Number Systems and Conversions Quiz

Number Systems

• Binary: Base-2 system. Uses only 0 and 1 (representing OFF and ON). MSB is the leftmost bit.
• Denary: Base-10 system. Uses values 0-9.
• Hexadecimal (Hex): Base-16 system. Uses 0-9 and A-F (A=10, B=11, ..., F=15).

Number Conversions

• Binary to Denary: Assign powers of 2 to each bit position (from right to left, starting with 2⁰). Add the values where the binary digit is 1.
• Denary to Binary: Repeatedly divide by 2, recording the remainders. Read the remainders from bottom to top to get the binary equivalent.
• Hexadecimal to Binary: Separate each hex digit, convert to denary, then convert each denary value to binary. Combine the binary values.
• Binary to Hexadecimal: Divide the binary string into groups of 4 bits (from right to left), add leading zeros if necessary. Convert each group to its hexadecimal equivalent.
• Hexadecimal to Denary: Convert the hexadecimal value to binary, then convert the binary value to denary following the power of 2 method.
• Denary to Hexadecimal: Convert the denary value to binary, then convert the binary value to hexadecimal as described above.

Binary Calculations

• Addition: Binary addition follows specific rules: 0+0=0, 1+0=1, 0+1=1, 1+1=0 (carry 1), 1+1+1=1 (carry 1).
• Overflow: Occurs when the result of an operation exceeds the number of bits available to store it.

Logical Shifts

• Left Shift: Shifts each bit to the left by a specified number of positions. Empty positions are filled with 0s.
• Right Shift: Shifts each bit to the right by a specified number of positions. Empty positions are filled with 0s.

Two's Complement

• Representing Negative Values: Used to represent negative values in binary. The most significant bit (MSB) is treated as the sign (1 for negative, 0 for positive). The magnitude of the negative number is calculated following the above denary to binary conversion rules.
• Conversion: To convert a negative binary value to two's complement, first find the binary equivalent of the value (ignoring the -). Invert all bits, then add 1 to the result. To convert a two's complement value to denary, follow the denary to binary conversion process.

Applications of Hexadecimal

• Defining Colors (HTML): Representing colors in web design.
• MAC Addresses: Unique identifiers for network devices.
• Assembly Language/Machine Code: Used in low-level programming.
• Memory Dumps: Displaying the contents of computer memory.
• Error Codes: Representing memory locations of errors.
• IP Addresses: Used to identify devices on the Internet.

Text, Sound, and Images

• ASCII: 7-bit character set, representing letters, numbers, and symbols.

• Unicode: Character set capable of representing many international languages using up to 4 bytes per character.

• Sound: Analog sound converted to digital through sampling.

  • Sampling Resolution (Bit Depth): Number of bits to represent each sample. Larger resolutions lead to more accurate sound but increased file size.
  • Sampling Rate: Number of samples taken per second (measured in Hertz). Higher rates lead to higher sound quality.

• Bitmap Images: Images made of pixels.

  • Color Depth: Number of bits per pixel, determining the number of colors the image can display. Higher depths increase file size.
  • Image Resolution: Number of pixels in the image, more pixels give more detail. Lower resolution images are 'pixelated'.

• File Size Calculation:

  • Images: Resolution × Color depth
  • Sound: Sampling rate × Sample resolution × Length (in seconds)

• File Types:

  • MIDI: Stores music data as commands; no actual audio is stored initially. Uses serial transmission.
  • MP3: Lossy format for audio compression, often reducing file size by up to 90%.
  • MP4: Multi-media file container format.
  • JPEG: Lossy format for image compression, reducing the original resolution.

• File Compression:

  • Lossless: Does not reduce quality, reconstructs the original file perfectly. Examples include Run-Length Encoding (RLE) for repeated data.
  • Lossy: Reduces file size by discarding data; quality is lost. Examples include MP3 and JPEG.

Memory Measurement

• Bit: Binary digit.
• Byte: 8 bits.
• Kilobyte (KB), Megabyte (MB), Gigabyte (GB), Terabyte (TB): Units of memory, based on powers of 1000 (Conventional System).
• Kibibyte (KiB), Mebibyte (MiB), Gibibyte (GiB), Tebibyte (TiB) ...: Units of memory, based on powers of 1024 (IECB System). This system is increasingly standard for accurate representation, especially in computing.