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Thought-Process to Discover Knowledge

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mathsComplex NumbersProperties of Complex Number Arithmetic

Properties of complex conjugate

Properties of complex conjugate is discussed in detail.



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Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, what is the conjugate of sum `bar(z_1+z_2)`?

  • `bar(z_1)+bar(z_2)`
  • `bar(z_1)+bar(z_2)`
  • `bar(z_1)-bar(z_2)`

The answer is '`bar(z_1)+bar(z_2)`'

 •  Conjugate of sum is sum of conjugates.
 •  Conjugate of difference is difference of conjugates.

Conjugate of Sum or Difference: For complex numbers `z_1, z_2 in CC`
`bar(z_1 +- z_2) = bar(z_1) +- bar(z_2)`

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2` what is the conjugate `bar(z_1 xx z_2)`?

  • `bar(z_1) xx bar(z_2)`
  • `bar(z_1) xx bar(z_2)`
  • `bar(z_1) -: bar(z_2)`

The answer is '`bar(z_1) xx bar(z_2)`'

 •  Conjugate of product is product of conjugates.
 •  Conjugate of quotient is quotient of conjugates.

Conjugate of product or quotient: For complex numbers `z_1, z_2 in CC`
`bar(z_1 xx z_2) = bar(z_1) xx bar(z_2)`
`bar(z_1 -: z_2) = bar(z_1) -: bar(z_2)`

Given `z = a+ib`, what is the conjugate `bar(z^n)`?

  • `(bar(z))^n`
  • `(bar(z))^n`
  • `(bar(z))^(1/n)`

The answer is '`(bar(z))^n`'

 •  Conjugate of a power is power of conjugate.
 •  Conjugate of a root is root of conjugate.

Conjugate of Power or Root: For a complex number `z in CC`
`bar(z^n) = (bar(z))^n `
`bar(z^(1/n)) = (bar(z_1))^(1/n) `

Given `z = a+ib`, what is the modulus `|bar(z)|`?

  • `|z|`
  • `|z|`
  • `-|z|`

The answer is '`|z|`'

 •  Modulus of a conjugate equals modulus of the complex number.

Modulus of a Conjugate: For a complex number `z in CC`
`|bar(z)| = |z| `

Given `z = a+ib`, what is the argument `text(arg) bar(z)`?

  • `text(arg)z`
  • `-text(arg)z`
  • `-text(arg)z`

The answer is '`-text(arg)z`'

 •  Argument of a conjugate equals negative of the argument of the complex number.

Argument of a Conjugate: For a complex number `z in CC`
`text(arg) bar(z) = - text(arg)z `

Given `z = a+ib`, what is the conjugate of conjugate `bar bar(z)`?

  • `z`
  • `z`
  • `-z`

The answer is '`z`'

 •  Conjugate of a conjugate is the complex number itself.

Conjugate or Conjugate: For a complex number `z in CC`
`bar (bar(z)) = z `

Given `z = a+ib`, what is the product `z bar(z)`?

  • `|z|`
  • `|z|^2`
  • `|z|^2`

The answer is '`|z|^2`'

 •  Product of a number and its conjugate is the square of the modulus.

Product of a number and its conjugate: For a complex number `z in CC`
`z bar(z) = |z|^2 `

Solved Exercise Problem:

Given `z = a+ib`, what is the argument `text(arg) bar(z)`?

  • `text(arg)z`
  • `-text(arg)z`
  • `-text(arg)z`

The answer is '`-text(arg)z`'

                            
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