Seiyun University CS Department Information

Introduction to Number Systems

• Digital logic systems can perform numerical computations using a binary number system.
• Humans use the decimal system, but digital machines use binary due to the availability of two-state devices (transistors).
• Other number systems used in conjunction with programmable logic devices include hexadecimal.

Characteristics of Numbering Systems

• Digits are consecutive.
• The number of digits equals the base size.
• Zero is always the first digit.
• The base number is never a digit.
• When 1 is added to the largest digit, a sum of zero and a carry of one results.

Significant Digits

• Most significant digit (MSD) and least significant digit (LSD) are important concepts in number systems.

Binary Number System

• Also known as the Base 2 system.
• Used to model electrical signals in computers.
• 0 represents no voltage or an off state, and 1 represents presence of voltage or an on state.

Binary Numbering Scale

• Each positional value is a power of 2.

Binary Addition

• There are 4 possible binary addition combinations.

Decimal to Binary Conversion

• The easiest way to convert a decimal number to binary is using the Division Algorithm.
• This method repeatedly divides a decimal number by 2 and records the quotient and remainder.

Binary to Decimal Conversion

• The easiest method for converting a binary number to decimal is using the Multiplication Algorithm.
• Multiply the binary digits by increasing powers of 2, starting from the right, and sum the products.

Octal Number System

• Also known as the Base 8 System.
• Uses digits 0-7.
• Readily converts to binary.
• Groups of three binary digits represent each octal digit.

Decimal to Octal Conversion

• Convert a decimal number to octal by repetitively dividing by 8 and recording the remainder.

Octal to Decimal Conversion

• Convert an octal number to decimal by multiplying each digit by its positional value and summing the products.

Octal to Binary Conversion

• Each octal digit converts to 3 binary digits.

Hexadecimal Number System

• Base 16 system.
• Uses digits 0-9 and letters A-F.
• Each hexadecimal digit represents 4 binary digits.

Decimal to Hexadecimal Conversion

• Convert a decimal number to hexadecimal by repetitively dividing by 16 and recording the remainder.

Hexadecimal to Decimal Conversion

• Convert a hexadecimal number to decimal by multiplying each digit by its positional value and summing the products.

Binary to Hexadecimal Conversion

• Use a substitution code to convert binary to hexadecimal.

Complementary Arithmetic

• 1's complement: switch all 0's to 1's and 1's to 0's.
• 2's complement: find the 1's complement and add 1.

Signed Numbers

• Need to represent both positive and negative numbers.

• Three representations: signed magnitude, signed 1's complement, and signed 2's complement.### Binary Operations

• Rules for adding two single-bit numbers:

  • A=0, B=0: Sum=0, Carry=0
  • A=0, B=1: Sum=1, Carry=0
  • A=1, B=0: Sum=1, Carry=0
  • A=1, B=1: Sum=0, Carry=1

• Rules for subtracting two single-bit numbers:

  • A=0, B=0: Difference=0, Borrow=0
  • A=0, B=1: Difference=1, Borrow=1
  • A=1, B=0: Difference=1, Borrow=0
  • A=1, B=1: Difference=0, Borrow=1

Binary Multiplication

• Rules for multiplying two single-bit numbers:
  • A=0, B=0: Product=0
  • A=0, B=1: Product=0
  • A=1, B=0: Product=0
  • A=1, B=1: Product=1

Conversion of Decimal Numbers to Any Radix Number

• Conversion of decimal numbers to any radix number involves two steps:
  • Step 1: Convert the integer part using successive division method
  • Step 2: Convert the fraction part using successive multiplication method