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summary of this topic

Properties of Complex Number Arithmetic

Properties of Complex Number Arithmetic

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 »  Complex conjugate

    →  distributes into addition, multiplication, and power
    →  `bar(z_1+z_2) ``= bar(z_1) + bar(z_2)`
    →  `bar(z_1xxz_2) ``= bar(z_1) xx bar(z_2)`
    →  `bar(z^n) ``= (bar(z))^n`

    →  modulus of the conjugate equals the modulus of the number
    →  `| bar(z)| ``= |z|`

    →  argument of the conjugate is negative of the argument of the number
    →  `arg (bar(z)) ``= - arg(z)`

Properties of complex conjugate

plain and simple summary

nub

plain and simple summary

nub

dummy

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

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

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

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

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

 •  Conjugate of a conjugate is the complex number itself.

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

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simple steps to build the foundation

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Properties of complex conjugate is discussed in detail.


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Starting on learning "Properties of complex conjugate". ;; In this page, properties of complex conjugate is discussed in detail.

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)`

The answer is '`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) -: bar(z_2)`

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

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

  • `(bar(z))^n`
  • `(bar(z))^(1/n)`

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

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

  • `|z|`
  • `-|z|`

The answer is '`|z|`'

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

  • `text(arg)z`
  • `-text(arg)z`

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

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

  • `z`
  • `-z`

The answer is '`z`'

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

  • `|z|`
  • `|z|^2`

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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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)`

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)`

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) `

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

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

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

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



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

  • `text(arg)z`
  • `-text(arg)z`

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

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Given z1 = a1+i b1 and z2 = a2+i b2, what is the conjugate of sum conjugate of z1+z2?
plus
conjugate of z1 plus conjugate of z2
minus
conjugate of z1 minus conjugate of z2
The answer is 'conjugate of z1 plus conjugate of z2'
Conjugate of sum is sum of conjugates.;; Conjugate of difference is difference of conjugates.
Conjugate of Sum or Difference: For complex numbers z1, z2 in complex numbers conjugate of z1 plus or minus z2 = conjugate of z1 plus or minus conjugate of z2.
Given z1 = a1+i b1 and z2 = a2+i b2, what is the conjugate of z1 multiplied z2?
multiplied
conjugate z1 multiplied conjugate z2
divided
conjugate z1 divided conjugate z2
The answer is 'conjugate z1 multiplied conjugate z2'
Conjugate of product is product of conjugates. ;; Conjugate of quotient is quotient of conjugates.
Conjugate of product or quotient: For complex numbers z1, z2 in complex numbers conjugate of z1 multiplied z2 = conjugate of z1 multiplied conjugate z2;; conjugate of z1 divided z2 = conjugate of z1 divided conjugate z2
Given z = a+i b, what is the conjugate of, z power n?
power n
conjugate of z, power n
1;one
conjugate of z power 1 by n
The answer is 'conjugate of z, power 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 complex numbers ;; conjugate of, z power n, equals, conjugate of z, power n ;; conjugate of, z power 1 by n, equals, conjugate of z, power 1 by n
Given z = a+i b, what is the modulus of conjugate of z?
z
modulus of z
minus
minus modulus of z
The answer is 'modulus of z'
Modulus of a conjugate equals modulus of the complex number.
Modulus of a Conjugate: For a complex number z in complex numbers, mod conjuage of z = mod of z
Given z equals a + i b, what is the argument of conjugate of z?
argument
argument of z
minus
minus argument of z
The answer is "minus argument of z"
Argument of a conjugate equals negative of the argument of the complex number.
Argument of a Conjugate: For a complex number z in complex numbers ;; arfument of conjugte of z = minus argument of z
Given z = a+i b, what is the conjugate of conjugate of z?
z
z
minus
minus z
The answer is ' z '
Conjugate of a conjugate is the complex number itself.
Conjugate or Conjugate: For a complex number z in complex numbers ;; conjugate of conjugate of z = z
Given z = a+i b, what is the product z multiplied conjugate of z?
mod;modulus
mod z
squared
mod z squared
The answer is 'mod z squared'
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 complex numbers ;; z multiplied by conjugate of z = mod z squared.
Given z = a+i b, what is the argument argument of conjugate z?
argument
argument of z
minus
minus argument of z
The answer is 'minus argument of z'.

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