Why Does the Complex Conjugate Involve Negating the Argument Theta?

In summary, this is true:-Euler's Formula, ##e^{i\theta}= \cos{\theta} + i\sin{\theta}##, is the most important equation in mathematics-If z=reiθ, why does z¯=re−iθ?
  • #1
Leo Liu
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Can someone please tell me why this is true? This isn't exactly the De Moivre's theorem. Thank you.
 

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  • #2
If you try to justify that formula, where do you get stuck?
 
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  • #3
$$\bar z^2=(x-yi)^2$$
$$\bar z^2=r^2(\cos\theta-i\sin\theta)^2$$
$$\bar z^2=r^2(\cos(2\theta)-i\sin(2\theta))$$
$$\bar z^2=r^2(\cos2\theta+i\sin(-2\theta))$$
$$\bar z^2=r^2(\cos(-2\theta)+i\sin(-2\theta))$$
I got it. Thanks.
 
  • #4
What about:
$$\bar z^2 = (re^{-i\theta})^2 = r^2e^{-2i\theta} =r^2(\cos (-2\theta) +i\sin(-2\theta))$$
 
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  • #5
PeroK said:
What about:
$$\bar z^2 = (re^{-i\theta})^2 = r^2e^{-2i\theta} =r^2(\cos (-2\theta) +i\sin(-2\theta))$$
I actually stated that I didn't want to use this method on the course chat haha.
1638798004769.png

Thank you though! It is very neat.
 
  • #6
Leo Liu said:
I actually stated that I didn't want to use this method on the course chat haha.
View attachment 293673
Thank you though! It is very neat.
You should study each part you doubt until it is intuitive to you. It is very fundamental and important.
In words explain why:
If ##z=re^{i\theta}##, why does ##\bar{z}=re^{-i\theta}##?
Then why does ##\bar{z}^2=re^{-i2\theta}##?
etc.
 
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  • #7
Many mathematicians consider Euler's Formula, ##e^{i\theta} = \cos{\theta} + i\sin{\theta}##, to be the most important equation in mathematics. You should get very comfortable with using it.
 
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  • #8
You can also just to algebra on the conjugate expressed in terms of ##\theta##, and then apply double angle trig formulas, if you want to see it verified brute force. It works out very straightforward this way, though, of course, using exponentials is far more elegant.
 
  • #9
FactChecker said:
If z=reiθ, why does z¯=re−iθ?
I guess it's because $$\text{cis}(-\theta)=e^{-i\theta}$$. :wink:
 
  • #10
Leo Liu said:
I guess it's because $$\text{cis}(-\theta)=e^{-i\theta}$$. :wink:
Yes. Negating the imaginary part of ##z=x+iy = r(\cos\theta+ i\sin\theta)## to get ##\bar{z}= x-iy = r(\cos\theta- i\sin\theta)## is the same as negating the argument, ##\theta##, to get ##\bar{z}=r(\cos{(-\theta)} + i\sin{(-\theta)})= r(\cos\theta- i\sin\theta)##.

It looks like every line in your original post is correct (I didn't look hard at it.), but it is not clear how you got some lines and if you understood it. At least in the beginning, it is good practice to really spell everything out. You can skip some details after you are well beyond that level, but not before.
 
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FAQ: Why Does the Complex Conjugate Involve Negating the Argument Theta?

What is the power of a complex conjugate?

The power of a complex conjugate is a mathematical operation that involves raising a complex number to a certain exponent. It is calculated by multiplying the complex number by itself a certain number of times, as determined by the exponent.

How is the power of a complex conjugate different from the power of a real number?

The power of a complex conjugate is different from the power of a real number because complex numbers have both a real and imaginary component, while real numbers only have a real component. This means that the power of a complex conjugate involves both multiplication and division, while the power of a real number only involves multiplication.

What is the significance of the complex conjugate in the power operation?

The complex conjugate is significant in the power operation because it allows us to simplify complex numbers and make calculations easier. By multiplying a complex number by its conjugate, we can eliminate the imaginary component and end up with a real number.

How does the power of a complex conjugate relate to the modulus of a complex number?

The power of a complex conjugate is related to the modulus of a complex number through the formula |z|^2 = z * z*, where z* is the complex conjugate of z. This means that the modulus of a complex number is equal to the complex number multiplied by its conjugate.

Can the power of a complex conjugate be negative or fractional?

Yes, the power of a complex conjugate can be negative or fractional. This means that we can raise a complex number to a negative or fractional exponent, just like we can with real numbers. The resulting answer will still be a complex number.

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