1. ## couple of proofs about complex numbers

okay so my teacher decides to give homework on things unrelated to class lecture, and the book really doesn't help at all. wish the dude could speak english.
anyhoo, i need help with a couple of proofs...
prove the following for complex numbers:
uno: u + v = v + u, and uv = vu for all u, v
dos: u+v = u + v and uv = u x v for all u, v (the underlines are supposed to be above the numbers...)
tres: u(vw) = (uv)w for all u,v,w

2. Originally Posted by mistykz
okay so my teacher decides to give homework on things unrelated to class lecture, and the book really doesn't help at all. wish the dude could speak english.
anyhoo, i need help with a couple of proofs...
prove the following for complex numbers:
uno: u + v = v + u, and uv = vu for all u, v
dos: u+v = u + v and uv = u x v for all u, v (the underlines are supposed to be above the numbers...)
tres: u(vw) = (uv)w for all u,v,w
Okay, I get what you need to do.

The first is proving commutivity of addition and multiplication of complex numbers, the second does the same for the complex conjugates, and the third is proving the associative law of multiplication.

In all of these cases let
$\displaystyle u = a + ib$
$\displaystyle v = c + id$
and work out each side and compare them.

For example, let's prove $\displaystyle uv = vu$:
$\displaystyle uv = (a + ib)(c + id) = ac + iad + ibc + i^2bd = (ac - bd) + i(ad + bc)$
and
$\displaystyle vu = (c + id)(a + ib) = ca + icb + ida + i^2db = (ac - bd) + i(ad + bc)$
(since da = ad, etc. We may assume that the addition and multiplication of real numbers is commutative.)

Thus $\displaystyle uv = vu$ where u and v are complex.

The other problems work the same way. I'd write out "tres" step by step since it's going to be a long one to work out.

-Dan

3. $\displaystyle \overline {a + bi} = a - bi$, $\displaystyle \overline {c + di} = c - di$.

$\displaystyle \left( {a + bi} \right) + \left( {c + di} \right) = \left( {a + c} \right) + \left( {b + d} \right)i$

$\displaystyle \overline {\left( {a + c} \right) + \left( {b + d} \right)i} = \left( {a + c} \right) - \left( {b + d} \right)i$.

Can you use these to complete the work?