Bounds

Understanding Bounds

• Bounds refer to the upper and lower limits of accuracy for an rounded or approximated measurement.
• The aim is to estimate the absolute possible range of a given value or result.
• For any number, the lower bound is typically slightly less than the number itself, while the upper bound will be slightly more. The exact difference is usually half of the rounding unit (the amount you are rounding to).

Significance of Bounds

• Bounds are particularly important when we are dealing with numbers that are rounded or approximated, as they can impact the accuracy of further calculations or decisions based on such values.
• If we know the bounds of a value, we are better informed about the possible range of the actual, unrounded value.
• Calculations involving bounds are often concerned with worst-case and best-case scenarios.

Calculation of Bounds

• The Upper Bound is calculated as the given number plus half of the unit of rounding.
• The Lower Bound is calculated as the given number minus half of the unit of rounding.
• For example, consider a number rounded to the nearest tenth, say, 3.6. The unit of rounding is 0.1 (a tenth). So, the upper and lower bounds would be:
  • Upper Bound = 3.6 + 0.05 = 3.65
  • Lower Bound = 3.6 - 0.05 = 3.55

Practice Problems

• Problem: What are the bounds of the number 12, rounded to the nearest whole number?
  • Solution: Lower Bound = 11.5, Upper Bound = 12.5
• Problem: Determine the bounds of the number 7.8, rounded to the nearest tenth.
  • Solution: Lower Bound = 7.75, Upper Bound = 7.85

Key Points to Remember

• Be comfortable with the concept of rounding and place values to calculate the unit of rounding.
• Practice often as this helps in understanding how to calculate bounds and when to apply the concept.
• Remember that bounds give an estimate of the range of possible values an approximated or rounded number can take.
• Be aware that bounds play a crucial role in analysing worst-case and best-case scenarios in various calculations.