Prove: If Z and W are complex numbers, then the conjugate of (Z+W) is equal to the conjugate of Z plus the conjugate of W.
My thought is that this is kind of like the distributive property, but I'm not sure. It doesn't help that I haven't written a proof in over 10 years. Help? Perhaps point me in the direction of a site that specializes in proofs?
Thanks,
Amy :)
Z = zx + zy i
W = wx + wy i
so
Z+W = (zx+wx) + (zy+wy)i
conjugate of Z+W = (zx+wx) - (zy+wy)i
= zx + wx - zy i - wy i
= (zx - zy i) + (wx - wy i)
which is of course conjugate of Z + conjugate of W
conjugate of
Thanks! One more question:
Prove: Let z be a complex number. Show that z is an element of the real number set if and only if the conjugate of z is equal to z.
My teacher said there would be two things to prove from this since it was an "if and only if" problem. So my question is, what am I proving? I think one proof is going to be that the conjugate of z is equal to z. I don't know what the other part is.
Help?
Thanks again,
Amy :)
idk man
Hi Amy! Proving this statement involves applying the properties of complex conjugates and basic algebra. I can help guide you through the proof step by step.
To start, let's recall the definition of the complex conjugate. If we have a complex number Z = a + bi, where a and b are real numbers, then its complex conjugate is denoted as Z* (read as "Z bar") and is given by Z* = a - bi.
Now, let's consider two complex numbers Z and W, so we can write them as Z = x + yi and W = u + vi, where x, y, u, and v are real numbers.
We want to prove that the conjugate of (Z + W) is equal to the conjugate of Z plus the conjugate of W, i.e., we want to show that (Z + W)* = Z* + W*.
To prove this, we can start by simplifying the left-hand side, (Z + W)*. Since (Z + W) = (x + yi) + (u + vi), we can expand it as follows:
(Z + W)* = (x + yi + u + vi)*
= (x + u) + (y + v)i.
Now, let's simplify the right-hand side, Z* + W*. Using the definition of complex conjugates, we have:
Z* + W* = (x - yi) + (u - vi)
= (x + u) + (-y - v)i.
Notice that both sides have the same real part (x + u), but the imaginary parts differ on the right-hand side. This difference occurs because the complex conjugate of a sum is equal to the sum of the conjugates of the individual terms.
Since the left-hand side and right-hand side are not equal, we conclude that the statement "the conjugate of (Z + W) is equal to the conjugate of Z plus the conjugate of W" is false. Therefore, the initial claim is not proven.
If you'd like to further explore proofs or find resources for proof-writing, I recommend checking out websites such as Khan Academy (www.khanacademy.org) or MathisFun (www.mathisfun.com). These platforms offer a range of lessons, explanations, and examples on various mathematical topics, including proofs.
I hope this helps! Let me know if you have any further questions.