Computer Science Quiz: Number Systems

Digital Logic Design Course Information

• Course title: Digital Logic Design
• Course code: 25241104
• Instructor: Dr. Abhishek Dixit
• Department: CSBS, MITS-DU
• Institution: Madhav Institute of Technology & Science, Gwalior-Deemed University

Digital Logic Design/Digital Electronics

• Digital Logic Design is fundamental to electrical and computer engineering.
• It is used to design hardware components like circuit boards and microprocessors.
• This hardware processes user input, system protocols, and data for various high-tech systems.
• Digital electronics encompasses areas like digital signals and engineering elements.
• These elements achieve high productivity through integrated circuits that contain multiple logic gates.

Digital Logic Design and Its Importance

• Most electronic devices are a combination of hardware and software.
• Hardware includes circuits that execute programs.
• Software consists of programs that manage hardware to fulfill user requests.
• Understanding Digital Logic Design is key to comprehending how electronic devices function.
• Computer science studies both hardware and software.

Conclusion

• Digital Logic Design (DLD/DE) is crucial for grasping how programs function within a machine.
• Computer engineers must understand both software and hardware components
• Hardware encompasses the CPU (ALU and MU), which uses integrated circuits and numerous logic gates such as adders, subtractors within.
• Understanding the internal workings of a computer provides a clearer comprehension of its overall operation.

Applications of Digital Logic Design

• Conventional computer design: CPUs, busses, and peripherals
• Networking and communications: Phones, modems, and routers
• Embedded products: Cars, toys, appliances, and entertainment devices

How Did It All Start?

• In 1850, George Boole invented Boolean algebra, which is a branch of mathematics that handles logical operations using binary variables.
• Boolean algebra is fundamental to modern digital computers.

Vacuum Tubes

• The first-generation computers used vacuum tubes for circuitry and magnetic drums for memory.
• These early computers were large and occupied entire rooms.
• Lee De Forest was a key innovator in the development of vacuum tubes.

Transistor Invention

• In 1947, Shockley, Brattain, and Bardeen invented the transistor.
• Transistors replaced vacuum tubes, leading to smaller and more efficient computer designs.
• Transistors enabled the integration of multiple devices onto a single chip.

Fully Transistorized Computer

• In 1955, Bell Labs announced the first fully transistorized computer (TRADIC).

Intel's First Microprocessor

• In 1971, Intel's 4004 microprocessor was introduced.

Number System

• Number systems are essential for representing data in computer systems.
• The base or radix of a number system represents the total number of digits.
• Common number system used in computer architecture are Binary, Octal, Decimal, Hexadecimal.

Binary Number System

• A binary number system uses only 0s and 1s to represent any numeric value.
• The base of the binary number system is 2.

Decimal Number System

• The decimal number system consists of 0 through 9 as digits for numerical representation.
• The base of the decimal number system is 10.

Octal Number System

• The octal system employs 0–7 as its digits for numerical representation.
• The base is 8.

Hexadecimal Number System

• The hexadecimal number system incorporates 0–9 and A-F alphanumeric digits for numerical representation.
• The base is 16 (A=10, B=11, C=12, D=13, E=14, F=15).

Number System Conversions

• Several techniques exist for converting between different number systems.

Binary to Decimal Conversion

• To convert binary to decimal, multiply each binary digit by its corresponding power of 2 and add the results.

Decimal to Binary Conversion

• To convert decimal to binary, repeatedly divide the decimal number by 2 and keep track of the remainders.

Binary to Octal Conversion

• Group binary digits into sets of three, starting from the least significant digit, and convert each group to its octal equivalent.

Binary to Hexadecimal Conversion

• Group the binary digits into sets of four, starting from the least significant digit, and convert each group to its hexadecimal equivalent.

Decimal to Octal Conversion

• Repeatedly divide the decimal number by 8 and record the remainder, with the least significant bit first.

Decimal to Hexadecimal Conversion

• Repeatedly divide the decimal number by 16 and record the remainder, with the least significant bit first.

Signed and Unsigned Numbers

• Signed numbers use a sign bit (most significant bit) to indicate the sign (positive or negative) of the number.
• Unsigned numbers are only positive numbers, and do not need a dedicated bit for sign.
• Sign-magnitude representation, 1's complement, and 2's complement are the three most common approaches.

Complements

• Complements in digital computers are used for simplifying subtraction operations and logical manipulations.
• There are radix complement- also called r’s complement, and diminished radix complement. or (r-1)’s complement.
• Systems like Binary have 1’s complement and 2’s complement.
• Decimal has 9’s and 10’s complement.
• Octal has 7’s and 8’s complement.
• Hexadecimal has F’s and 16’s complement.

1's Complement

• The 1's complement of a binary number reverses each bit (0 becomes 1, 1 becomes 0)

2's Complement

• The 2's complement of a binary number is the 1's complement plus 1.

9's Complement

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

10's Complement

• The 10's complement is calculated by adding 1 to the 9's complement of the number.

Binary Arithmetic

• Binary Arithmetic operations include addition, subtraction, multiplication, and division.

Binary Addition/Subtraction/multiplication/division

• The addition/subtraction/multiplication/division procedures follow specific algorithms.

Subtraction using 1's Complement

• The subtraction process uses one's complement arithmetic.
• The process involves taking the one’s complement of the second number.
• It is also necessary to account for end-around carry.

Subtraction using 2's Complement

• In 2's complement subtraction, add the 2's complement of the subtrahend.
• The final outcome is considered positive/negative depending on the MSB.

Binary Codes

• Binary codes represent numbers, letters, or words with specific groups of symbols or binary bits).
• These codes are fundamental in digital data representation, storage, and transmission.

Classification of Binary Codes

• Binary codes are categorized based on various principles. This includes numerical codes, alphabetical codes (e.g., ASCII) and various other groupings (2421, 5211, etc...).

Weighted Codes

• These codes use positional weights. (e.g., 8421, 2421, etc)

Non-Weighted Codes

• These codes (e.g., Excess-3 and Gray codes) do not assign fixed weights to positions.

Excess-3 Code

• A non-weighted code, this code is used to represent decimal numerals.

Gray Code

• Commonly known as unit-distance codes. Consecutive numbers differ by only one bit in its binary code form.

ASCII Code

• The American Standard Code for Information Interchange, or ASCII.
• It is a widely used code that encodes characters, symbols, and numbers for use with devices that interface with small computer systems, printers, and terminals.

EBCDIC Code

• The Extended Binary Coded Decimal Interchange Code, or EBCDIC. It is an 8-bit character encoding. This is commonly used in mainframe computers to save data.