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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

Argument

» Angle with positive real-axis

» `theta = tan^(-1) (b/a)`

» Arguments `= 2npi+theta` where `n=0,1,2,...`

» Principal argument `= theta` where `0 <= theta < 2pi`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

The angle made by the line segment with the real axis is the argument of the complex number.

Principal argument of a complex number is in the range `-pi` to `pi`.

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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Argument is defined for a complex number represented as a point in Argand plane.

Starting on learning "Argument of a complex number". ;; Argument is defined for a complex number represented as a point in Argand plane.

A Complex number `a+i b` is represented on complex plane as shown in the figure. What is the angle the line segment `OP` makes with the real axis?

- `tan^(-1)(b/a)`
- `0`

The answer is '`tan^(-1)(b/a)`'

The angle between the real axis in positive direction and the line segment `OP` is called the argument of the complex number. The argument of a complex number is given as `text(arg) z`.

What does "argument" mean?

- an independent element that plays a role in determining the value of something
- complete absence and avoidance of mentioning of something

The answer is 'an independent element that plays a role in determining the value of something'.

For example the arguments of a function `f() = 2xy+3` are the variables `x` and `y`. In this case, the arguments define the value of the function.

In complex numbers, the modulus provides the absolute value. In addition to the modulus, to completely specify the complex number, the additional independent element required is the "argument".

For `z=a+i b in CC`, argument is given by `text(arg) z = tan^(-1)(b/a)`

arg(z) is called...

- Practice Saying the Answer

The answer is 'Argument or arg'.

Find the argument of `1+i`

- `tan^(-1)1`
- `sqrt(2)`

The answer is '`tan^(-1)1`'

What are the possible values of `tan^(-1)1`?

- only one possible value `pi/4`
- Two possible values `pi/4` and `(3pi)/4`
- `(n pi)/4` where `n = 1,3,5,cdots`

The answer is '`(n pi)/4` where `n = 1,3,5,cdots`'.

The possible values of `tan^(-1)1` are `(n pi)/4` where `n = 1,3,5,cdots`.

What are the possible values of `tan^(-1)1`, if the opposite side and adjacent sides are given as `1`?

- Answer is same : `(n pi)/4` where `n = 1,3,5,cdots`
- Answer is different : `(n pi)/4` where `n = 1,5,9,cdots`

'Answer is different : `(n pi)/4` where `n = 1,5,9,cdots`'

Note that: `tan^(-1)1 = (n pi)/4` where `n = 1,3,5,cdots` but

`tan^(-1) (text(opposite-side)=1)/(text(adjacent-side)=1) = (n pi)/4` where `n = 1,5,9,cdots` and

`tan^(-1) (text(opposite-side)=-1)/(text(adjacent-side)=-1) = (n pi)/4` where `n = 3,7,11,cdots`

The argument of a complex number can be given in the form `2n pi + theta` where `n=0,1,2...`. The argument corresponding to `n=0` is called the principal argument. The range of principal argument is `-pi <= theta <= pi`.

Given the complex numbers `z_1, z_2, z_3, z_4` as shown in figure. The value of argument of complex numbers are

• `0 < text(arg) z_1 < pi/2`

• `pi/2 < text(arg) z_2 < pi`

• `0< text(arg) z_3 <-pi/2`

• `-pi/2< text(arg) z_4 <-pi`

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Argument: **For a complex number `z=a+i b in CC`, the argument of `z` is given as ` text(arg) z = tan^(-1)(b/a)`

**Principal Argument: **For a complex number `z=a+i b in CC`, the principal argument of `z` is given as ` text(arg) z = tan^(-1)(b/a)`, where `-pi <= text(arg) z <= pi`

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Find the principal argument of `1-i`

- `pi/4`
- `-pi/4`
- `+-pi/4`

The answer is '`-pi/4`'

*your progress details*

Progress

*About you*

Progress

A Complex number a+i b is represented on complex plane as shown in the figure. What is the angle the line segment O P makes with the real axis?

tan;inverse;b;a

tan inverse b by a

0;zero

0

The answer is "tan inverse b by a"

The angle between the real axis in positive direction and the line segment O P is called the argument of the complex number. The argument of a complex number is given as arg of z.

What does "argument" mean?

independent;element;plays

an independent element that plays a role in determining the value of something

complete;absence;avoidance

complete absence and avoidance of mentioning of something

The answer is 'an independent element that plays a role in determining the value of something'. ;; For example the arguments of a function f = 2x y+ 3 are the variables x and y. In this case, the arguments define the value of the function.

In complex numbers, the modulus provides the absolute value. In addition to the modulus, to completely specify the complex number, the additional independent element required is the "argument". ;; For z=a+i b, argument is given by arg of z = tan inverse b by a

arg(z) is called...

argument;arg

The answer is 'Argument or arg'.

The angle made by the line segment with the real axis is the argument of the complex number.

For a complex number z = a+i b, the argument of z is tan inverse b by a.

Find the argument of 1 plus i

tan;inverse;1

tan inverse 1

square;root;2

square root 2

The answer is 'tan inverse 1'

What are the possible values of tan inverse 1?

only;one;1

only one possible value, pi by 4

two;2

Two possible values pi by 4 and 3 pi by 4

n;equals;5

n pi by 4 where n equals 1 3 5 et cetera

The answer is 'n pi by 4 where n equals 1 3 5 et cetera'

The possible values of tan inverse 1 are n pi by 4 where n equals 1, 3, 5, et cetera. ;; What are the possible values of tan inverse 1, if the opposite side and adjacent sides are given as 1?

same

Answer is same : n pi by 4 where n equals 1 3 5 et cetera

different

Answer is different: n pi by 4 where n equals 1 5 9 et cetera

Answer is different: n pi by 4 where n equals 1 5 9 et cetera

Note that: tan inverse 1 ; = n pi by 4 ; where n = 1,3,5,et cetera ;; but ; tan inverse opposite-side =1 by adjacent-side =1 ; = n pi /4 ; where n = 1,5,9,et cetera ;; and; tan inverse opposite-side =-1 by adjacent-side =-1 ; = n pi /4 ; where n = 3,7,11,et cetera

The argument of a complex number can be given in the form 2n pi + theta ; where n=0,1,2 et cetera. ;; The argument corresponding to n=0 is called the principal argument. The range of principal argument is minus pi to pi.

Principal argument of a complex number is in the range minus pi to pi

For a complex number z=a+i b, the principal argument of z is given as tan inverse b by a, where argument of z is in the range minus pi to pi.

Given the complex numbers z 1, z 2, z 3, z 4, as shown in figure. The value of argument of complex numbers are ;; argument of z 1 is, 0 to pi by 2 ;; argument of z 2 is, pi by 2 to pi ;; argument of z 3 is, 0 to minus pi by 2 ;; argument of z 4 is, minus pi by 2 to minus pi

Find the principal argument of 1 minus i.

pi by 4

pi by 4

minus

minus pi by 4

plus or minus

plus or minus pi by 4

The answer is 'minus pi by 4'