Introduction to Computer Systems

Tutorial 01 - Data Representation

Week 2

Preparation Exercises [1 mark]

Complete the following questions on a separate sheet of paper, with your name and student number clearly written. Please ensure your writing is legible. Hand in to your tutor at the beginning of your tutorial / laboratory session.

1. How many different values can be represented in a 4-bit word? An 8-bit word? A 16-bit word? In a 32-bit word?

   What range of unsigned integers can be stored in each of these words?

   What range of two's complement numbers can be stored in each of these words?

   Please give all your answers (except for the 32 bit words) as decimal numbers.

2. Convert the following numbers to decimal:
   • 76₈
   • B11₁₆
   • 101010₂ as an UNSIGNED integer.

     This means we are only considering positive binary integers from 000001 to 111111₂.

   • 101110₂ as a SIGNED integer.

     This means we are using two's complement numbers with 000001 to 011111 representing positive numbers and 111111 to 100000 representing negative numbers.

   (as will be discussed, the C language supports both signed and unsigned integers)

3. What is the unsigned binary representation of 231 ₁₀?

   Convert the result from binary to octal. How many binary digits make up each octal digit?

   Convert the binary value to hexadecimal. How many binary digits make up each hexadecimal digit?

Tutorial Exercises [4 marks]

After an introduction of the class and going through solutions to the preparation exercises, work through the following questions in your session. Finish any uncompleted exercises for homework.

1. Assuming unsigned numbers, what are the results of the following binary additions / subtractions? (give the result in binary)

   0101 00112 +	0101 00112 -
   0110 11102	0001 01012

2. Now assume that ALL numbers in Question 1 are in two's complement form. Try to work out all results from Question 1 using binary arithmetic. Then convert the numbers and results to decimal. Are the results correct? If not, explain why not.

3. What are the octal representations of the following numbers? Hint: you might find it easier to first translate to binary.

   011 110 101 010 011₂     1001 0111 1111 0001₂     AE3C9F₁₆     30123221₄

4. One byte is defined as 8 bits. Can the value from Preparation Exercise 3, (i.e. 231₁₀) be represented as an unsigned integer in a single byte?

   Can the value be represented as a two's complement number in a single byte? Explain your answer.

5. Represent the number -1.125 as an IEEE single precision floating-point number.

6. What are the decimal values of the following IEEE single precision floating point numbers whose bit patterns are represented by the following hexadecimal numbers.

   42E48000₁₆   3F880000₁₆

7. Can all numbers that can be represented by a finite fractional decimal value be exactly represented using a finite number of binary digits? Concisely substantiate your claim.

8. Suppose we had an integer array of length 256, and the size of an integer is 4 bytes. How many such arrays could fit into an 8 KB area of memory? Hint: express all quantities as powers of two.

Homework!

1. Repeat Questions 1 and 2 above for the following:

   0001 00012 +	1010 10112 -
   1101 00112	1001 01102

2. Represent pi (=3.142) as an IEEE single precision floating-point number.