Upper & Lower Bounds | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Bounds & Error Intervals

What are bounds?

• Bounds refer to the smallest (lower bound, LB) and largest (upper bound, UB) numbers that a rounded number can lie between.
  • They indicate how low or high the number could have been before it was rounded.
• The bounds for a number, x, are expressed as LB ≤ x < UB.
  • Note that the lower bound is included in the range of possible values for x, but the upper bound is not.

How do we find bounds when a number has been rounded?

• The basic rule is "Half Up, Half Down."
  • Upper Bound: To find the upper bound, add half the degree of accuracy.
  • Lower Bound: To find the lower bound, subtract half the degree of accuracy.
  • Error Interval: LB ≤ x < UB.
• It's tempting to think that the upper bound should end in a 9, or 99, etc., but the Error Interval LB ≤ x < UB does not include the upper bound.
  • The upper bound is the cut-off point for the greatest value that the number could have been rounded from but will not actually round to the number itself.

Calculations using Bounds

• Adding Numbers: $𝑇=𝑎+𝑏$T = a + b
  • Upper Bound: Add the upper bound of $𝑎$a and the upper bound of $𝑏$b.
  • Lower Bound: Add the lower bound of $𝑎$a and the lower bound of $𝑏$b.
• Subtracting Numbers: $𝑇=𝑎-𝑏$T = a − b
  • Upper Bound: Use the upper bound of $𝑎$a and subtract the lower bound of $𝑏$b.
  • Lower Bound: Use the lower bound of $𝑎$a and subtract the upper bound of $𝑏$b.
• Multiplying Numbers: $𝑇=𝑎\times 𝑏$T = a × b
  • Upper Bound: Multiply the upper bound of $𝑎$a and the upper bound of $𝑏$b.
  • Lower Bound: Multiply the lower bound of a and the lower bound of $𝑏$b.
• Dividing Numbers: $𝑇=𝑎÷𝑏$T = a ÷ b
  • Upper Bound: Use the upper bound of $𝑎$a and divide it by the lower bound of $𝑏$b.
  • Lower Bound: Use the lower bound of $𝑎$a and divide it by the upper bound of $𝑏$b.

How can bounds help with calculations?

• You can utilize bounds to determine the level of accuracy in a calculation.
• This helps in deciding how to round your answer. 
  • For example, if the lower bound of a value is 8.33217 and the upper bound is 8.33198, the true value lies between these bounds.
  • When both bounds round to 8.332 to 4 significant figures, any discrepancies when rounded to 5 significant figures become apparent.
  • This ensures that the answer unequivocally rounds to 8.332 with 4 significant figures.