4.5.4.9 Underflow and overflow Quiz: AQA Computer Science, Unit 5

The 20 questions

1. What is overflow in a number representation?

   • A result whose magnitude is too small to be told apart from zero
   • An error caused by a sign bit set to zero
   • A result whose magnitude is too large to be stored in the available bits
   • A result that has been rounded to fewer digits

2. What is underflow in a number representation?

   • A result whose magnitude is too small to be represented, so it becomes zero or is lost
   • A carry out of the most significant bit, which shows that the true sum needs an extra bit
   • A result that is too large for the exponent field, so the stored value wraps around to zero
   • A value stored with more bits than it needs, which wastes space in the representation used

3. In floating point, when does underflow most commonly occur?

   • When a number is converted from binary to decimal, which can lose the final digits in the output
   • When two positive values are added together and the sum is larger than the format can hold
   • When a mantissa has more bits than the format allows, so the extra bits are simply dropped
   • When a very small value needs an exponent below the smallest that can be stored

4. In which circumstance does overflow occur in unsigned integer addition?

   • When the sum needs more bits than are available, so a carry out of the top bit is produced
   • When the sum is less than zero, because an unsigned result cannot be stored in the register at all
   • When the two numbers being added are equal, because the adder then produces a carry every time
   • When a fraction is added to an integer, since the fraction must be stored in a separate register

5. What happens in two's complement when a result exceeds the largest positive value?

   • The result is rounded to the nearest representable value without error
   • The result wraps around and gives an incorrect negative value, which is overflow
   • The result is stored correctly as a larger positive number
   • The sign bit is ignored and the result stays positive

6. Which is an example of underflow?

   • A timestamp stored with more bits than it needs, so the extra bits are never used by the system
   • A large population stored in a 16-bit unsigned integer that cannot hold the full count of people
   • The sum of 255 and 1 in 8-bit unsigned binary, which needs more bits than the eight available
   • A very small probability stored as a floating point value below the smallest representable magnitude

7. Which is an example of overflow?

   • Adding 1 to 255 in 8-bit unsigned binary, which needs 9 bits to store
   • Converting 1010 from binary to decimal, which gives the value 10 without any loss of precision
   • Storing 0.5 in a fixed point format that has enough fractional bits to hold it exactly
   • Storing 0 in unsigned binary, where the all-zero pattern is the smallest value the format holds

8. Does 9 + 8 overflow in 4-bit unsigned binary?

   • Yes, since 17 exceeds the 4-bit maximum of 15
   • No, since 17 is less than 32, the value of two to the power of five
   • Only if the sum is negative, because unsigned binary can only overflow below zero
   • No, since 9 + 8 equals 1 in 4 bits when the carry is kept as part of the sum

9. A value of 10^-41 is stored in a floating point format whose smallest positive magnitude is 10^-38. What happens?

   • It overflows into the sign bit
   • It underflows and is stored as zero or lost
   • It overflows and becomes infinity
   • It is stored exactly because it is small

10. A 4-bit two's complement register holds 0111 (7) and adds 1. What is the result?

    • 0111, unchanged
    • 1000, which is -8, so the result overflows
    • 1000, which is +8 correctly
    • 0000, with a carry out and no error

11. Subtract 5 from 3 in 4-bit unsigned binary. What happens?

    • The result cannot be negative, so it wraps to 1110, which is 14
    • The result is 1110, which is -2 correctly, because the subtraction is performed in two's complement form
    • The result is 0010, which is 2, because the smaller operand is subtracted from the larger in the register
    • The result is 0000 with no error, since the subtraction saturates at zero for any negative value

12. A floating point exponent is too large to store in its field. What is this error called?

    • Overflow
    • Normalisation
    • Underflow
    • Rounding error

13. An 8-bit two's complement value is 127 and 1 is added. What is the result?

    • 128, which is correct
    • -127
    • -128, which is an overflow
    • 0, with no overflow

14. Does adding 64 and 63 overflow in 8-bit unsigned binary?

    • No, since 127 is within the 8-bit range
    • Yes, since 64 + 63 exceeds 64
    • Yes, since 127 needs 8 bits and overflows
    • No, because a carry is never an overflow

15. Which check detects overflow in unsigned addition of two n-bit numbers?

    • The sign bit of the result being 1
    • The result having an odd number of 1s
    • A carry out of the most significant bit
    • The result being less than the first operand

16. Why can overflow in two's complement be detected when two positive numbers are added?

    • Two positive numbers cannot sum to a negative value, so a negative sign bit signals that the true sum is out of range
    • Overflow is detected by checking whether the result is zero, since a zero result can only come from wrapping
    • A carry out of the top bit always means overflow in two's complement, so carries are checked first
    • A negative sign bit always shows a correct answer, so the sign alone confirms the result is valid

17. Why do rounding and underflow both matter in scientific computing?

    • They are identical errors, so a single check for one of them also covers the other kind
    • They never occur together, since rounding removes underflow entirely from any stored result
    • They only affect integers, so fractions can be ignored safely in any scientific calculation
    • Very small values can be lost to zero and rounding accumulates, so results may be inaccurate

18. An 8-bit unsigned counter is at 254 and increments three times. What value is shown, and what happens?

    • 257, which needs 9 bits and is stored correctly
    • 0, with no overflow
    • 255, since the counter stops at its maximum
    • 1, after wrapping past 255, with an overflow event

19. Which statement about underflow in floating point is correct?

    • Underflow is avoided entirely by normalising the mantissa, since the leading bit is always kept
    • Underflow always produces a larger value than the true result, because the stored value is rounded up
    • Values below the smallest representable magnitude are lost, often becoming zero
    • Underflow only happens with negative exponents in fixed point, never in any floating point format

20. An accumulator is 16 bits wide, and a running sum exceeds 65535. What is the best description?

    • Normalisation, because the running sum must be shifted left until its leading bit is one again
    • Underflow, because the running sum is too small to store in the sixteen bits available to it
    • Rounding error only, since no bit was lost when the running sum was held in the accumulator
    • Overflow of the accumulator, since the running sum needs more bits than the 16-bit container

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