CECS 229 Programming Assignment #1

1/12/24, 3:01 PM CECS 229 Programming Assignment #1 localhost:8888/nbconvert/html/OneDrive/Documents/Teaching/1. Materials/2024/CECS 229/2. Programming Assignments/PA %231/0. Jupyter Files/C… 1/5 CECS 229: Programming Assignment #1 Due Date: Sunday, 2/11 @ 11:59 PM Submission Instructions: Complete the programming problems in the file named pa1.py . You may test your implementation on your Repl.it workspace by running main.py . When you are satisfied with your implementation, download pa1.py and submit it to the appropriate CodePost auto- grader folder. Objectives: 1. Find all integers in a given range that are congruent to an integer under some modulo . 2. Find the -representation of a given integer. 3. Apply numerical algorithms for computing the sum of two numbers in binary representation. 4. Apply numerical algorithms for computing the product of two numbers in binary representation. Problem 1: Complete the function equiv_to(a, m, low, high) that returns a list of all the integers in the range such that . EXAMPLES: Finding all integers such that : IN: equiv_to(3, 5, -10, 15) OUT: [-7, -2, 3, 8, 13] Finding all integers such that : IN: equiv_to(12, 15, -29, -11) OUT: [-18] Finding all integers such that : IN: equiv_to(-20, 4, 3, 21) OUT: [4, 8, 12, 16, 20]

1/12/24, 3:01 PM CECS 229 Programming Assignment #1 localhost:8888/nbconvert/html/OneDrive/Documents/Teaching/1. Materials/2024/CECS 229/2. Programming Assignments/PA %231/0. Jupyter Files/C… 2/5 HINT: By definition, all integers that are equivalent to under modulo must satisfy that Hence, Notice that if all the values must to be in the range , then What lower- and upper-bound does this place on ? How do these -values allow us to find the goal -values? Problem 2: Complete the function b_rep(n, b) that computes the base -representation of an integer n given in decimal representation (i.e. typical base 10 representation). Your implementation must use ALGORITHM 1.2.1 of the "Integer Representations & Algorithms" lecture notes. No credit will be given to functions that employ any other implementation. The function can not use built-in functions that already perform some kind of base -representation. For example, the function implementation can not use the functions bin() or int(a, base=2) . The function should satisfy the following: 1. INPUT: n - a positive integer representing a number in decimal representation b - an integer representing the desired base 1. OUTPUT: a string containing the -expansion of integer a . EXAMPLES: IN: b_rep(10, 2) OUT: 1010 IN: b_rep(10, 8) In [ ]: def equiv_to ( a , m , low , high ): k_low = # FIXME: update k_low k_high = # FIXME: update k_high k_vals = list ( range ( k_low , k_high + 1 )) x_vals = # FIXME: update x_vals return x_vals

1/12/24, 3:01 PM CECS 229 Programming Assignment #1 localhost:8888/nbconvert/html/OneDrive/Documents/Teaching/1. Materials/2024/CECS 229/2. Programming Assignments/PA %231/0. Jupyter Files/C… 4/5 OUT: '1000110' Problem 4: Complete function binary_mul(a, b) that computes the product of the binary numbers and using ALGORITHM 1.2.4 of the "Integer Representations & Algorithms" lecture notes. No credit will be given to functions that employ any other implementation. The function can not use built- in functions that already perform some kind of binary representation or addition of binary numbers. For example, the function implementation can not use the functions bin() or int(a, base=2) . The function should satisfy the following: 1. INPUT: a - a string of the 0's and 1's that make up the first binary number. Assume the string contains no spaces. b - a string of the 0's and 1's that make up the second binary number. Assume the string contains no spaces. In [42]: def binary_add ( a , b ): # removing all whitespace from the strings a = a . replace ( ' ' , '' ) b = b . replace ( ' ' , '' ) # padding the strings with 0's so they are the same length if len ( a ) < len ( b ): diff = len ( b ) - len ( a ) a = "0" * diff + a elif len ( a ) > len ( b ): diff = len ( a ) - len ( b ) b = "0" * diff + b # addition algorithm result = "" carry = 0 for i in reversed ( range ( len ( a ))): a_i = int ( a [ i ]) b_i = int ( b [ i ]) result += # FIXME: Update result carry = # FIXME: Update carry if carry == 1 : result += # FIXME: Update result return # FIXME return the appropriate string

1/12/24, 3:01 PM CECS 229 Programming Assignment #1 localhost:8888/nbconvert/html/OneDrive/Documents/Teaching/1. Materials/2024/CECS 229/2. Programming Assignments/PA %231/0. Jupyter Files/C… 5/5 1. OUTPUT: the string of 0's and 1's that is the result of computing . EXAMPLE: IN: binary_mul( '101011' , '11011') OUT: '10010001001' In [75]: def binary_mul ( a , b ): # removing all whitespace from the strings a = a . replace ( ' ' , '' ) b = b . replace ( ' ' , '' ) # multiplication algorithm partial_products = [] i = 0 # tracks the index of the current binary bit of string 'a' beginning at 0, r for bit in reversed ( a ): if bit == '1' : partial_products . append ( """FIXME: Append the correct object to partial pro i += 1 result = '0' while len ( partial_products ) > 0 : result = binary_add ( "FIXME: Input the correct arguments" ) del partial_products [ 0 ] return # FIXME: Return the appropriate result