- #1
Jbreezy
- 582
- 0
Homework Statement
How is arg(z1/z2) = arg(z1z2) ?Where the bold z2 represents the conjugate.
Jbreezy said:Homework Statement
How is arg(z1/z2) = arg(z1z2) ?Where the bold z2 represents the conjugate.
Homework Equations
The Attempt at a Solution
Jbreezy said:Yeah I tried. I just took an example say z1= 2+2i and z2= 3+4i
So arg (z1/z2) = .56-.08i
now arg(z1z2) = 14-2i
I don't get how this person wrote that arg(z1/z2) = arg(z1z2)
oay said:So you are happy that your particular example satisfies the identity [itex]arg(z_1/z_2)=arg(z_1 \bar{z_2})[/itex], but do you understand why it is an identity - ie it is satisfied by any [itex]z_1[/itex] and [itex]z_2[/itex] (with [itex]z_1,z_2 \neq 0[/itex])?
If not, think about how you can represent each of the following:
[tex]arg(z_1/z_2)[/tex][tex]arg(z_1 z_2)[/tex][tex]arg(\bar{z_2})[/tex] in terms of [itex]arg(z_1)[/itex] and [itex]arg(z_2)[/itex].
Of course, that just verifies it for those two particular values in that example. What happens if you write for general z1 and z2 in polar form and try it?
LCKurtz said:Of course, that just verifies it for those two particular values in that example. What happens if you write for general ##z_1## and ##z_2## in polar form and try it?
What do you mean?
That's right, you've answered all three correctly.Jbreezy said:[tex]arg(z_1/z_2)= arg(z1)-arg(z2)[/tex]
[tex]arg(z_1 z_2)= arg(z1) +arg(z2)[/tex]
[tex]arg(\bar{z_2})= -arg(z2)[/tex]but you said in terms of ##arg(z1) ##and ## arg(z2)##
So I don;t know about the last one.
oay said:That's right, you've answered all three correctly.
I took that to mean that he/she was unsure about it simply because I'd said "in terms of [itex]arg(z_1)[/itex] and [itex]arg(z_2)[/itex]" and only one of these terms was necessary. A problem of the wording really, rather than not understanding the answer, IMO.LCKurtz said:But didn't he say he didn't understand the third one?
LCKurtz said:##z = re^{i\theta}## form.
Any complex number can be represented this way.Jbreezy said:I don't know this form. You represent a complex number like a + ib like this? what?
Jbreezy said:I don't know this form. You represent a complex number like a + ib like this? what?
Conjugating z2 does not affect the argument of z1/z2. The argument of a complex number is determined by its angle with the positive real axis, and conjugation only changes the sign of the imaginary part, not the angle.
This is because the argument of a product is the sum of the arguments of the factors. When z2 is conjugated, its argument changes by 180 degrees, but the argument of z1 remains the same. Therefore, the argument of z1z2 will be equal to the argument of z1 plus 180 degrees, which is the same as the argument of z1/z2.
Yes, this can happen if the argument of z2 is already 180 degrees. In this case, conjugating z2 would not change the argument, so the argument of z1/z2 and z1z2 would be equal without conjugation.
This property allows us to easily calculate the argument of a quotient by conjugating the denominator. It is also useful in simplifying complex expressions involving arguments, as we can replace the original expression with one that is easier to work with.
Yes, there are several other properties of complex numbers that involve conjugation, such as the fact that the modulus of a conjugate is equal to the modulus of the original number, and the fact that the product of a complex number and its conjugate is always a real number.