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Thought-Process to Discover Knowledge

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mathsComplex NumbersIntroduction to Complex Numbers

Complex Numbers: Equivalent representations

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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Which of the following equals `2`?

  • `4-2`
  • `1+1`
  • `6/3`
  • all the 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?

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

  • `sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)`
  • `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,...`
  • `sin (theta + 2n pi)` where `n=0,1,2,...`
  • it is not the above

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

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

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)`

                            
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