1. Convert the following binary exponential expressions to their 'English'

Transcription

1 Answers to Practice Problems Practice Problems - Integer Number System Conversions 1. Convert the decimal integer into the following number systems: a c b d. 1AB Convert the following numbers to decimal integers. a = c = b = d. A4E 16 = Convert the following decimal integers to binary: a. 2 = 10 2 c. 27 = e. 128 = g. 31 = b. 245 = d. 127 = f. 17 = h. 257 = Which of the following are not valid numbers in the radix indicated? a d g (N/G) b. DEF 16 e (N/G) h c f i (N/G) Practice Problems - Fraction Number System Conversion 1. Convert the following fractions to binary: a..875 = b..2 = c..71 = Convert the following fractions from the base indicated to decimal: a = 13/32 c = 1/32 e = 94/343 b..a26 16 = 2598/4096 d = 16/27 f = 124/125 = Practice Problems - Binary-to-English Conversions 1. Convert the following binary exponential expressions to their 'English' NTC 1/23/05 54

2 counterparts: Answers: 8 8K 8M 8G 128M 32K 2G 1M 256G 2. Convert the following 'English' expressions to their binary exponential equivalents: 1K M 32K 128G 2M 4G 512K 8M Answers: Word problems a. How much memory is supported by a system with 24-bit memory addresses? Ans.: 2 24 = 16MB b. How large an address is needed to address a memory of 4GB? Ans.: 32 bits c. How many op code bits are need in an instruction which supports 128 different operations? Ans.: 7 d. How many registers are supported by an instruction with a register ID field of 6 bits? Ans.: 64 Practice problems - 2's Complement Numbers: 1. Given the binary number , what is its decimal value if it is a a. unsigned binary number Ans: 219 b. sign and complement number Ans: -91 c. 2's complement number Ans: Convert +24 into a 2's complement number NTC 1/23/05 55

3 Ans: Note the leading zero! 3. Convert -24 into a 2's complement number Ans: Convert -1 into a 2's complement number (assume a 4-bit result) Ans: 1111 Note that -1 is regardless of the size of the register. Note also that any positive or negative 2's complement number can have its leading digit propagated to the left without changing the number. 5. What is the minimum (maximum negative) number which can be represented using a 6-bit 2's Complement representation? What is the maximum (positive) number? Ans: +31, What is the minimum (maximum negative) number which can be represented using a n-bit 2's Complement representation? Ans.: -2 n-1 What is the maximum (positive) number? Ans..: 2 n-1-1 Convert, if possible, the following decimal numbers to 2's complement assuming an 8-bit binary representation for all = = = = = = = N/G (N/G = Convert the following 2's complement numbers to decimal = = = = = = 1 Practice Problems - Biased Binary Numbers 1. What bias should you choose for biased binary representations with each of the following number of bits? NTC 1/23/05 56

4 a. 8 bits b. 2 bits c. 24 bitsd. 13 bits Bias = M 4K 2. What bias should you choose for biased binary representation of each of the following decimal numbers? a. -10 b c d. +75 Bias = Convert the following decimal numbers to the appropriate biased binary format. a. -34 b. -12 c d Ans.: Convert the following biased binary numbers to decimal a. 101 b c d Ans: Practice Problems - BCD format 1. Express the following Decimal numbers in packed BCD format: a. 0 = b. 392 = c = d. -3 = Express the following Decimal numbers in unpacked BCD format: a. 1 = b. 201 = c. 99 = d = What decimal number, if any, is represented by each of the following packed BCD numbers? a c = 740 = 9 b d = -767 N/G (1100 is not a valid BVD digit 4. What is the maximum unsigned decimal number which can be represented in NTC 1/23/05 57

5 a. 4 bytes unsigned packed BCD notation?ans.: b. 12 byte unsigned unpacked BCD notation? Ans.: Practice Problems - Binary Floating Point Numbers 1. Using the floating point register format given above for the examples, show the register contents (in binary, not hex) for the following binary floating point numbers: a x 2 3 b x 2 8 c x 2-6 Ans.: What is the minimum number of exponent bits required to accommodate the following binary numbers (review the section on biased binary numbers, if necessary.) a. 1.0 x b x Ans.: 10 exponent bits 8 exponent bits 3. What is the binary floating point number (in the form b.bbb... x 2 e ) represented by each of the following FLP register contents a (five bit exponent, bias = 15) x 2-1 b (eight bit exponent, bias = 127) -1.0 x c. 3B1CFF 16 (six bit exponent, bias = 31) x Express the following binary floating point number in IEEE floating point format x The following hex number represents the contents of an IEEE floating point register. Express this in normalized binary floating point form ( b.bbb... x 2 e ) 9F = x 2-65 NTC 1/23/05 58

6 Practice Problems - Unsigned Arithmetic 1. Do the following binary additions a b c Ans Do the following binary subtractions a b c Ans.: Practice Problems - 2's Complement Arithmetic 1. Do the following 2's complement additions. In all cases assume results are placed in an 8-bit register. For each problem, indicate whether the addition resulted in an overflow. a b c d Ans.: No Overflow Overflow No Overflow No Overflow 2. Do the following 2's complement subtractions. In all cases assume results are placed in an 8-bit register. For each problem, indicate whether the subtraction resulted in an overflow. a b c d Ans.: No Overflow Overflow Overflow No Overflow Practice Problems - BCD Arithmetic 1. Do the following BCD additions. For each problem, state which digit(s), if any, NTC 1/23/05 59

7 required adjustment. a b c Ans.: Subtracted: Do the following BCD subtractions. For each problem, state which digit(s), if any, required adjustment. a b d Ans.: Subtracted: Practice problems - Binary Floating Point Arithmetic 1. Add the following Binary floating point numbers. Normalize the result x and x Ans.: x Add the following IEEE floating point numbers. Express the result in IEEE hexadecimal format 659B8000 and E = Show the hexadecimal IEEE format for the values zero and one. Zero = One = 3F NTC 1/23/05 60