Quantum Fundamentals: Winter-2024
Complex Number Practice : Due Day 4 Th 2/15 Math Bits

  1. Complex Arithmetic: Rectangular Form S0 4959S For the complex numbers \(z_1=3-4i\) and \(z_2=7+2i\), compute:
    1. \(z_1-z_2\)
    2. \(z_1 \, z_2\)
    3. \(\frac{z_1}{z_2}\)
  2. Circle Trigonometry and Complex Numbers S0 4959S

    Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

  3. Complex Number Algebra, Exponential to Rectangular--Practice S0 4959S

    If \(z_1=5e^{7i\pi/4}\), \(z_2=3e^{-i\pi/2}\), and \(z_3=9e^{(1+i\pi)/3}\), express each of the following complex numbers in rectangular form, i.e. in the form \(x+iy\) where \(x\) and \(y\) are real.

    1. \( z_1 +z_2 \)
    2. \( z_1 z_2 \)
    3. \( \frac{z_2}{z_3} \)

  4. Complex Numbers, All Forms--Practice S0 4959S Represent the following four complex numbers in rectangular form \(a + ib\), exponential form \(|z|e^{i\phi}\) , and on an Argand diagram:

    1. \(e^{i\pi}\)

    2. \(i\)

    3. \(\sin\frac{\pi}{2}\)

    4. \(\cos\frac{\pi}{4}-i\sin\frac{\pi}{4}\)
  5. Exponential Form of Complex Numbers--Practice S0 4959S For each of the following complex numbers \(z\), find \(z^2\), \(\vert z\vert^2\), and rewrite \(z\) in exponential form, i.e. as a magnitude times a complex exponential phase:

    • \(z_1=i\),

    • \(z_2=2+2i\),

    • \(z_3=3-4i\).
  6. Complex Number Algebra--Practice S0 4959S Express each of the following complex numbers in rectangular form, i.e. in the form \(x+iy\) where \(x\) and \(y\) are real.
    1. \(3e^{2(1+i\pi)}\)
    2. \(3e^{i\pi}+3e^{-i\pi}\)
    3. \((1-i)^8\)
    4. \(\left(1+i\sqrt{3}\right)^{6}\)
    5. \(\frac{2+3i}{1-i}\)
  7. Graphs of the Complex Conjugate S0 4959S

    For each of the following complex numbers, determine the complex conjugate, square, and norm. Then, plot and clearly label each \(z\), \(z^*\), and \(|z|\) on an Argand diagram.

    1. \(z_1=4i-3\)
    2. \(z_2=5e^{-i\pi/3}\)
    3. \(z_3=-8\)
    4. In a few full sentences, explain the geometric meaning of the complex conjugate and norm.