4.5.4.5 Rounding errors Quiz: AQA Computer Science, Unit 5

The 20 questions

1. Why can the decimal value 0.1 not be stored exactly in binary?

   • Computers always round every number to the nearest integer before storing it in memory
   • Binary cannot store any digit that comes after the decimal point in a fixed number of bits
   • Its binary form, 0.0001100110011..., is a repeating fraction that never terminates
   • 0.1 is an irrational number, so it has no finite binary form and cannot be stored exactly

2. What is a rounding error?

   • The time taken by the processor to round a number before it is stored in memory
   • The error caused by reading the wrong sign bit when a value is decoded from memory
   • The number of bits lost when a value overflows its available field in the representation
   • The difference between the value stored and the exact value

3. Which representations may produce inaccurate values for decimal numbers?

   • Floating point only
   • Fixed point only
   • Both fixed point and floating point
   • Neither, since computers store decimals exactly

4. For a real number to be represented exactly in binary in a given number of bits, what must be true?

   • It must be an integer, so that every digit after the decimal point is zero
   • It must be a natural number, so that it can be counted exactly in binary form
   • It must have no more than four decimal places, so that rounding is never needed at all
   • It must be representable as a binary fraction in that number of bits

5. What is the absolute error of an approximation?

   • The magnitude of the difference between the exact value and the approximation
   • The square of the difference between the exact value and the approximation, which is always positive
   • The approximation multiplied by one hundred, which gives the size of the error as a percentage
   • The difference divided by the exact value, giving a fraction that is always less than one

6. What is the relative error of an approximation?

   • The absolute error divided by the magnitude of the exact value
   • The absolute error minus one, which gives the size of the error relative to the answer
   • The exact value divided by the absolute error, which shows how many errors fit in the answer
   • The absolute error multiplied by the magnitude of the exact value, which scales the size of the error

7. A value of 0.8 is stored as 0.79. What is the absolute error?

   • 0.02
   • 0.79
   • 0.01
   • 0.1

8. A value of 0.8 is stored as 0.79. What is the relative error?

   • 0.8
   • 0.0125
   • 1.25
   • 0.125

9. An approximation of 1000 is given as 1003. What is the relative error?

   • 0.003
   • 0.3
   • 3
   • 0.03

10. An error of 0.1 occurs in a value of 0.5, and an error of 0.1 occurs in a value of 1000. Which has the larger relative error?

    • Both have equal relative error because the absolute errors match
    • The value 1000, with relative error 0.0001
    • The value 1000, since its absolute error is the same and its magnitude is greater
    • The value 0.5, with relative error 0.2

11. Using 2 fractional bits in fixed point, which value can be stored exactly?

    • 2.1
    • 2.75
    • 2.3
    • 2.6

12. A value of 0.2 is stored as 0.1875 (0.0011 in binary). What is the absolute error?

    • 0.0125
    • 0.3875
    • 0.0075
    • 0.0250

13. A value of 2.6 is stored in fixed point with 2 fractional bits, using the nearest representable value. What is stored, and what is the absolute error?

    • 2.6, with absolute error 0
    • 2.5, with absolute error 0.5
    • 2.75, with absolute error 0.15
    • 2.5, with absolute error 0.1

14. A measurement of 50,000 has an absolute error of 5. What is the relative error?

    • 0.001
    • 0.0001
    • 0.1
    • 0.01

15. A value of 1/3 is stored as 0.333333. Which pair gives the absolute error and relative error approximately?

    • Both are exactly zero because 0.333333 is a finite decimal
    • Absolute error 1.0 x 10^-6, relative error 3.3 x 10^-7
    • Absolute error 0.000333, relative error 0.001
    • Absolute error about 3.3 x 10^-7, relative error about 1.0 x 10^-6

16. Why is relative error more informative than absolute error when comparing a value near 1 with a value near 1,000,000?

    • Absolute error is always zero for numbers near one, so the comparison depends only on large values
    • Absolute error is larger for numbers near one, so it is always the less useful measure to use
    • Relative error does not depend on magnitude at all, so it is never useful for comparing accuracy
    • Relative error scales with the magnitude, so the same absolute error can mean very different accuracy

17. Why might a long sum of fractional values drift away from the exact total?

    • Errors cancel out perfectly whenever the number of terms being added is an even number
    • Each stored value carries a small representation error, and these errors accumulate across operations
    • Floating point addition removes all errors after each step, so the total stays exact throughout
    • Fractions are truncated to integers before they are added, which loses the decimal part entirely

18. A fixed point format with 3 fractional bits stores 1.3 as the nearest representable value. What are the stored value and the absolute error?

    • 1.5, with absolute error 0.2
    • 1.25, with absolute error 0.25
    • 1.375, with absolute error 0.075
    • 1.25, with absolute error 0.05

19. A value of 0.0001 is stored as 0.0002. What is the relative error?

    • 0.5, or 50 percent
    • 1.0, or 100 percent
    • 0.0001, or 0.01 percent
    • 2.0, or 200 percent

20. Which statement about rounding errors is correct?

    • Rounding errors are impossible whenever the number has at least thirty-two bits of storage available
    • Rounding errors can grow when many operations are chained, so results are approximate
    • Rounding errors occur only with integers and never with fractions, so decimals are always exact
    • Rounding errors disappear once a value is stored in binary, because binary has no rounding at all

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