Bounds and Error Intervals

TL;DR

When a number is rounded, it actually represents a range of possible values. We find the lower bound and upper bound to define this range, creating an error interval. This helps us understand the maximum and minimum possible values when calculations involve rounded numbers.

1. The Mental Model

Think of rounding like putting a fence around a number. The "fence posts" are the lower and upper bounds, and everything inside that fence is a possible value the original unrounded number could have been.

2. The Core Material

When a number is rounded, it means we don't know its exact value, only that it falls within a certain range. We use bounds to define this range.

Understanding Lower and Upper Bounds

Let's say a number has been rounded to a certain degree of accuracy (e.g., to the nearest whole number, to one decimal place, or to the nearest 10).

• Lower Bound (LB): This is the smallest possible value the original number could have been before rounding up to the given rounded value.
• Upper Bound (UB): This is the largest possible value the original number could have been before rounding down to the given rounded value.

The error interval is written using inequalities: LB ≤ x < UB. Notice that the upper bound uses < because if the number was exactly the upper bound, it would round up to the next rounded value, not the one we're currently considering.

How to Find the Bounds

Here's a step-by-step process:

1. Identify the degree of accuracy: What was the number rounded to? (e.g., nearest 1, nearest 0.1, nearest 10, nearest 0.01).
2. Calculate the 'half-unit': Divide the degree of accuracy by 2. This is how far above and below the rounded number your bounds will be.
3. Find the Lower Bound: Subtract the half-unit from the rounded number.
4. Find the Upper Bound: Add the half-unit to the rounded number.

Let's look at an example:

Example 1: Rounded to the nearest whole number
A length is measured as 5 cm (to the nearest whole number).

1. Degree of accuracy: Nearest 1 cm.
2. Half-unit: 1 cm / 2 = 0.5 cm.
3. Lower Bound: 5 - 0.5 = 4.5 cm.
4. Upper Bound: 5 + 0.5 = 5.5 cm.
5. Error Interval: 4.5 cm ≤ length < 5.5 cm

This means the actual length was anything from 4.5 cm (inclusive) up to, but not including, 5.5 cm.

Example 2: Rounded to one decimal place (1 d.p.)
A weight is measured as 3.2 kg (to 1 d.p.).

1. Degree of accuracy: Nearest 0.1 kg.
2. Half-unit: 0.1 kg / 2 = 0.05 kg.
3. Lower Bound: 3.2 - 0.05 = 3.15 kg.
4. Upper Bound: 3.2 + 0.05 = 3.25 kg.
5. Error Interval: 3.15 kg ≤ weight < 3.25 kg

Example 3: Rounded to the nearest 10
A crowd size is 230 people (to the nearest 10).

1. Degree of accuracy: Nearest 10 people.
2. Half-unit: 10 / 2 = 5 people.
3. Lower Bound: 230 - 5 = 225 people.
4. Upper Bound: 230 + 5 = 235 people.
5. Error Interval: 225 people ≤ crowd < 235 people

graph TD
    A["Start with Rounded Number and Accuracy"] --> B["Identify Degree of Accuracy (DOA)"]
    B --> C{"Is DOA 'nearest whole', '1 d.p.', 'nearest 10' etc.?"}
    C --> D["Calculate Half-Unit (DOA / 2)"]
    D --> E["Lower Bound = Rounded Number - Half-Unit"]
    D --> F["Upper Bound = Rounded Number + Half-Unit"]
    E --> G["Form Error Interval (LB ≤ x < UB)"]
    F --> G
    G --> H["End"]

Performing Calculations with Bounds

When you perform calculations (add, subtract, multiply, divide) with numbers that are themselves rounded, you need to think about how to get the minimum and maximum possible results.

• To find the Maximum Result: Use the largest possible values (UBs) and smallest possible values (LBs) in a way that maximises the outcome.
• To find the Minimum Result: Use the smallest possible values (LBs) and largest possible values (UBs) in a way that minimises the outcome.

Here's a quick guide:

Notice for subtraction and division, to get the maximum result, you want the first number to be as big as possible (UB) and the second number to be as small as possible (LB). To get the minimum result, it's the opposite.

3. Worked Example

A rectangular field has a length of 50 m and a width of 20 m, both measured to the nearest metre.
Calculate the maximum possible area of the field.

Step 1: Find the bounds for the length.
Length = 50 m (nearest metre).
Degree of accuracy = 1 m.
Half-unit = 1 / 2 = 0.5 m.
Lower Bound for Length (LB_L) = 50 - 0.5 = 49.5 m.
Upper Bound for Length (UB_L) = 50 + 0.5 = 50.5 m.
Error interval for Length: 49.5 m ≤ L < 50.5 m

Step 2: Find the bounds for the width.
Width = 20 m (nearest metre).
Degree of accuracy = 1 m.
Half-unit = 1 / 2 = 0.5 m.
Lower Bound for Width (LB_W) = 20 - 0.5 = 19.5 m.
Upper Bound for Width (UB_W) = 20 + 0.5 = 20.5 m.
Error interval for Width: 19.5 m ≤ W < 20.5 m

Step 3: Calculate the maximum possible area.
Area = Length x Width.
To get the maximum area, we need to multiply the maximum possible length by the maximum possible width.
Maximum Area = UB_L x UB_W
Maximum Area = 50.5 m x 20.5 m
Maximum Area = 1035.25 m²

So, the maximum possible area of the field is 1035.25 m².

4. Key Takeaways

• Every rounded number represents a range, not a single exact value.
• The lower bound (LB) is the smallest value the original number could have been.
• The upper bound (UB) is the largest value the original number could have been before rounding down.
• The error interval is always written as LB ≤ x < UB.
• To find bounds, halve the degree of accuracy, then add/subtract it.
• When calculating with bounds, think about whether you need the maximum or minimum possible outcome for the final result.

Common Mistakes to Avoid:
- Forgetting the 'less than' symbol (<) for the upper bound in the error interval.
- Confusing which bounds to use (LB or UB) when calculating max/min results for subtraction or division.
- Miscalculating the 'half-unit' by not correctly identifying the degree of accuracy (e.g., using 1 for nearest 0.1).
- Rounding your final bound calculations; bounds are usually very precise.

5. Now Try It

A time is recorded as 4.7 seconds, rounded to one decimal place. Another time is recorded as 12 seconds, rounded to the nearest second.
Calculate the maximum possible difference between these two times.

To succeed, you'll need to:
1. Find the lower and upper bounds for the 4.7-second measurement.
2. Find the lower and upper bounds for the 12-second measurement.
3. Determine which combination of bounds (from the subtraction table) will give you the maximum possible difference.
4. Calculate that maximum difference.