Complex Number equivalency Watch

nugiboy
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#1
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Hey guys,

I've got a question about a complex number z involving it's modulus and conjugate:

let z\in\mathbb{C}\setminus\left\{{0  }\right\}

Show that \frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}

I've been manipulating both sides for a while and can't quite get them to work out to each other.

Can anyone point me in the right direction?
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Felix Felicis
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(Original post by nugiboy)
Hey guys,

I've got a question about a complex number z involving it's modulus and conjugate:

let z\in\mathbb{C}\setminus\left\{{0  }\right\}

Show that \frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}

I've been manipulating both sides for a while and can't quite get them to work out to each other.

Can anyone point me in the right direction?
No idea if this is rigorous enough/ the kind of thing you're looking for but let z = re^{i\theta} \implies \overline{z} = re^{-i\theta}.
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brianeverit
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(Original post by nugiboy)
Hey guys,

I've got a question about a complex number z involving it's modulus and conjugate:

let z\in\mathbb{C}\setminus\left\{{0  }\right\}

Show that \frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}

I've been manipulating both sides for a while and can't quite get them to work out to each other.

Can anyone point me in the right direction?
Can you start from  z\overline{z}\equiv|z|^2 and do anything with that?
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nugiboy
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(Original post by brianeverit)
Can you start from  z\overline{z}\equiv|z|^2 and do anything with that?
Wow that made it easier than I thought!

So starting with the given equivalence:

\frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}

\frac{z\overline{z}}{|\overline{  z}|} \equiv |z|

\frac{|z|^2 }{|\overline{z}|} \equiv |z|

\frac{|z|^2 }{|z|} \equiv |z|

|z| \equiv |z|

Is that suitably thorough in your opinion?
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davros
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(Original post by nugiboy)
Wow that made it easier than I thought!

So starting with the given equivalence:

\frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}

\frac{z\overline{z}}{|\overline{  z}|} \equiv |z|

\frac{|z|^2 }{|\overline{z}|} \equiv |z|

\frac{|z|^2 }{|z|} \equiv |z|

|z| \equiv |z|

Is that suitably thorough in your opinion?
It's the right argument, but in the wrong direction - you just need to reverse all these steps so you end up proving the relation you're given
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around
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(Original post by nugiboy)
Wow that made it easier than I thought!

So starting with the given equivalence:

\frac{z}{|\overline{z}|} \equiv \frac{|z|}{\overline{z}}

\frac{z\overline{z}}{|\overline{  z}|} \equiv |z|

\frac{|z|^2 }{|\overline{z}|} \equiv |z|

\frac{|z|^2 }{|z|} \equiv |z|

|z| \equiv |z|

Is that suitably thorough in your opinion?
You can't prove an equivalence by assuming it's true and then deriving something true. As davros says, just write the steps backwards and you have a decent argument.
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nugiboy
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(Original post by davros)
It's the right argument, but in the wrong direction - you just need to reverse all these steps so you end up proving the relation you're given
Ok sweet I've got it then - without using the end of the relation given.

Can you explain why it is not suitable to prove that the given relation is equivalent using the whole relation?
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davros
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(Original post by nugiboy)
Ok sweet I've got it then - without using the end of the relation given.

Can you explain why it is not suitable to prove that the given relation is equivalent using the whole relation?
If you're asked to prove something, you cannot assume that thing to begin with and use it in your proof - you need to start with something else and use that to derive the required statement.

Of course, we don't actually know what you're allowed to assume - you could have derived the answer using the x+iy form for z or by using the exponential form for z
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brianeverit
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(Original post by nugiboy)
Ok sweet I've got it then - without using the end of the relation given.

Can you explain why it is not suitable to prove that the given relation is equivalent using the whole relation?
You can only prove something true by starting from something you KNOW to be true and finishing up with the thing you are trying to prove.
To prove that something is not true, a standard method is to assume that it is and show that that assumption leads to a contradiction, i.e. to something that is obviously NOT true.
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