Understanding the Binary Number System

• Binary is a base-2 numeral system that uses only two digits: 0 and 1.
• Computers use binary to represent all data and instructions.
• Each binary digit is called a bit.

Binary Number System

• It's a positional notation system, meaning each bit's value depends on its position.
• The rightmost bit is the least significant bit (LSB), and the leftmost bit is the most significant bit (MSB).
• Each position represents a power of 2, starting from 2^0 on the right.
• To convert a binary number to decimal, sum the values of each bit position that has a 1. For example, 1011 binary is (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 8 + 0 + 2 + 1 = 11 decimal.

Binary Representation

• Binary is used to represent various types of data in computers.
• Integers can be represented directly in binary.
• Characters are typically represented using standardized binary codes like ASCII or Unicode.
• Instructions for the CPU are also encoded in binary.
• Colors can also be represented.
• Different encodings and interpretations are context-dependent.

Bits, Bytes, and Words

• A bit is a single binary digit (0 or 1).
• A byte is a group of 8 bits.
• A word is a group of bits that a CPU processes as a single unit. The size of word can vary (e.g. 16-bit, 32-bit, 64-bit).

Binary Arithmetic

• Addition, subtraction, multiplication, and division can all be performed in binary.
• Binary addition follows simple rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10 (carry-over 1).
• Subtraction uses borrowing when necessary.
• Multiplication and division are similar to decimal arithmetic but simplified by using only 0 and 1.

Logic Gates

• Logic gates are fundamental building blocks of digital circuits.
• They perform basic logical operations on binary inputs.
• Common logic gates include AND, OR, NOT, XOR, NAND, and NOR.
• Each gate has a truth table that defines its output for all possible input combinations.
• Logic gates are implemented using transistors.

Boolean Algebra

• Boolean algebra is a branch of algebra that deals with binary variables and logical operations.
• It provides a formal way to analyze and simplify digital circuits.
• Boolean algebra uses operators like AND (.), OR (+), and NOT (').
• Key concepts include truth tables, Boolean expressions, and simplification techniques (e.g., Karnaugh maps).

Binary Encoding Schemes

• ASCII: A character encoding standard that uses 7 bits to represent characters (128 characters).
• Extended ASCII: Uses 8 bits (256 characters), adding more symbols and characters.
• Unicode: A more comprehensive standard that uses a variable number of bits (UTF-8, UTF-16, UTF-32) to represent a much wider range of characters from different languages.
• UTF-8 is a popular encoding that uses one byte for ASCII characters and multiple bytes for other characters.

Binary Conversion

• Decimal to Binary: Repeatedly divide the decimal number by 2, noting the remainders. The remainders, read in reverse order, form the binary equivalent.
• Binary to Decimal: Multiply each bit by its corresponding power of 2 and sum the results.
• Binary to Hexadecimal: Group the binary digits into sets of 4 (starting from the right), and then convert each group into its hexadecimal equivalent (0-9, A-F).
• Hexadecimal to Binary: Convert each hexadecimal digit to its 4-bit binary equivalent.

Signed Number Representation

• Sign and Magnitude: The most significant bit (MSB) represents the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude.
• One's Complement: To get the one's complement of a number, invert all the bits (0 becomes 1, and 1 becomes 0).
• Two's Complement: To get the two's complement, invert all the bits (one's complement) and add 1. This is the most common method for representing signed integers in computers because it simplifies arithmetic operations.

Floating-Point Representation

• Used to represent real numbers (numbers with fractional parts).
• Follows the IEEE 754 standard.
• Consists of three parts: sign, exponent, and mantissa (or significand).
• Single-precision (32-bit) and double-precision (64-bit) formats are common.
• The exponent is biased to allow representation of both positive and negative exponents.

Bitwise Operations

• Operations performed on individual bits of a number.
• Common bitwise operators: AND (&), OR (|), XOR (^), NOT (~), left shift (<<), and right shift (>>).
• AND: Sets a bit to 1 only if both corresponding bits are 1.
• OR: Sets a bit to 1 if either of the corresponding bits is 1.
• XOR: Sets a bit to 1 if the corresponding bits are different.
• NOT: Inverts all the bits.
• Left Shift: Shifts bits to the left, filling with zeros.
• Right Shift: Shifts bits to the right (arithmetic or logical shift).

Uses of Binary in Computing

• Data Storage: All data on computers is stored in binary format.
• Networking: Data is transmitted over networks in binary form.
• CPU Operations: CPUs perform calculations and execute instructions using binary arithmetic and logic.
• Memory Addressing: Memory locations are addressed using binary numbers.
• Image Representation: Images are stored as a collection of binary values representing pixel colors.
• Audio Representation: Sound is digitized and stored as binary data.
• Programming: Programs are ultimately compiled or interpreted into machine code, which is binary instructions for the CPU.

Advantages of Binary

• Simplicity: Easy to implement in hardware with only two states (on/off, high/low voltage).
• Reliability: Less susceptible to noise and interference compared to systems with more states.
• Universality: Can represent any type of data or instruction.
• Logical Operations: Binary logic is the foundation for digital circuits and computer logic.

Disadvantages of Binary

• Verbosity: Requires more digits to represent the same value compared to decimal.
• Readability: Binary numbers can be difficult for humans to read and write, leading to the use of hexadecimal or decimal representations in many contexts.