Ex. 1 → Write a method simplify to the class RationalNumber. This method should return the simplified version of the fraction as a tuple.

Ex. 2 → To provide results with confidence intervals a special calculus, so-called interval arithmetic is introduced in numerical mathematics; (refer to [3, 14]). Define a class called Interval and provide it with methods for addition, subtraction, division, multiplication, and power (with positive integers only). These operations obey the following rules:

.

Provide this class with methods that allow operations of the type a + I, a I, I + a, I a, where I is an interval and a an integer or float. Convert an integer or float to an interval [a,a] first. (Hint: you may want to use function decorators for this; (refer to section Function as decorators in Chapter 7, Functions). Furthermore, implement the __contains__ method, which enables you to check if a given number belongs to the interval using the syntax x in I for an object I of type Interval. Test your class by applying a polynomial f=lambda x: 25*x**2-4*x+1 to an interval.

Ex. 3 → Consider the example under section Classes as decorators. Extend this example to obtain a function decorator that counts how often a certain function is called.

Ex. 4 → Compare the two ways to implement a method for reverse addition __radd__ in the class RationalNumber: the one given in the example in section Special methods and the one given here:

class RationalNumber:
    ....
    def __radd__(self, other):
        return other + self

Do you expect an error in this version? What is the error and how do you explain it? Test your answer by executing:

q = RationalNumber(10, 15)
5 + q

Ex. 4 → Consider the decorator class CountCalls as in example in section Classes as decorators. Provide this class with a method, reset, which sets the counters of all functions in the dictionary, CountCalls.instances, to zero. What would happen if the dictionary were replaced by an empty dictionary instead?