Upper and lower bounds Notes

Lower and Upper Bounds in Measurement

From a Single Number to an Interval

1. When a value is reported as a rounded measurement (for example, “16 m to the nearest metre”), the true value is not exactly known.
2. Instead, it lies in an interval of possible values.
   1. The lower bound is the smallest possible value.
   2. The upper bound is the largest possible value (not included for continuous data).
3. Together, these are called the limits of accuracy.

Continuous Measurements Create Ranges, Not Single Values

1. Many real measurements are continuous: they can take any value in an interval (for example, height, mass, time, temperature).
2. This is different from discrete quantities, which come in fixed steps (like number of students in a class).
3. Because continuous measurements are recorded to a stated accuracy (nearest metre, nearest 0.1 kg, 2 significant figures, etc.), you should treat the recorded number as a label for an interval of possible true values, not as an exact value.

How to Find Bounds from a Rounded Value

1. Rounding works by grouping values around a midpoint.
2. Suppose a measurement is rounded to the nearest unit of size $a$.
3. Examples of step size $a$:
   1. Nearest 10 → $a=10$
   2. Nearest 0.1 → $a=0.1$
   3. Nearest 0.01 → $a=0.01$
4. Then the half-step is $\frac{a}{2}​$.
5. If the recorded (rounded) value is $x$, then
   1. Lower bound = $x−\frac{a}{2}$
   2. Upper bound = $x+\frac{a}{2}$
6. So the true value TT satisfies $$x-\frac{a}{2} \leq T<x+\frac{a}{2}$$
7. We usually include the lower bound but exclude the upper bound.
8. This matches the rule that values exactly halfway ($\frac{a}{2}$ ​above the midpoint) round up.

Choosing the Correct Inequality Signs

The inequality signs matter because of how rounding behaves at the midpoint.

Bounds for Decimals and Significant Figures

Nearest Decimal Place

1. If a length is 1.6 m to the nearest 0.1 m, then
   1. Step size: $a=0.1$
   2. Half-step: $\frac{a}{2}=0.05$
2. So the true length $L$ satisfies $1.55 \leq L < 1.65$$
3. The lower bound is $1.55 \text{ m}$.

Significant Figures

With significant figures, the step size depends on the place value of the last significant digit.

Using Bounds in Calculations

1. Once you have lower and upper bounds for quantities, you can use them to find maximum and minimum possible results of calculations.
2. Let $A$ and $B$ be quantities with bounds. To:

Maximize the result

1. $A+B$: use upper bound + upper bound
2. $A−B$: use upper bound − lower bound
3. $A\times B$: use upper bound × upper bound (if both are positive)
4. $\frac{A}{B}$: use upper bound ÷ lower bound (if both are positive)

Minimize the result

1. $A+B$: use lower bound + lower bound
2. $A−B$: use lower bound − upper bound
3. $A\times B$: use lower bound × lower bound (if both are positive)
4. $\frac{A}{B}$: use lower bound ÷ upper bound (if both are positive)

These rules match the summary table: to make an answer as big as possible, combine the bounds in a way that pushes the result up; to make it as small as possible, combine them to push the result down.

Why Bounds Matter in Real Decisions

1. A lower bound is often the safest guaranteed minimum you can claim.
2. If a crate is labelled 40 kg to the nearest 0.5 kg, the minimum possible mass is the lower bound, $39.75 \text{ kg}$.
3. When measurements are used for safety limits, capacity planning or costs, working with lower and upper bounds helps avoid over-estimating what is guaranteed.

Building and Interpreting Bounds on a Number Line

1. A number line helps you visualize the interval idea.
2. Mark the stated rounded value in the centre.
3. Move left and right by half the rounding step.
4. Mark:
   1. a solid (closed) point at the lower bound (included),
   2. an open point at the upper bound (excluded).