HOMEWORK # 2 SOLUTIO

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1 HOMEWORK # 2 SOLUTIO Problem 1 (2 points) a. There are 313 characters in the Tamil language. If every character is to be encoded into a unique bit pattern, what is the minimum number of bits required to do this? 8 bits can used to encode 2 8 = 256 characters and 9 bits can be used to encode 2 9 = 512 characters. So, we would need 9 bits. b. How many more characters can be accommodated in the language without requiring additional bits for each character? = 199 Problem 2 (4 points) Convert the following 2's complement binary numbers to decimal numbers. a First bit is 1. So it is a ve number. 2 s complement of 1010 = = So the answer is -6. b This is a +ve number since it starts with 0 Answer is 2. c This is a ve number since it starts with 1. Its 2 s complement is = So the answer is -1 d This is a +ve number since it starts with 0. The answer is 31. Problem 3 (4 points) a. What is the largest positive number one can represent in a 16-bit 2's complement code? Write your result in binary and decimal binary and = decimal

2 b. What is the greatest magnitude negative number one can represent in a 16-bit 2's complement code? Write your result in binary and decimal binary and = decimal c. What is the largest positive number one can represent in a 16-bit signed magnitude code? Write your result in binary and decimal binary and = decimal d. What is the greatest magnitude negative number one can represent in a 16-bit signed magnitude code? Write your result in binary and decimal binary and (2 15 1) = decimal Problem 4 (2 points) What are the 8-bit patterns used to represent each of the characters in the string "This Is Easy!"? (Only represent the characters between the quotation marks.) Character Hex (from ASCII table) Binary equivalent T h i s Space I s Space E a s y ! Problem 5 (4 points) Convert the following decimal numbers to 8-bit 2's complement binary numbers. If there is problem while doing this, describe it. a

3 b. 64 c Does not fit in an 8-bit signed number d Problem 6 (4 points) The following binary numbers are 4-bit 2's complement binary numbers. Which of the following operations generate overflow? Justify your answers by translating the operands and results into decimal. a No overflow Answer is 0100 binary = 4 decimal [7 + (-3)] b Overflow Answer is 0111 binary = 7 decimal. But actual answer is -9 [(-7) + (-2)]

4 c No overflow Answer is 1000 binary = -8 decimal [(-1) + (-7)] d Overflow Answer is 1000 binary = -8 decimal. But actual answer is 8 [3 + 5] Problem 7 (2 points) A computer programmer wrote a program that adds two numbers. The programmer ran the program and observed that when 5 is added to 8, the result is the character m. Explain why this program is behaving erroneously. The error that is occurring here is that 5 and 8 are being interpreted as characters 5 and 8 respectively. As a result, the addition that is taking place is not 5 + 8; rather, it is If we look up values in the ASCII table, 5 is 0x35 and 8 is 0x38. 0x35 + 0x38 = 0x6d, which is the ASCII value for m.

5 Problem 8 (2 points) Compute the following: a. OT(1011) OR (1011) NOT(1011) = 0100 Answer = (0100) OR (1011) = 1111 b. OT(1001 A D (0100 OR 0110)) 0100 OR 0110 = AND 0110 = 0000 Answer = NOT(0000) = 1111 Problem 9 (4 points) Write the decimal equivalents for these IEEE floating point numbers. a Sign bit is 0 (+ve). Exponent = 127. Fraction = 1* *2-2 = 0.75 Answer = (+) 1.fraction * 2exponent 127 = 1.75 * 2 0 = 1.75 b Sign bit is 1 (-ve). Exponent = 125. Fraction = 1*2-1 = 0.5 Answer = (-) 1.fraction * 2exponent 127 = * 2-2 =

6 Problem 10 (2 points) Given a black box which takes n bits as input and produces one bit for output, what is the maximum number of unique functions that the black box can implement? (Hint: Try to visualize a truth table for a single function of n bits. Determine how many rows such a truth table has. Then determine how many combinations are possible with the number of rows that you just found) Consider a single function that this black box implements. If there are n binary inputs, the truth table contains 2 n rows. Now, each of these rows in the truth table can be filled with 0 or 1. The number of ways in which we can fill in these rows (using 0 and 1) gives us the number of unique functions. Since each of the rows can be filled in using 2 possible values and since the number of rows is 2 n, the number of ways = 2 power (2 n ).

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