🪴 TJ's Notes 1.0

Digital Logic Design Lecture 3

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Digital Logic Design Lecture 3

Index

  1. Looking back to conversion methods
  2. 8 Bit Signed Number & Signed Magnitude Representation
  3. Ones Complement Representation
  4. Two’s Complement Basics
  5. Summary of Two’s Bit Complement

Content

Looking back to conversion methods

Refer DLD Lecture 2

8 Bit signed Number & Signed Magnitude Representation

To represent a positive number or a negative number we use a bit.

Positive

For Positive we use bit zero ( 0 ) in the MSB Position to represent a positive binary number.

Negative

For Negative we use a bit ( 1 ) in the MSB Position to represent a negative binary number.

Ones Complement Representation

We can have this only for binary numbers only. In this complement representation the bits are flipped for lets say a not gate. for example 1011000110 would be 0100111001

In a two’s complement system, the binary number 1000 does not represent -7. It represents -8. Let’s clarify this with more detail.

Two’s Complement Basics

  1. Positive Numbers:

    • Positive numbers are represented in their straightforward binary form.

  2. Negative Numbers:

    • Negative numbers are represented by taking the two’s complement of their absolute value.

Example: Representing +7 and -7

  1. +7 in 4-bit Binary:

    • 0111

  2. -7 in 4-bit Two’s Complement:

    • Start with the binary of +7: 0111
    • Invert all bits: 1000
    • Add 1: 1000 + 1 = 1001
    • So, -7 is represented as 1001.

Correct Representation of 1000

  • 1000 in a 4-bit two’s complement system represents -8, not -7.

Range of 4-bit Two’s Complement

In a 4-bit two’s complement system, the values range from -8 to +7:

  • 0000 = 0
  • 0001 = 1
  • 0010 = 2
  • 0011 = 3
  • 0100 = 4
  • 0101 = 5
  • 0110 = 6
  • 0111 = 7
  • 1000 = -8
  • 1001 = -7
  • 1010 = -6
  • 1011 = -5
  • 1100 = -4
  • 1101 = -3
  • 1110 = -2
  • 1111 = -1

Summary of Two’s Bit Complement

  • +7: 0111
  • -7: 1001
  • +8 is not representable in a 4-bit two’s complement system, as the range is from -8 to +7.

For an 8-bit two’s complement system, the range is from -128 to +127:

  • +127: 0111 1111
  • -128: 1000 0000

This system ensures that each binary number has a unique interpretation, avoiding any confusion.

References

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