Week 1 Topic 3: Binary representation

• Data representation on computers involves encoding, storing, and manipulating information using binary representation (0s and 1s).

Binary Representation

• Binary representation uses bits, the smallest unit of data, with a value of either 0 or 1.
• Typically, computers store data in bytes, which consist of eight bits.
• The value of a bit is stored as an electrical charge in a capacitor within a memory device.

Number Systems

• Computers use number systems like binary, decimal, octal, and hexadecimal to represent numbers.
• Character encoding schemes like ASCII and Unicode map characters to numeric codes.
• Computers handle images, audio, and video using specialized formats like JPEG and MP3.
• Data structures, such as arrays and linked lists, organize data in memory.
• File formats dictate data structure within files.

Logic Gates

• Logic gates are fundamental components of digital circuits, performing logical operations on binary inputs to produce binary outputs.
• Types of logic gates include:
  • AND: Output is true only if all inputs are true.
  • OR: Output is true if any input is true.
  • NOT: Inverts the input.
  • NAND: AND gate followed by a NOT gate.
  • NOR: OR gate followed by a NOT gate.
  • XOR: Output is true if the number of true inputs is odd.
• Connecting logic gates allows for the implementation of complex logical functions.

Transistors

• Transistors are semiconductor devices that act as switches to control current flow based on voltage.
• PMOS transistors switch on when the gate has a low voltage.
• NMOS transistors switch on when the gate has a high voltage.

Truth Tables

• A truth table is a tabular representation showing the outputs of a logical expression for all possible input combinations.
• Each column represents an input variable, and the final column represents the output.
• Rows enumerate all possible input combinations.
• Truth tables are used to define the behavior of logic gates, logical expressions, and Boolean functions.

AND Gate

• Produces a true output (1) only when all inputs are true.
• Used for logical multiplication, signal processing, and control applications.

NOT Gate

• Also known as an inverter, it inverts the input signal.
• If the input is 1, the output is 0, and vice versa.
• Can be created using a NAND gate.

OR Gate

• Produces a true output (1) when at least one input is true.
• If all inputs are false, the output is false (0).
• Used for logical summation, signal conditioning, and control applications.

NAND Gate

• A universal gate can form most other gate types.
• Outputs a low voltage (0) if both inputs are high voltage (1).
• Can be constructed using four transistors.

XOR Gate

• Produces a true output (1) when the number of true inputs is odd.
• Used in arithmetic operations, data encryption, error detection, and communication protocols.

Adders

• Adders are digital circuits used to perform the addition of binary numbers.
• They are essential components of arithmetic logic units (ALUs).
• A half adder adds two single binary digits and produces a sum bit and a carry bit.
  • Sum bit: Result of adding inputs A and B using a XOR gate.
  • Carry bit: Result of the AND operation between inputs A and B.

Full Adder

• A full adder handles an incoming carry bit from the previous bit addition.
• It adds three binary digits and produces a sum bit and a carry bit.
• Can be created using two half adders and an OR gate.

Data Types

• Common data types include:
  • Integer (int): Whole numbers, typically 4 bytes long.
  • Character (char): Individual characters, typically 1 byte long.
  • String: A collection of characters.
  • Floating-point number (float): Numbers with decimal places, typically 4 bytes.
  • Double-precision floating-point number (double): Numbers with more precision than float, typically 8 bytes.

Number Representation

• The decimal system uses ten symbols (0-9).
• Binary uses base 2, with numbers 0 and 1.
• Each bit's place has a value based on powers of 2.
  • Converting binary to decimal involves multiplying each bit by its corresponding power of 2.
  • Converting decimal to binary involves repeatedly dividing the decimal number by 2 and noting the remainder.

Hexadecimal System

• Hexadecimal (hex) has a base of 16 and uses 16 symbols: 0-9, A-F.
• Used as a more readable representation of binary-coded values.
• Each digit's place has a value based on powers of 16.
• One byte is always two hex digits.

Text Representation

• Characters are represented as ASCII or Unicode.
  • ASCII uses 1 byte per character and is limited to English letters.
  • Unicode represents thousands of different characters.
    • UTF-16 uses 2 bytes per character.
    • UTF-8 uses 1 byte if possible and is backward compatible with older systems.

Image Representation

• Computer screens have millions of pixels, each with three sub-pixels: red, green, and blue.
• Different colors are due to different combinations and intensities of these primary colors.
• Typically, 3 bytes are assigned per pixel, written in hex (e.g., #ffffff for white).
• Human eyes have three types of color-sensitive optical cells (cone cells).