `6/3`
- all the above

The answer is 'All the above'.

A number can be equivalently represented with numerical expressions of various forms.

Given hypotenuse is `c`. Which of the following represent the length of side opposite to the angle `theta` in a right angle triangle?

- `c sin theta`
- `c tan theta cos theta`
- `c sqrt(1 - cos^2 theta)`
- all the above

The answer is 'all the above'.

A constant can be represented as expressions.

It was established that solution to quadratic equation is in the form `a+b i`. What is an equivalent expression of this?

- `sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)`
- The option given above is not correct.

The answer is '`sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)`'

A number `a+bi` is equivalently given as

`color(darkorange)(sqrt(a^2+b^2)) (color(blue)(a/(sqrt(a^2+b^2)))`

`quad quad quad + color(green)(b/(sqrt(a^2+b^2))) i)`

This form looks similar to `color(blue)(cos)` and `color(green)(sin)` of a right angled triangle.

It is equivalently represented as `color(darkorange)(r)(color(blue)(cos theta) + i color(green)(sin theta))` where `color(darkorange)(r = sqrt(a^2+b^2))` and `theta = tan^(-1)(b/a)`.

`sin theta` equals ...

- `sin (theta + 2n pi)` where `n=0,1,2,...`
- it is not the above

The answer is '`sin (theta + 2n pi)`'

A number in the form `a+bi` is equivalently given as

`quad quad = r (cos (theta+ 2n pi) `

`quad quad quad quad + i sin (theta+ 2n pi))`

where `r = sqrt(a^2+b^2), theta = tan^(-1)(b/a)`

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

*your progress details*

Progress

*About you*

Progress

Which of the following equals 2 ?

minus;4;2

4 minus 2

plus;1

1+1

by;6;3

6 by 3

all;above

all the above

The answer is 'All the above'. ;; A number can be equivalently represented with numerical expressions of various forms.

Given hypotenuse is c . Which of the following represent the length of side opposite to the angle theta in a right angle triangle?

sine;

c sine theta

tan;cos;cause

c tan theta cos theta

square;root;minus;1

c square root of 1 minus cos squared theta

all;above

all the above

The answer is 'all the above'. ;; A constant can be represented as expressions.

It was established that solution to quadratic equation is in the form a+b i . What is an equivalent expression of this?

square;root;plus;a;b;

square root of a squared plus b squared ;; multiplied by ;; a by square root ; + ; b by square root ;; into i

option;above;correct;not

The option given above is not correct.

The answer is the expression given with a and b.

A number a plus b i is equivalently given as square root of a squared plus b squared ;; multiplied by ;; a by square root ; + ; b by square root ;; into i ;; This form looks similar to cos and sine of a right angled triangle. ;; It is equivalently represented as r cos theta plus i sine theta ;; where r equals square root of a squared plus b squared and theta = tan inverse (b by a)

sine theta equals?

sine; theta;plus;2

sine theta plus 2 n pi, n is a whole number

not;above

it is not the above

The answer is sine theta plus 2n pi

A complex number in the form a+i b is equivalently given in the form r (cos theta + i sine theta)

A number in the form a+b i, is equivalently given as r cos theta + 2 n pi , plus i sine (theta+ 2 n pi ;; where r = square root a squared plus b squared ;; , theta = tan inverse b by a