Complex Numbers

Reading

Section 1.1, 1.2

Problems

• Practice Problems (page 18): 1, 2, 3, 4, 7, 9, 12
• Problems to be ready to present: 8
• Challenge: 13, 14
• More practice problems:
  1. Recall the definitions of \(\bar z\) and \(z^{-1}\). Show that \(\bar z = z^{-1}\) if and only if \(z\) lies on the unit circle (all complex numbers with length \(1\)).
  2. Show that if \(z\) is an \(n\)-th root of \(1\), then \(z\) must lie on the unit circle.
  3. Suppose \(z\) is an \(n\)-th root of \(1\). Show that \(z^{-1} = z^{n-1}\).
  4. Suppose \(z\neq 1\) is a third root of \(1\).
     • Use the result of problem 13 to show that \(1 + z + \bar z = 0\).
     • Use this equation, and the fact that \(z\) has length \(1\), to show that \(z = \frac{-1\pm i\sqrt{3}}{2}\).
  5. Suppose \(z\neq 1\) is a fifth root of \(1\).
     • Show that \(z = 1 + z + z^2 + \frac{1}{z^2} + \frac{1}{z} = 0\).
     • Use this to show that if we set \(w = z + \frac{1}{z}\) then we have \(w + w^2 = 1\).
     • From that last equation find \(w\), and then from the previous equation find \(z\). You should be finding 4 different solutions this way.
     • Examine the four numbers on the plane, and with that information in hand find \(\cos\frac{2\pi}{5}\) and \(\sin\frac{2\pi}{5}\). Use Wolfram Alpha http://www.wolframalpha.com/ to verify the formula you found.
  6. Consider the vectors based at the origin and ending at the complex numbers \(z\) and \(w\) respectively. Show that the dot product between the two vectors equals the real part of \(z\bar w\).
  7. True or false: The vectors based at the origin and ending at the complex numbers \(z\) and \(w\) respectively are perpendicular if and only if \(z/w\) is a purely imaginary number.

Topics to know

1. Definition of Complex Numbers as pairs of real numbers
2. Properties of \(i\)
3. Real numbers are embedded into the Complex Numbers
4. Finding the square root of a number (Find roots of \(\pm i\), then find their roots)
5. Complex Numbers as points on a plane. Addition as vector addition
6. Multiplication by \(i\) amounts to rotation by 90 degrees
7. Conjugate of a number, \(\bar z\)
8. Modulus/Absolute value \(|z|\)
9. Polar coordinates representation of a complex number
10. Multiplication and division via polar representation
11. Use of polar representation for roots