Talk:Complex conjugate

Asterisks versus overlines

The article should probably be converted from asterisk notation (a*) to overline notation (${\overline {a}}$) to avoid confusion with the conjugate transpose. I haven't seen anything but conjugation use overline notation, has anyone else? -FunnyMan 18:02, Oct 23, 2004 (UTC)

I strongly second this proposal: overline is the standard notation in complex analysis and geometry.--FpWl 18:18, 29 Apr 2005 (UTC)

I also endorse this proposal, if nothing else it helps the article agree with the complex numbers entry. I went ahead and made the change. Thenub314 21:07, 19 July 2006 (UTC)

"if w is non-zero" removed (small item)

I removed "if w is non-zero" from the following:

$\left({\frac {z}{w}}\right)^{*}={\frac {z^{*}}{w^{*}}}$ if w is non-zero

because I am almost certain that the equality still holds when w is zero. Even though dividing by zero is not desirable in many cases, it is still mathematically *defined* as "undefined" (in most number systems). Please correct me if i'm wrong. Fresheneesz 08:40, 8 March 2006 (UTC)

LaTeX template removed

Hey -- I removed the template, since I did the job. If I missed anything, just let me know! --NicApicella 16:06, 13 March 2006 (UTC)

typography

I think that the initial definition of z ought to be in a bigger font, but short of doing something hacky, I don't know how to make it so. --anon

Done. Oleg Alexandrov (talk) 18:55, 22 August 2006 (UTC)

"Idempotence", huh?

Its obviously NOT idempotent because in particular '${\overline {\overline {\overline {z}}}}\neq z\!\$

I've never heard this usage of the term idempotence. Could someone please point me to some references? (Because I haven't found any.) I think this is confusing (if not bogus), and should be removed.

(I know this is stated in the idempotence article, too, and I think it should be removed from there as well.) --Matt Kovacs (talk) 17:08, 24 January 2009 (UTC)

I have not seen this particular word used before. Most frequently you see statements like "blah is an idempotent". Often it is first defined in group theory (or abstract algebra) texts. Thenub314 (talk) 18:56, 24 January 2009 (UTC)

The $\mathbb {R}$-isomorphisms of $\mathbb {C}$ form a (pretty small) group $\{id,{\bar {}}\ \}$. The only NON-idempotent there is the conjugation. I don't see any other link to idempotency.

Idempotency is a word used to describe something that when applied over and over again results in the same thing. The only example I can think right now is the powers of 1, that is 1=1²=1³=... Idempotency is also used in computer science to describe functions that always produce the same results, such as decreasing the amount of money in an account. I don't know if this property of the conjugate is called "idempontency" or not, but when you apply the conjugate over and over again it sure doesn't result in the same thing. --Fred —Preceding unsigned comment added by 89.181.13.221 (talk) 03:02, 27 July 2010 (UTC)

xkcd

I'm sorry, I still don't get it. http://xkcd.com/849/ --204.246.229.130 (talk) 15:59, 20 January 2011 (UTC)

Explaining a joke usually means to destroy it, but … In theoretical physics, the wavefunction is a complex function. If you multiply any complex number by its complex conjugate, the result is a real number (namely the magnitude squared). The same is true for the wavefunction: multiplying it with its complex conjugate turns it into a real-valued probability density, which actually has physical meaning. (The complex wavefunction itself arguably doesn’t – it’s merely a theoretical construct and cannot be measured in a physical experiment.) I guess the funny part comes from the play with words: on the one hand “real” means it’s a real (and not a complex) number. On the other hand, “real” means we’re now talking about hands-on physics instead of elusive theoretical calculations. Physical observables are always real values, and they can – in principle – be measured. Complex values cannot. Experimental physicists might say that complex numbers only make sense in mathematics, which they’d consider “merely” an auxiliary discipline. Compare http://xkcd.com/435/ including its hover-tooltip comment. --78.48.0.203 (talk) 21:49, 14 September 2011 (UTC)

History

Where did the concept of complex conjugates came from? It would be great if someone with that knowledge would make an addition to the article. — Preceding unsigned comment added by 101.208.119.211 (talk) 08:39, 3 March 2012 (UTC)

Please, cf. Nahin, Paul J. (1998), An Imaginary Tale: The story of $\scriptstyle {\sqrt {-1}}$, Princeton University Press , Princeton, ISBN 978-0-691-12798-9.

Mgvongoeden (talk) 12:37, 4 March 2012 (UTC)

(Complex) multiplication is always commutative

I don't get the follwing statement from the article:

"${\overline {(zw)}}={\overline {w}}\;{\overline {z}}\!\$ (note the reversed arguments if z and w don't commute)"

For complex numbers, multiplication is always commutative so in no way z and w will not commute. Maybe this may be true for non-scalar or matrix algerba, but for complex numbers, the order of operands is unimportant.89.137.186.101 (talk) 20:05, 6 September 2012 (UTC)

"... a complex number is equal to its complex conjugate if its imaginary part is [...] -i."

No it is not i.e. nonsense; introduced by this edit 14-Apr-2015 Nxavar. Fixing it means resplicing/reverting the relevant rearranged material - see the linked edit for the original paragraph. — Preceding unsigned comment added by 2.99.166.53 (talk) 15:23, 7 June 2015 (UTC)

You are right, it is total gibberish. (The author of that sentence seemed to confuse equality with "can be interchanged by a field isomorphism," or something.) I have removed the offending sentence. --JBL (talk) 15:33, 7 June 2015 (UTC)

Power of conjugate and conjugate of power

The article states that

${\begin{aligned}{\overline {z^{n}}}&=\left({\overline {z}}\right)^{n},\quad \forall n\in \mathbb {Z} \\\end{aligned}}$

, meaning it's only valid for integer n. However, empirically testing with Wolfram Mathematica shows all real n satisfy the equation. Has it just never been proven so, or is it a mistake?

31.217.16.25 (talk) 18:15, 9 September 2015 (UTC)

The issue is complications from multiple values: when you raise complex numbers to non-integer powers, you get many values back, and so it's not trivial to explain what you mean by an equality of the sort in question if n isn't an integer -- choosing a branch of the logarithm, etc. etc. See Exponential_function#Complex_plane for more about this. Probably there's no consistent way to choose a branch of the logarithm to make it true once and for all for all z and (complex) n, though maybe with natural choices of branch cuts it is true for all real n. --JBL (talk) 18:33, 9 September 2015 (UTC)