Warm up problems for Complex Variables

Here are a few warm up problems for Complex Variables. I will show you on Wednesday how you will turn HW assignments like this in so be sure to bring a laptop to class this Wednesday!

The problems

  1. Express the following numbers in the form \(a+bi\).

    1. \((7-2i)(3i+5)\)
    2. \(i^{275}\)
    3. \((1+i)^4\)

  2. Let \(z_1=(x_1, y_1)\) and \(z_2=(x_2, y_2)\) be complex numbers.

    1. Show that \[\mathrm{Re}(z_1+z_2)=\mathrm{Re}(z_1)+\mathrm{Re}(z_2).\]
    2. Find a counter example to the claim that \[\mathrm{Re}(z_1z_2)=\mathrm{Re}(z_1)\mathrm{Re}(z_2).\]

  3. Let \(P(z)=a_nz^n + a_{n-1}z^{n-1} \cdots +a_1z + a_0\) be a polynomial and suppose that \(z_0\) is a root of \(P\). Show that \(\overline{z_0}\) is also a root of \(P\).
    Hint: The algebraic properties of the conjugate imply that \[\overline{P(\overline{z})} = P(z).\]
    You may use this fact.