uc3m_cs

Materials used for Computer Structure

• License CC BY-NC 4.0
• Curse 2022-2023

Exercise 1

(statement) The following numbers are represented in two’s complement using 6 bits. Indicate their corresponding decimal value:

• 100000
• 010011

(solution) The corresponding decimal value:

• 100000: - (011111 + 1) = -32
• 010011: +16+2+1 = 19

Exercise 2

(statement)

A computer has a word width that allows you to represent numbers in complement to two in the range [-234, 234-1]. Respond:

1. What is the number of bits that store the registers of this computer?
2. How much KB of memory can address at most this computer?

(solution)

For this computer:

1. [2^n-1...-1, 0... 2^n-1-1] = [-234, 234-1] -> 2^n-1=234 -> 9 bits
2. If this is a 9 bits computer then 2⁹ bytes (memory address by bytes) -> 2⁹/2¹⁰ KiB -> half a kilobyte

Exercise 3

(statement) Indicate the decimal value of the following numbers represented in the single precision IEEE 754 standard: 0xBF400000, 0x7F804000 and 0x00180000.

(solution)

• 0xBF400000 -> 1011 1111 0100 0000 0000… -> 1 01111110 100000000… -> -1.1 * 2^-1 -> 0,110 -> -0,75
• 0x7F804000 -> 0111 1111 1000 0000 0100… -> 0 11111111 000000001… -> NaN
• 0x00180000 -> 0000 0000 0001 0100 0000… -> 0 00000000 001010000… -> no-normalize: +0.001100.. x 2^-126

Exercise 4

(statement) Represent in the single precision IEEE 754 standard the numbers 15 and 3,75.

(solution) The corresponding value:

• 15 -> 8+4+2+1 -> 1111.00 * 2⁰ -> 1.1110 * 2³ -> 0 10000010 111000…
• 3.75 -> 2+1+0.5+0.25 -> 11.11 * 2⁰ -> 1.1110 * 2¹ -> 0 10000000 111000…

Exercise 5

(statement) Add the previous numbers represented in the IEEE 754 standard using the IEEE 754 representation.

(solution) The corresponding value:

                                 1 11
   * 15    <-> 1.1110 * 2^3 <->  1.111000 * 2^3
   *  3.75 <-> 1.1110 * 2^1 <->  0.011110 * 2^3
   * 18.75                      10.010110 * 2^3 -> 10010.11

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