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Number Systems: Question 3

Syllabus 1.1

Structured 8 marks

A weather station stores each sensor reading as an 8-bit unsigned binary number in its onboard memory, and its processor sometimes needs to add two readings together.

(a) Add the two 8-bit binary numbers 01011011 and 00100101 using binary addition. Show your working, and give your answer in binary. [3]

(b) Add the two 8-bit binary numbers 10110110 and 01101011 using binary addition. Show your working, and give your answer in binary. [3]

(c) State whether an overflow error occurs for the addition in part (b), and explain your answer. [2]

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Worked solution

Part (a): Adding 01011011 and 00100101

Add the two 8-bit numbers column by column from bit 0 (rightmost) to bit 7 (leftmost), carrying a 1 into the next column whenever a column totals 2 or more.

    01011011
+   00100101
  • bit 0: 1 + 1 = 0, carry 1
  • bit 1: 1 + 0 + carry 1 = 0, carry 1
  • bit 2: 0 + 1 + carry 1 = 0, carry 1
  • bit 3: 1 + 0 + carry 1 = 0, carry 1
  • bit 4: 1 + 0 + carry 1 = 0, carry 1
  • bit 5: 0 + 1 + carry 1 = 0, carry 1
  • bit 6: 1 + 0 + carry 1 = 0, carry 1
  • bit 7: 0 + 0 + carry 1 = 1, carry 0 (nothing carried out of the register)

Result: 10000000

Check using denary: 01011011 = 91, 00100101 = 37, and 91 + 37 = 128, which equals 10000000. Correct, and no carry was produced out of bit 7.

Part (b): Adding 10110110 and 01101011

    10110110
+   01101011
  • bit 0: 0 + 1 = 1, carry 0
  • bit 1: 1 + 1 = 0, carry 1
  • bit 2: 1 + 0 + carry 1 = 0, carry 1
  • bit 3: 0 + 1 + carry 1 = 0, carry 1
  • bit 4: 1 + 0 + carry 1 = 0, carry 1
  • bit 5: 1 + 1 + carry 1 = 1, carry 1
  • bit 6: 0 + 1 + carry 1 = 0, carry 1
  • bit 7: 1 + 0 + carry 1 = 0, carry 1 (a 1 is carried out beyond the register)

Result stored in the 8-bit register: 00100001, with an extra 1 carried out beyond bit 7.

Check using denary: 10110110 = 182, 01101011 = 107, and 182 + 107 = 289. Since 289 needs 9 bits to represent exactly, only the lowest 8 bits fit in the register: 289 - 256 = 33, which equals 00100001. The extra 1 (worth 256) is lost.

Part (c): Overflow in part (b)

An overflow error does occur in part (b). The true sum, 289, is larger than the largest value an 8-bit register can hold (255), so a 1 had to be carried out beyond the bit 7 column during the addition. Since the register can only store 8 bits, that carried-out bit is lost, leaving the register holding 00100001 (33) instead of the correct value, 289. (By contrast, part (a)‘s addition produced a final carry of 0 out of bit 7, so no overflow occurs there.)

Final answers

  • (a) 01011011 + 00100101 = 10000000 (no overflow)
  • (b) 10110110 + 01101011 = 00100001, with a 1 carried out of the register
  • (c) Yes, an overflow error occurs, because the correct sum (289) needs 9 bits and cannot fit in the 8-bit register.