Exercises on Basics

Use the following complex numbers in your explorations.

In[11]:=

  z1=2+3 I
  z2=3-5 I
  z3=6 I
  z4=6
  z5=-2 I
  z6=-3+2 I
  z7=-5-2 I

Out[11]=

  2 + 3 I

Out[12]=

  3 - 5 I

Out[13]=

  6 I

Out[14]=

  6

Out[15]=

  -2 I

Out[16]=

  -3 + 2 I

Out[17]=

  -5 - 2 I

1. Compute z*zbar and the modulus of z for the given complex numbers and use them to make a conjecture about the relationship between z*zbar and the modulus of z.

2. Compute the real part of z, the imaginary part of z, z+zbar, and z-zbar for the given complex numbers and use them to make a conjecture about the relationships between these quantities.

3. Compute the modulus of zi*zj ( |zi*zj| ) and the modulus of zi times the modulus of zj ( |zi| |zj| )for i, j = 1 to 7 using the given complex numbers and use them to make a conjecture about the relationships between these quantities.

4. Is there a similar relationship between moduli of quotients and quotients of moduli? Use the above complex numbers to explore this.

5. Without using Mathematica, explain why the modulus of z and the modulus of zbar are equal and also why (zbar)bar is equal to z.

6. Evaluate the moduli of the real and imaginary parts of z for the above complex numbers and conjecture on their relationship to the modulus of z.

7. Evaluate the modulus of the sum of two complex numbers for various combinations of the complex numbers above and conjecture on the relationship between that and the sum of the moduli.

8. Evaluate the modulus of the difference of two complex numbers for various combinations of the complex numbers above and conjecture on the relationship between that and the difference of the moduli.

Up to Complex Numbers