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Algebra of Complex Numbers

Algebra of Complex Numbers

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Exponent of Complex Numbers


 »  Play with the forms of the complex number

    →  component form `a+ib`

    →  polar form `r(cos theta + i sin theta)`

    →  exponent form `re^(i theta)`

 »  Convert to the standard form of complex numbers `a+ib`

    →  eg: `(a+ib)^(c+id)`: convert `(a+ib)` to polar form `re^(i theta)`

    →  eg: `r^(id)` : convert `r` to `e^(ln r)`

Exponent of a complex Number

plain and simple summary

nub

plain and simple summary

nub

dummy

To find exponent and root of complex numbers, the rules of numerical expression is used to arrive at the coordinate form `a+ib`.

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simple steps to build the foundation

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Exponent of a complex number is explained.


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Starting on learning "Exponent of a complex number". ;; In this page, Exponent of a complex number is explained.

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, Can the exponent `z_1^z_2` be written in `a+ib` form?

  • No. Exponent to the power of complex number is not possible
  • Yes. By converting to polar form `z_1 = r e^(i theta)`.

The answer is 'Yes. By converting to polar form `z_1 = r e^(i theta)'

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, exponent
`z_1^z_2`
`quad quad = (r_1e^(i theta_1) )^(a_2+ib_2)`
`quad quad = r_1^a_2 `
`quad quad quad quad xx r_1^(ib_2)`
`quad quad quad quad xx e^(i theta_1 a_2)`
`quad quad quad quad xx e^(i theta_1 i b_2)`

`quad quad = r_1^a_2 `
`quad quad quad quad xx e^(ib_2 ln r_1) (`
`quad quad quad quad xx e^(i theta_1 a_2)`
`quad quad quad quad xx e^(- theta_1 b_2)`

`quad quad = r_1^a_2 e^(- theta_1 b_2) `
`quad quad quad quad xx e^(i(b_2 ln r_1 + theta_1 a_2))`

The result is in the polar form and can be converted to coordinate form.

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, Can the root `root(z_2)(z_1)` be computed in `a+ib` form?

  • No. root to a complex number is not possible
  • Yes. By considering root as power of `(1/z_2)`

The answer is 'Yes. By considering root as power of `(1/z_2)`'

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, root
`root(z_2)(z_1)`
`quad quad = z_1^(1/z_2)`
`quad quad = z_1^(bar(z_2)/(|z_2|^2))`
By following the rules of exponent of a complex number, the root can be solved.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Exponent and Roots of Complex number
 •  For `z_1^(z_2)`, convert `z_1` to polar form `re^(i theta)`
 •  For `a^(ib)`, convert `a` to `e^(ln a)` form
 •  For `z_1^(1/z_2)`, convert `1/z_2` to a complex number in numerator `bar(z_2)/(|z_2|^2)`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

What is `(1+i)^(3/2)`

  • `3/2+3/2i`
  • `2^(3/4)(cos (3 pi/8)+i sin (3 pi/8))`
  • `2^(3/2)(cos (3 pi/4)+i sin (3 pi/4))`
  • `2^(3/4)(cos (3 pi/2)+i sin (3 pi/2))`

The answer is '`2^(3/4)(cos (3 pi/8)+i sin (3 pi/8))`'

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Given z1 = a1+i b1 and z2 = a2+i b2, Can the exponent z1 to the power z2 be written in a+i b form?
no;exponent;power
No. Exponent to the power of complex number is not possible
yes;s;converting;by;buy
Yes. By converting to polar form z1 = r e power i theta.
The answer is 'Yes. By converting to polar form z1 = r e power i theta'.
Given z1 = a1+i b1 and z2 = a2+i b2, exponent ;; z1 power z2 ;; equals r1 e power i theta1 power a2+i b2 ;; equals r1 power a2 into r1 power i b2 into e power i theta1 a2 into e power i theta1 i b2 ;; equals r1 power a2 into e power i b2 ln r1 into e power i theta1 a2 into e power - theta1 b2 ;; equals r1 power a2 e power - theta1 b2 into e power i b2 ln r1 + theta1 a2 ;; The result is in the polar form and can be converted to coordinate form.
Given z1 = a1+i b1 and z2 = a2+i b2, Can the root z2 root of z1 be computed in a+i b form?
no;not;possible
No. root to a complex number is not possible
yes;s;considering;1
Yes. By considering root as power of one by z2
The answer is "Yes. By considering root as power of one by z2"
Given z1 = a1+i b1 and z2 = a2+i b2, ;; z2 root of z1 ;; equals z1 to the power 1 by z2 ;; equals z1 to the power conjugate of z2 by mod of z2 squared. ;; By following the rules of exponent of a complex number, the root can be solved.
To find exponent and root of complex numbers, the rules of numerical expression is used to arrive at the coordinate form a+i b
Exponent and Roots of Complex number: ;; For z1 power z2, convert z1 to polar form ;; For a power ib, convert A to e^ln a form ;; For z1 power 1 by z2, convert 1 by z2 to a complex number in numerator.
What is 1+ i power 3 by 2
by 2;buy 2
3 by 2 plus 3 by 2 i
pi by 8
2 power 3 by 4 cos 3 pi by 8 plus i sine 3 pi by 8
pi by 4
2 power 3 by 2 cos 3 pi by 4 plus i sine 3 pi by 4
pi by 2
2 power 3 by 4 cos 3 pi by 2 plus i sine 3 pi by 2
The answer is "2 power 3 by 4 cos 3 pi by 8 plus i sine 3 pi by 8"

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