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summary of this topic

Introduction to Complex Numbers

Introduction to Complex Numbers

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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`

Complex Numbers: Equivalent representations

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

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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.


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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

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Progress

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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

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