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Number Systems

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Why Computers Use Binary

  • Computers process data using logic gates that have only two states (on/off).
  • Binary (base-2) uses digits 1 (on) and 0 (off) to match these states.
  • All data must be converted to binary before a computer can process it.
  • Examples: magnetic hard drives use North/South polarity; optical disks use land/pit.
  • Binary allows fast processing and efficient storage.

Number Systems Overview

  • Denary (base-10): digits 0–9, each column is a power of 10.
  • Binary (base-2): digits 0–1, each column is a power of 2.
  • Hexadecimal (base-16): digits 0–9 and letters A–F (A=10, B=11, C=12, D=13, E=14, F=15).
  • One hex digit represents exactly 4 bits (a nibble).
  • Largest 16-bit denary value: 65,535 (binary 1111111111111111).

Converting Between Binary & Denary

  • Denary → Binary: write column headings (powers of 2) from largest ≤ number down to 1; for each column, put 1 if the value fits, subtract it, else 0.
  • Binary → Denary: write column headings, multiply each 1-bit by its heading, sum the results.
  • If a binary number ends in 1, the denary result is odd (quick check).
  • Pad with leading zeros if a specific bit-length is required (e.g., 8-bit).

Converting Between Hexadecimal & Denary

  • Denary → Hex: Method 1: convert denary to binary, split into nibbles, convert each nibble to hex. Method 2: divide by 16; quotient = first hex digit, remainder = second hex digit.
  • Hex → Denary: Method 1: convert each hex digit to a nibble, join to 8-bit binary, convert to denary. Method 2: multiply first digit by 16, add second digit.
  • Exam is non-calculator; use binary method if unsure of multiplication/division by 16.

Converting Between Hexadecimal & Binary

  • Binary → Hex: split 8-bit binary into two nibbles (4 bits each); convert each nibble to its hex digit (0–F).
  • Hex → Binary: convert each hex digit to its 4-bit binary nibble; join the nibbles.
  • Always write out the full 8-bit binary result for clarity.

Uses of Hexadecimal

  • Hex is used because it is shorter than binary and easier for humans to read with fewer copying errors.
  • MAC addresses: 12 hex digits (e.g., AA:BB:CC:DD:EE:FF) instead of 48 binary digits.
  • Colour codes: 6 hex digits (e.g., #66FF33) instead of 24 binary digits.
  • URLs: non-standard characters are encoded as hex preceded by % (e.g., %20 for space).

Binary Addition

  • Rules: 0+0=0, 0+1=1, 1+0=1, 1+1=0 carry 1, 1+1+1=1 carry 1.
  • Add from rightmost bit to left, carrying over when sum ≥ 2.
  • Overflow occurs when the result exceeds the available bits (e.g., 8-bit sum > 255).
  • Always show carry bits clearly; marks are awarded for correct carries even if final answer is wrong.

Binary Shifts

  • Logical left shift multiplies by 2 for each shift (e.g., shift left 1 → ×2).
  • Logical right shift divides by 2 for each shift (e.g., shift right 1 → ÷2).
  • Bits shifted out are lost; empty positions are filled with 0.
  • Overflow occurs when a 1 is shifted out of the most significant bit (MSB) on a left shift.

Two's Complement

  • Two's complement represents signed (positive and negative) binary numbers.
  • The MSB has a negative weight (−128 in 8-bit); remaining bits are positive.
  • To convert a negative denary number: write the positive binary, then copy bits from right until the first 1, then invert all remaining bits.
  • Example: −76 → positive 01001100 → copy rightmost bits (100) → invert rest → 10110100.

Slide

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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 55 câu hỏi. Đăng ký để xem tất cả.

  1. 1.Why do computers use binary to process data?

    Easy
    • ABecause logic gates have only two states, on and off
    • BBecause binary is easier for humans to read than denary
    • CBecause binary uses digits 0-9
    • DBecause binary is the only number system that can represent negative numbers

  2. 2.What is the denary value of the binary number 1011?

    Easy
    • A11
    • B13
    • C9
    • D14

  3. 3.Convert the denary number 45 to binary.

    Medium
    • A101101
    • B111101
    • C101011
    • D110101

  4. 4.Convert the hexadecimal number B9 to denary.

    Medium
    • A185
    • B177
    • C169
    • D191

  5. 5.What is the result of adding the binary numbers 00011001 and 10001001?

    Medium
    • A10100010
    • B10000010
    • C10101010
    • D10010010

  6. 6.What is the denary value of the two's complement binary number 11111111?

    Hard
    • A-1
    • B255
    • C-127
    • D0

  7. 7.What is the two's complement representation of -76?

    Hard
    • A10110100
    • B11001100
    • C10101100
    • D10010100

  8. 8.Which of the following is a valid use of hexadecimal in computing?

    Easy
    • ARepresenting MAC addresses
    • BStoring text files
    • CPerforming arithmetic in spreadsheets
    • DDisplaying images

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