data representation in computer systems, Study notes of Computer Science

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Chapter 5

Data representation

Learning outcomes

 By the end of this Chapter you will be able to:

• Explain how integers are represented in computers using:
  • Unsigned, signed magnitude, excess, and two’s complement notations
• Explain how fractional numbers are represented in computers
  • Floating point notation (IEEE 754 single format)
• Calculate the decimal value represented by a binary sequence in:
  • Unsigned, signed notation, excess, two’s complement, and the IEEE 754 notations.
• Explain how characters are represented in computers
  • E.g. using ASCII and Unicode
• Explain how colours, images, sound and movies are represented

Introduction

 In Chapter 3

• How information is stored
  • in the main memory,
  • magnetic memory,
  • and optical memory

 In Chapter 4:

• We studied how information is processed in computers
• How the CPU executes instructions

 In Chapter 5

• We will be looking at how data is represented in computers
• Integer and fractional number representation
• Characters, colours and sounds representation

Binary numbers

 Binary number is simply a number comprised of only 0's and 1's.

 Computers use binary numbers because it's easy for them to communicate using electrical current -- 0 is off, 1 is on.

 You can express any base 10 (our number system -- where each digit is between 0 and 9) with binary numbers.

 The need to translate decimal number to binary and binary to decimal.

 There are many ways in representing decimal number in binary numbers. (later)

Example

•  3040510 = 30000 + 400 +
  • = 310^4 +410^2 +5*10
•  101012 =10000+100+
  • =12^4 +12^2 +1*2

Examples: decimal -- binary

 Find the binary representation of 12910.

 Find the decimal value represented by

the following binary representations:

• 10000011
• 10101010

Number representation

 Representing whole numbers

 Representing fractional numbers

Integer Representations

• Unsigned notation
• Signed magnitude notion
• Excess notation
• Two’s complement notation.

Advantages and disadvantages of

unsigned notation

 Advantages:

• One representation of zero
• Simple addition

 Disadvantages

• Negative numbers can not be represented.
• The need of different notation to represent negative numbers.

Representation of negative

numbers

 Is a representation of negative numbers

possible?

 Unfortunately:

• you can not just stick a negative sign in front of a binary

number. (it does not work like that)

 There are three methods used to represent

negative numbers.

• Signed magnitude notation
• Excess notation notation
• Two’s complement notation

Example

 Suppose 10011101 is a signed magnitude representation.

 The sign bit is 1, then the number represented is negative

 The magnitude is 0011101 with a value 2^4 +2^3 +2^2 +2^0 = 29

 Then the number represented by 10011101 is – 29.

position 7 6 5 4 3 2 1 0 Bit pattern 1 0 0 1 1 1 0 1

contribution - 24 23 22 20

Exercise 1

1. 3710 has 0010 0101 in signed magnitude notation. Find the signed magnitude of – 3710?
2. Using the signed magnitude notation find the 8-bit binary representation of the decimal value 24 10 and -
3. Find the signed magnitude of – 63 using 8-bit binary sequence?

Signed-Summary

 In signed magnitude notation,

• The most significant bit is used to represent the sign.
• 1 represents negative numbers
• 0 represents positive numbers.
• The unsigned value of the remaining bits represent The magnitude.

 Advantages:

• Represents positive and negative numbers

 Disadvantages:

• two representations of zero,
• Arithmetic operations are difficult.

Excess Notation

 In excess notation:

• The value represented is the unsigned value with a fixed value subtracted from it. - For n-bit binary sequences the value subtracted fixed value is 2 (n-1).
• Most significant bit:
  • 0 for negative numbers
  • 1 for positive numbers