Computer Fundamentals - Number Systems

Computer Fundamentals - Number Systems

• The presentation introduces four number systems: binary, octal, decimal, and hexadecimal.
• It discusses conversions between these systems, along with representations of signed and unsigned numbers.

Data, Program and Information

• A number is a string of one or more digits used for counting, quantifying, and measuring.
• A digit is a single numerical symbol of a number, or the smallest symbol of numbers.

Number Systems - Types

• Binary (base 2): Uses only 0 and 1.
• Octal (base 8): Uses digits 0-7.
• Decimal (base 10): Uses digits 0-9.
• Hexadecimal (base 16): Uses digits 0-9 and A-F.

Positional Number System

• Each digit in a number has a value based on its position.
• In decimal, the value of each digit is multiplied by increasing powers of 10.

Decimal Number System

• Represents numbers using decimal digits (0-9).
• Each digit's position determines its power of 10 multiplier.
  • Example: 83 = (8 x 10¹) + (3 x 10⁰).
• Applies to both integers and fractions (negative exponents).

Decimal Number System - Most & Least Significant Digits

• The leftmost digit is the most significant digit (highest value).
• The rightmost digit is the least significant digit (lowest value).
  • Example: 3501.51 -> 3 (most significant) , 1 (least significant)

Binary Number System

• Uses only two digits: 0 and 1.
• Each digit's position determines its power of 2 multiplier.
  • Example: 10₂ = (1 * 2¹) + (0 * 2⁰) = 2₁₀
• Often uses subscripts (2) to indicate binary representation (e.g., 1001₂).

Binary Number System - Conversion to Decimal

• Convert binary to decimal by adding the weights of each digit with a value of 1, based on its position (powers of 2).
  • Example: 00101010₂ = 02⁰ + 12¹ + 02² + 12³ + 02⁴ + 12⁵ + 02⁶ + 02⁷ = 42₁₀

Binary Number System - Conversion from Decimal

• Divide the decimal number by 2 repeatedly, recording the remainders.
• The remainders, read in reverse order form, are the binary equivalents.
  • Example: 75₁₀ converts to 1001011₂

Hexadecimal Number System

• Shortens binary representations to be more human-readable.
• Uses digits 0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, F=15.
• Example: 1AF₁₆ = 1⋅16² + 10⋅16¹ + 15⋅16⁰ = 431₁₀.

Hexadecimal to Decimal Conversion

• 1. Convert the hexadecimal number to binary
• 2. Convert binary to decimal (using the powers of 2 method)

Octal Number System

• Uses digits 0-7.
• Each position multiplies its digit by increasing powers of 8

Signed and Unsigned Integers

• Unsigned: Represents only positive values.
  • The range of an N-bit unsigned integer is from 0 to 2^N - 1.
• Signed: Represents both positive and negative values. Common approaches include sign-magnitude, one's complement, and two's complement.
  • Two's complement is the common method. The MSB acts as a sign bit.

Coding Systems for Text-Based Data

• ASCII (American Standard Code for Information Interchange): Used primarily with personal computers.
• EBCDIC (Extended Binary-Coded Decimal Interchange Code): Primarily used for mainframes.
• Unicode: A universal coding standard replacing ASCII in many applications. Includes UTF-8, UTF-16, and UTF-32.

Coding Systems for Other Data Types

• Images (pixels): Use different numbers of bits depending on the color depth (e.g., 16-color, 256-color, True Color).
• Audio: Typically involves converting analog audio signals to digital bits using Analog-to-Digital (ADC) converters, and then back to analog for output by Digital-to-Analog (DAC) converters.
• Video: Can represent images using bits and pixel codes.
• Machine Language: Uses binary codes to represent machine operations.