2.6 Exercises

Ex. 1: Check whether  is a zero of the function:

Ex. 2: According to de Moivre's formula, the following holds:

Choose numbers n and x and verify that formula in Python.

Ex. 3: Complex numbers. Verify Euler's formula in the same way:

Ex. 4: Suppose we are trying to check the convergence of a diverging sequence (here, the sequence is defined by the recursive relation  and ):

u = 1.0 # you have to use a float here!
uold = 10. 
for iteration in range(2000):
   if not abs(u-uold) > 1.e-8:
      print('Convergence')
      break # sequence has converged
   uold = u
   u = 2*u
else:
   print('No convergence')

1. Since the sequence does not converge, the code should print the
    No convergence message. Execute it to see what happens.

2. What happens if you replace the line

      if not abs(u-uold) > 1.e-8:

with

      if abs(u-uold) < 1.e-8:

It should give exactly the same result, shouldn't it? Run the code again to see what happens.

3. What happens if you replace u=1.0 with u=1 (without a decimal point)? Run the code to check your predictions.
4. Explain the unexpected behavior of this code.

Ex. 5: An implication C = (A ⇒ B) is a Boolean expression that is defined as

• C is True if A is False, or A and B are both True
• C is False otherwise

Write a Python function implication(A, B).

Ex. 6: This exercise is to train Boolean operations. Two binary digits (bits) are added by using a logical device called a half adder. It produces a carry bit (the digit of the next higher value) and the sum as defined by the following table, and half adder circuit:

Definition of the half adder operation:

A full adder consists of two half adders and sums two bits and an additional carry bit on the input (see also the following figure):

Write a function that implements a half adder and another that implements a full adder. Test these functions.