Complex Numbers and Conjugates (1 Viewer)

[_luke_]

If you have an equation that is true, if you take the conjugate of both sides is it also true?

For example, take z to be an imaginary number.

az^2 + bz + c = 0

Now the question asked to prove this true for the conjugate of z. Am i justified in doing this?

Also, are there any other 'tricks' you can think of straight away in regards to complex numbers?

Thanks.

 

[Paradoxica]

If you have an equation that is true, if you take the conjugate of both sides is it also true?

For example, take z to be an imaginary number.

az^2 + bz + c = 0


Now the question asked to prove this true for the conjugate of z. Am i justified in doing this?

Also, are there any other 'tricks' you can think of straight away in regards to complex numbers?

Thanks.

All statements that are true where the imaginary unit exists, remain true when all instances of the imaginary unit are sign-flipped.

 

[InteGrand]

All statements that are true where the imaginary unit exists, remain true when all instances of the imaginary unit are sign-flipped.

That's a bit risky to say, e.g. Im(i) = 1 is true. When we flip i to -i, we get Im(-i) = 1, which is false.

 

[Paradoxica]

That's a bit risky to say, e.g. Im(i) = 1 is true. When we flip i to -i, we get Im(-i) = 1, which is false

Excluding statements that require the use of the Imaginary Extraction Operator.

 

[Carrotsticks]

If you have an equation that is true, if you take the conjugate of both sides is it also true?

For example, take z to be an imaginary number.

az^2 + bz + c = 0

Now the question asked to prove this true for the conjugate of z. Am i justified in doing this?

Also, are there any other 'tricks' you can think of straight away in regards to complex numbers?

Thanks.

In general, no. Usually, you have to add in a fairly strong condition in order for a property to hold for both Z and Z conjugate.

For your above statement, we need A, B and C to either be all real and/or all equal in order for equation to also work for Z bar.

Excluding statements that require the use of the Imaginary Extraction Operator.

What is this 'operator'? I have not heard of such a thing before.

 

[KingOfActing]

@Original question

Yes, you can take the imaginary conjugate of both sides of an equation.

However, in this case a direct proof is simple enough:



Note that we've assumed a, b and c are real, and only used elementary properties of complex conjugates. (Thanks for the correction, FrankXie )

Late edit:
If you conjugate both sides, you're going to end up doing the same thing I did above, but backwards.

 

[FrankXie]

If you have an equation that is true, if you take the conjugate of both sides is it also true?

Yes it is always true. Namely, if f(z)=g(z), then the conjugate of f(z)=conjugate of g(z).

 

[FrankXie]

You've assumed a, b and c are all real.

 

[KingOfActing]

You've assumed a, b and c are all real.

Yes I have, whoopsies