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Binary arithmetic, shifts and hexadecimal

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Notes de leçon

Binary addition

  • Rules: 0+0 = 0; 0+1 = 1; 1+1 = 10 (write 0, carry 1); 1+1+1 = 11 (write 1, carry 1).
  • Example: 0101 + 0011 = 1000 (5 + 3 = 8).
  • Overflow: if the result needs more bits than are available (e.g. an 8-bit register gets a 9th bit), the extra bit is lost, giving a wrong answer.

Adding unsigned values

Adding unsigned valuesOperationBinaryDenary00000101 + 00000011000010005 + 3 = 811111111 + 00000001100000000255 + 1 = 2568-bit result after overflow000000000; carry cannot fit

Binary shifts

  • A left shift moves bits left, filling with 0s: it multiplies an unsigned value by 2 for each place, provided no significant bit is lost. 00001100 (12) shifted left 2 = 00110000 (48).
  • A right shift moves bits right, filling with 0s: it divides by 2 (rounding down) for each place. 00001100 shifted right 1 = 00000110 (6).
  • Bits shifted off the end are lost.

Unsigned eight-bit shifts

Unsigned eight-bit shiftsOperationBit patternValueOriginal0000110012Left one0001100024Right one000001106Left two0011000048

Hexadecimal

  • Hexadecimal (base 16) uses digits 0-9 and A-F (A=10 ... F=15).
  • Binary to hex: split into 4-bit groups (nibbles). 1010 1111 → A F → AF. Hex to binary: convert each digit: 3C → 0011 1100.
  • Hex to denary: 3C = 3×16 + 12 = 60. Denary to hex: 255 = FF.

Four bits per hex digit

Four bits per hex digitHex digitFour bitsDenary300113601106A101010C110012F111115

Why hex?

  • Hex is shorter and easier for humans to read and write than long binary: colour codes (#FF0000), memory addresses, MAC addresses and error codes.

Diapos

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Questions d'entraînement

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  1. 1.What base does hexadecimal use?

    Easy
    • A16
    • B10
    • C2
    • D8

  2. 2.What is binary 1 + 1?

    Easy
    • A10
    • B2
    • C11
    • D0

  3. 3.What does overflow mean?

    Easy
    • Athe result is too big for the bits available
    • Bthe result is zero
    • Cthe result is negative
    • Dthe loop ends

  4. 4.What is F in denary?

    Easy
    • A15
    • B16
    • C14
    • D10

  5. 5.For an unsigned binary value, with no significant bit lost, what is the effect of a one-place left shift?

    Easy
    • Adoubles the value
    • Bhalves the value
    • Cadds 1
    • Dsubtracts 1

  6. 6.What is 00000110 shifted left 1?

    Easy
    • A00001100
    • B00000011
    • C00001010
    • D00001000

  7. 7.What is 00001000 shifted right 2?

    Easy
    • A00000010
    • B00100000
    • C00000100
    • D00000001

  8. 8.What is 0011 + 0110 in binary?

    Easy
    • A1001
    • B1010
    • C0111
    • D1000

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