Appendix D — Problem Set 2 | What Kinds of Numbers?

Companion to §1.1 — kinds of numbers and closure

A. Recognition and closure

1. Smallest sufficient number system

For each quantity, give the smallest system among \mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C that contains it.

  1. 17 molecules
  2. a charge imbalance of -3e
  3. an exact mole fraction 2/7
  4. the diagonal length of a unit square
  5. the root of z^2+4=0

2. Which operation forces extension?

State the operation and the required extension.

  1. 3-8 starting in \mathbb N
  2. 5/12 starting in \mathbb Z
  3. \sqrt 2 starting in \mathbb Q
  4. \sqrt{-9} starting in \mathbb R

3. Counterexamples to closure

For each false claim, give one counterexample and then repair the claim.

  1. \mathbb N is closed under subtraction.
  2. \mathbb Z is closed under division.
  3. \mathbb R is closed under square roots.
  4. \mathbb C is closed under division.

B. Complex arithmetic as geometry

4. Cartesian operations

Let z_1=3-4i and z_2=-2+i. Calculate:

  1. z_1+z_2
  2. z_1z_2
  3. z_1/z_2
  4. |z_1| and z_1^*
  5. verify z_1z_1^*=|z_1|^2.

5. Polar form

Convert each number to re^{i\theta} using the principal argument -\pi<\theta\le\pi.

  1. 1+i
  2. -\sqrt3+i
  3. -2i
  4. -3-3i

6. Multiplication rotates and scales

Let z=2e^{i\pi/6} and w=3e^{-i\pi/4}.

  1. Compute zw in polar form.
  2. Compute z/w in polar form.
  3. Convert zw to Cartesian form.
  4. State separately what happened to magnitude and angle during multiplication.

C. Chemistry and measurement

7. Spectroscopic phase

An ideal signal is represented by Z(t)=4.00e^{i(20.0t+\pi/3)}.

  1. Write the observed real channel x(t)=\Re Z(t).
  2. Write the quadrature channel y(t)=\Im Z(t).
  3. Evaluate both channels at t=0.
  4. Prove that x(t)^2+y(t)^2 is time-independent and interpret the result.

8. Interference requires relative phase

Two equal complex amplitudes are \psi_1=A and \psi_2=Ae^{i\phi}. Observable intensity is proportional to |\psi_1+\psi_2|^2.

  1. Derive |\psi_1+\psi_2|^2=2A^2(1+\cos\phi).
  2. Evaluate it for \phi=0,\pi/2,\pi.
  3. Explain why replacing each amplitude by its magnitude before addition destroys physical information.

9. Roots and symmetry

Solve z^3=8 completely.

  1. Write 8 in polar form including the multivalued phase.
  2. Find all three roots.
  3. Plot their locations conceptually on the complex plane.
  4. Show that their vector sum is zero.

10. Error diagnosis

A student calculates \frac{1}{2-i}=\frac{1}{2}-\frac{1}{i}.

  1. Identify the invalid algebraic step.
  2. Calculate the correct value.
  3. Verify your result by multiplication.