Computer Number Systems Quiz

Computer Number Systems

• Computer number systems are used to represent and work with numbers in digital computing.
• These systems are essential for data encoding, calculations, and communication within and across digital devices.
• The four primary computer number systems are binary, decimal, octal, and hexadecimal.

Binary Number System (Base-2)

• Binary is the fundamental number system in computing.
• It uses only two digits: 0 and 1.
• Each digit in a binary number represents a power of 2.
• The rightmost digit represents 2⁰ (1), the next digit to the left represents 2¹ (2), the next 2² (4), and so on.
• Binary numbers are used in all digital devices and computer systems to represent data and perform logical operations like addition, subtraction, and bitwise operations.
• Example: Binary 1010 represents 1 * 2³ + 0 * 2² + 1 * 2¹ + 0 * 2⁰ = 10 in decimal.

Decimal Number System (Base-10)

• Decimal is the number system most commonly used by humans.
• It employs ten digits: 0 to 9.
• Each digit in a decimal number represents a power of 10.
• The rightmost digit represents 10⁰ (1), the next digit to the left represents 10¹ (10), the next 10² (100), and so on.
• Decimal numbers are used for everyday calculations, mathematics, and most human-oriented contexts.
• Example: Decimal 345 represents 3 * 10² + 4 * 10¹ + 5 * 10⁰ = 345.

Octal Number System (Base-8)

• Octal uses eight digits: 0 to 7.
• Each digit represents a power of 8.
• The rightmost digit represents 8⁰ (1), the next 8¹ (8), next 8² (64), and so on.
• Octal numbers are less common in modern computing but were historically used to represent machine code and memory addresses.
• Example: Octal 52 represents 5 * 8¹ + 2 * 8⁰ = 42 in decimal.

Hexadecimal Number System (Base-16)

• Hexadecimal employs sixteen digits: 0 to 9, and A to F (A=10, B=11, C=12, D=13, E=14, F=15).
• Each digit represents a power of 16.
• The rightmost digit represents 16⁰ (1), the next 16¹ (16), the next 16² (256), and so on.
• Hexadecimal numbers are widely used in computing for representing binary data compactly and memory addresses.
• Example: Hexadecimal 1A7 represents 1 * 16² + 10 * 16¹ + 7 * 16⁰ = 423 in decimal.

Converting Decimals to Other Number Systems

• Decimal to Binary: Repeated division by 2.
• Decimal to Octal: Repeated division by 8.
• Decimal to Hexadecimal: Repeated division by 16.

Arithmetic Operations on Binary Numbers

• Binary Addition: Follows the same rules as decimal addition but carries over when the sum is 2 (not 10).
  • 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0 (carry-over 1).
• Binary Subtraction: Similar to decimal subtraction but borrows when the minuend is smaller than the subtrahend.
  • 0 - 0 = 0, 1 - 0 = 1, 1 - 1 = 0, 0 - 1 = Borrow.
• Binary Multiplication: Similar to decimal multiplication but only involves 0 and 1.
  • 0 * 0 = 0, 0 * 1 = 0, 1 * 0 = 0, 1 * 1 = 1.
• Binary Division: Similar to decimal division, involving a dividend, divisor, quotient, and remainder.
  • 0 ÷ 1 = 0 remainder 0, 1 ÷ 1 = 1 remainder 0, 0 ÷ 0 = undefined, 1 ÷ 0 = undefined.