Number Systems and Memory

• The lecture addresses aspects of information storage with focus on bits, bytes, words and hexadecimal representation, and also addresses data addressing and byte ordering, strings and code representation, including bitwise and logical operations

Bits, Bytes, Words, Paragraphs, and Pages

• The bit is the smallest unit of information in computing, storing two values: 0 (no electric current) and 1 (electric current).
• Computer memory consists of elements that each store one bit.
• Bits are small, so they are accessed in packages: bytes, words, paragraphs, or pages.
• A byte is the smallest standard package of bits, often hardware-dependent, but is commonly 8 bits today but has ranged from 6-9 bits.
• Bytes are uniquely addressable in memory.
• A word is the unit of information accessed by a CPU, which depends on the CPU (8, 16, 32, 64, or 128 bits).
• A word is the amount of data transferred between the CPU and main memory in one go.
• A paragraph is typically 16 bytes (128 bits).
• A page is typically 4KB (4096 bytes) or 8KB (8192 bytes) and also hardware/system dependent.
• A page is the amount of data transferred between main memory and secondary storage (hard disk or SSD).
• Memory is a physical byte array with each element having a unique address in the range of 0 to n-1 where n is the size of the memory.
• Most programming languages can access information at the byte level, and programs can manipulate each byte in memory because each byte has a unique address in physical memory.

Hexadecimal Representation

• Bits, Bytes, Words and Paragraphs are powers of 2, so hexadecimal is useful
• Computations often involve dividing or multiplying by a power of 2
• Multiplying and dividing by two in binary is easy as you can add or remove a 0
• Octal (base 8) is another option
• Octal is a base 8 number system: Digital 8 is Octal 10
• An octal digit can be represented by 3 bits, and multiplying a number by 8 adds a 0 to the end
• In early computing, bytes were 6 or 9 bits long, bytes could be represented by two or three octal digit, which explains why early computer system used Octal
• Today the most common byte is 8 bits long, and a byte requires 3 Octal digits
• The common and most signifcant digit is not used
• We want something where # of bits per digit divides nicely into 8
• Other Options are
  • Base 2 (Binary) with 1 bit per digit
  • Base 4 with 2 bits per digit
  • Base 16 with 4 bits per digit -Base 256 with 8 bits per digit
• Hexadecimal is base 16 number system that uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Multiplying a number by 16 adds a 0 to the end
• A hex Digit can be represented by 4 bits and every 4-bit number represent a single hexagon
• A byte is 8 bits and can be represented by 2 hex digits
• 216 (base 10) = D8 (base 16) = 11011000 (base 2)
• 1048576 (base 10) = 100000 (base 16)
• It's best to do everything in hex as it can avoid conversion mistakes given all arithmetic rules apply, and uses fewer digits
• To convert from decimal to hex: repeatedly divide the decimal value by 16, with the remainder being the digit in the hex representation
• To convert from hex to decimal: Multiply each hex digit by incresing power of 16 and add result
• Hex representations are typically prefixed with 0x, and should be internalized without conversions

Addressing and Byte Ordering

• Addressing refers to how data is located in memory, and is relatively standard
• Each byte has a unique address, in increasing order
• A piece of data in memory consists of 1 or more bytes
• The address of that piece of data is the address of the first byte of the data.
• Byte ordering refers to the order of bytes in a piece of data
• For data such as pieces of strings, the order of the bytes only depends on the data itself
• There are two forms of byte orders for numerical data types (integers and floats) and pointers:
  • Big Endian: bytes are ordered from most significant to least significant
  • Little Endian: bytes are ordered from least significant to most significant
• Byte ordering is hardware (CPU) dependent, and neither is better than the other
• Byte ordering matters when reading / writing binary data to files, when receiving / sending binary data over the network, for exchanging data between different computer architectures, and when debugging
• Debugging requires looking at raw data in memory, and knowing whether system is bit or little endian, and to use this kowledge to interpret values

Strings and Code in Memory

• Strings are stored as sequences of bytes in memory, and most use nul-terminated strings, of 0 or more bytes- none of which is nul (0)
• A string of n characters takes n+1 bytes to store as strings are terminated by a nul (0) chracter
• Each character is assumed to be ACSII which is 1 byte per character
• In Java, Unicode for string Encoding is used that is 2 bytes per character
• ASCII (American Standard Code for Information Interchange), all characters are one byte in size, originally of 7 bits are used
• ASCII encoding does not contain any non-english characters or special symbols Unicode can encode characters from most known languages, as UTF-16 uses 2 bytes per character, and UTF-8 is variable length encoding that is same as ASCII for first 128 characters.
• Each program instruction is a sequences of one or more bytes, and the CPU executes by moving a pointer in memory

Bitwise and Logical Operations

• We can't address a single bit as we cannot read or set one
• Must access the byte of word containing the bit to be set or read
• A byte us denoted by the char type in C, and a word is denoted by the int type
• Implement function which takes a byte and integer, and returns to value of the ith bit in that byte
• Bit 0 is the least significant bit, and bit 7 is the most significant bit
• The >> operator right shifts the desired bit to position 0, and the & operator computes the biswise AND of 000000001 and the result of right shift
• The result is the bit moved to positition 0
• List of Bitwise and Logical Operators

Further Problems to Solve

• Implement function that sets unsigned b with int i, returns byte b with ith bit set to 1
• Implement function to create a reset functing with same inputs, that returns with ith bit set to 0

Explore number systems like binary and octal, their conversions, and challenges. Learn about memory organization, bytes, and data transfer units between the CPU and main memory like pages.