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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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