Upper and lower bounds (Higher Tier)

Understanding Upper and Lower Bounds

Bounds are values which are known to be either larger or smaller than a given number.
The upper bound of a number is the smallest number that is greater than the original number.
The lower bound of a number is the largest number that is less than the original number.
Every real number has both an upper and lower bound.

When dealing with measurements which are rounded, the rounded measurement cannot provide the exact value but offers a range. This is where the concept of bounds can be useful.
Mostly, when the bounds of a measurement are referred to, it is regarding the maximum error or the possible range of the true value due to rounding.
For example, if a length is measured to be 5cm to the nearest cm, the lower bound of the length is 4.5cm and the upper bound is 5.5cm. This suggests the true value lies somewhere within 4.5cm and 5.5cm.

Depending on how a number is rounded (whether to a significant figure or a decimal place), the upper and lower bounds might differ.
When rounding to a significant figure, the bounds usually result in varying numbers of decimal places. eg. if 127 is rounded to 100 (to 1 significant figure), the lower bound is 50 and the upper bound is 150.
When rounding to a decimal place, the last decimal place of the upper and lower bounds is always one-half of the place value you’re rounding to.

When performing calculations with measurements that have been rounded, the result is subject to the same uncertainties as the original measurements. Therefore, the accuracy of the result can also be described in terms of bounds.
A general rule is that the maximum possible value (upper bound) of a result is calculated using the maximum possible values of the inputs, and the minimum possible value (lower bound) is calculated using the minimum possible values of the inputs.