Problem 1 (25 points) In this problem, you will prove that the absolute value of the true relative round-off error lEt/xl associated with the approximate representation of any number x in any normalized floating-point system binary, decimal,...) that relies on chopping is always less than the machine epsilon e of that system. Consider a normalized (regular normalization as described in the textbook, not IEEE normalization) floating-point system with t significant figures: in that svstem, a numberxrp can be represented exactly as where the nj are the t integers of the normalized mantissa, b is the base (which can be anything: 2,8, 10,..) of the system and e is the exponent of the number. The number x that you are trying to represent can either be among the finite number of xFp values (in which lucky case Et0), or be located in between 2 consecutive xfp values with the same exponent e, xm and xm-1, and it will be chopped to xm as illustrated below: 4x For all questions, clearly show your work and reasoning to get full credit. a. (10 points) For a given exponent e, express the absolute value of the spacing |Ar between 2 consecutive numbers in terms of t, b and e. Hint: write x in the general positive number format b. (10 points) For the same exponent e as in part a), calculate the largest possible value of |Ax/x| in this normalized system, and express your result in terms of t and b (5 points) Use your class notes and part b) to prove that, for any value of X, Et/|