Digital Logic and Coding Schemes Quiz

Course Outline

• DCIT 207 Computer Organisation and Architecture course outline.
• Lecturer: Dr. Ernest Gyebi.

Data Representation

• Data representation refers to the form in which data is stored, processed, and transmitted.
• Data refers to the symbols that represent people, events, things, and ideas.
• Data can be a name, a number, or a photograph.
• All data stored and processed by digital computers is in binary code (1s and 0s).
• The 1s and 0s are referred to as bits.
• The 1s are represented by voltages in the computer (+5 volts = binary 1), and 0s by voltages (0 volts = binary 0).
• Devices like smartphones, iPods, and computers store data in digital formats that electronic circuitry can handle.

Number Systems

• Number systems are based on positional number systems.
• The position of a digit within the number determines the digit's value.
• For example, in the number 1234₁₀, the 2 represents 2 x 10² = 200.
• The decimal number system has positional values: 10³, 10², 10¹, 10⁰.
• The radix or base determines the number of different values in the number system.
• For example, decimal radix = 10 digits (0-9).
• Binary radix = 2 digits (0, 1).
• Octal radix = 8 digits (0-7).
• Hexadecimal radix = 16 digits (0-F).

Binary Numbers

• Radix/Base = 2
• Symbols: 0, 1
• Positional values: 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
• Decimal equivalent: 64, 32, 16, 8, 4, 2, 1
• Examples:
  • 101₂ = 1 x 2² + 0 x 2¹ + 1 x 2⁰ = 4 + 0 + 1 = 5₁₀.
• To indicate a number is binary, use the prefix % or the subscript 2. e.g., %1110 or 1110₂

Conversion from Decimal to Binary

• Repeatedly divide by 2 to convert a decimal number to binary until the result is less than 2
• Example (53):
  • 53 ÷ 2 = 26 remainder 1
  • 26 ÷ 2 = 13 remainder 0
  • 13 ÷ 2 = 6 remainder 1
  • 6 ÷ 2 = 3 remainder 0
  • 3 ÷ 2 = 1 remainder 1
  • 1 ÷ 2 = 0 remainder 1
• To get the result work backwards, gives: 110101₂
• To check result convert back to base 10 (check your arithmetic!):
  • 1 x 2⁵ + 1 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2⁰ = 32 + 16 + 0 + 4 + 0 + 1 = 53.

Using Sum of Weights Method for Decimal to Binary Conversion

• Used when repeatedly dividing by 2 is cumbersome.

1. Write the number in powers of 2. The highest power + 1 gives the number of bits in the binary.
2. Check the positions and place 1s in the appropriate weight positions where there are powers of 2.
3. Place 0s in the remaining positions (where there are no powers of 2).

Octal and Hexadecimal Numbers

• Octal (base 8) and hexadecimal (base 16) are used for communication between people and computers because binary numbers are unsuited for routine handling by humans.

Hexadecimal Numbers

• Base or Radix=16
• Symbols: 0-9, A-F
• Positional values: 16⁴, 16³, 16², 16¹, 16⁰
• Decimal values: 65,536, 4096, 256, 16, 1
• Example: 1D7E₁₆ = 1 x 16³ + 13 x 16² + 7 x 16¹ + 14 x 16⁰ = 4096 + 3328 + 112 + 14 = 7550₁₀

Conversion from Hexadecimal to Binary

• Each hexadecimal digit is equivalent to four binary digits (2⁴ = 16).
• To convert a hexadecimal number to binary, write each hexadecimal digit as a 4-bit binary number.
• Example: $FD69A = %1111 1101 0110 1001 1010

Signed Numbers

• Digital systems need to handle both positive and negative numbers.
• A signed binary number includes sign and magnitude information.
• The sign indicates whether the number is positive or negative.
• There are three forms for representing signed integers in binary: sign-magnitude, 1's complement, and 2's complement.

The Sign Bit

• The left-most bit of a signed number is the sign bit.
• Positive = 0, Negative = 1
• Also referred to as the Most Significant Bit (MSB).

Sign-Magnitude Form

• The left-most bit is the sign bit.
• The remaining bits represent the magnitude.

Complements

• Complements are used in digital circuits for faster subtraction by adding.
• Binary complement = reversing all bits and adding 1.
• Carry from the high-order position is discarded.

9's and 10's Complements

• The 9's complement of a decimal number is obtained by subtracting each digit from 9.
• The 10's complement = 9's complement + 1.

1's and 2's Complement

• For binary representation, the problem is easier since the number of digits is only 2, and the number of digits is always known.
• 10's complement = 9's complement + 1. Thus, 2's complement = 1's complement + 1.

1's Complement Form

• Positive numbers are represented the same as in sign-magnitude.
• Negative numbers are the 1's complements of the corresponding positive numbers.
• Finding the 1's complement of a number involves inverting each bit.

2's Complement Form

• The computer finds the 1's complement and adds 1 to get the 2's complement.
• Positive numbers are represented the same as in sign-magnitude/1's complement.
• Negative numbers are the 2's complements of the corresponding positive numbers.
• In all arithmetic operations with negative integers, computers use 2's complement.

Binary Fractions

• Binary fractions can also be represented.
• Positional values: 2⁻¹, 2⁻², 2⁻³, 2⁻⁴, 2⁻⁵, etc
• Decimal values are: .5, .25, .125, .0625, .03125, 1/32
• Conversion of binary fractions to decimal: Repeated multiplication by 2.

Fixed-Point and Floating-Point Numbers

• A fixed-point number has a fixed decimal point location.
• Integers are fixed at far right; fractions are fixed at far left.
• Floating-point numbers (real numbers) store integers and fractional parts.
• A floating-point number uses a mantissa (fractional part) and an exponent to represent a number.
• Examples 53.95 can be stored as: (0.5395 × 10²)

IEEE 754 Floating Point Format

• A standard way to represent floating-point numbers.
• Used by computers to increase performance because it frees up the CPU.
• There are different precisions for floating-point numbers, leading to a tradeoff between accuracy and storage requirements.

Alphameric Codes

• Scheme to store letters, punctuation marks, and special characters as binary numbers.
• Character data involves letters, symbols, and numerals.
• The EBCDIC, ASCII and Unicode character codes.

Bits, Bytes, and Words

• Data handled by computers is ultimately reduced to bits (0 or 1).
• A computer word is a group of adjacent bits used by the central processor, manipulated as a unit.
• Byte = 8-bits.
• Nibble = 4-bits.
• The number of bits in a word can vary from computer to computer.

Memory Operations

• Write operation: stores data into a designated memory location.
• Read operation: copies data out of a designated memory location without destroying the data.

The Memory Stack

• Storage device that uses Last-In, First-Out (LIFO) methodology.
• Implemented in computing systems.
• Data to be retrieved is retrieved in the reverse order.

RAMStack

• The ramstack memory allocation is used in microprocessor based systems, rather than the use of dedicated registers
• A special register called a stack pointer(SP) is in charge of the top of the stack.