Exploring Upper and Lower Bounds: Defining Precision

When we measure or round numbers, we often don't get the exact value. Upper and lower bounds help us understand the range within which the true value lies. Think of it as knowing that your friend is somewhere between $1.45 \text{meters}$ and $1.55 \text{meters}$ tall if they tell you they are about $1.5 \text{meters}$.

What are Upper and Lower Bounds?

• Lower Bound: The smallest value that rounds up to the given number.
• Upper Bound: The largest value that rounds down to the given number.

By knowing both bounds, we can say with certainty that the true value lies somewhere between them.

How to Find Upper and Lower Bounds

Step-by-Step Guide

1. Identify the Level of Accuracy: Determine the unit to which the number has been rounded (e.g., nearest whole number, decimal place, or significant figure).

2. Find the Half Unit:

   • Calculate half of the unit to which the number was rounded.
   • Example: $\text{Half Unit} = \frac{\text{Unit}}{2}$
     
     • If rounded to the nearest whole number, half unit is $0.5$
     • If rounded to the nearest tenth (0.1), half unit is $0.05$
3. Calculate the Lower Bound: $$\text{Lower Bound} = \text{Rounded Value} - \text{Half Unit}$$
4. Calculate the Upper Bound: $$\text{Upper Bound} = \text{Rounded Value} + \text{Half Unit}$$

Why are Upper and Lower Bounds Important?

Understanding the bounds of a value is crucial in fields requiring high precision. It allows engineers, scientists, and mathematicians to assess the reliability of their measurements and calculations, ensuring that structures are safe, medicines are effective, and scientific data are accurate.

Example

Find the upper and lower bounds of 5.6 rounded to the nearest tenth.

1. Level of Accuracy: Nearest tenth (0.1).

2. Half Unit: $$\text{Half Unit} = \frac{0.1}{2} = 0.05$$

3. Lower Bound: $$\text{Lower Bound} = 5.6 - 0.05 = 5.55$$
4. Upper Bound: $$\text{Upper Bound} = 5.6 + 0.05 = 5.65$$

So, the true value lies between $5.55$ and $5.65$

 

Using Bounds in Calculations

When performing calculations with rounded numbers, we need to consider how the bounds affect the result.

Example: Adding Measurements

Two lengths are measured as 3.2 meters and 4.7 meters, both to the nearest tenth. Find the upper and lower bounds of their sum.

1. Find Half Unit: $$\text{Half Unit} = \frac{0.1}{2} = 0.05$$

2. Find Bounds for Each Measurement:

   • First Length: $$\text{Lower Bound}_1 = 3.2 - 0.05 = 3.15 \\ \text{Upper Bound}_1 = 3.2 + 0.05 = 3.25$$

   • Second Length: $$\text{Lower Bound}_2 = 4.7 - 0.05 = 4.65 \\ \text{Upper Bound}_2 = 4.7 + 0.05 = 4.75$$
     
     

3. Calculate Bounds of Sum:

   • Lower Bound of Sum: $$\text{Lower Sum} = \text{Lower Bound}_1 + \text{Lower Bound}_2 = 3.15 + 4.65 = 7.8$$
   • Upper Bound of Sum: $$\text{Upper Sum} = \text{Upper Bound}_1 + \text{Upper Bound}_2 = 3.25 + 4.75 = 8.0$$

So the sum lies between $7.8$ meters and $8.0$ meters

 

Example: Multiplying Measurements

A rectangle has a length of 8 cm (nearest cm) and a width of 5 cm (nearest cm). Find the upper and lower bounds for the area.

1. Find Half Unit:  $$\text{Half Unit} = \frac{1}{2} = 0.5 \text{ cm}$$

2. Find Bounds for Each Measurement:

   • Length: $$\text{Lower Length} = 8 - 0.5 = 7.5 \text{ cm} \\ \text{Upper Length} = 8 + 0.5 = 8.5 \text{ cm}$$

   • Width: $$\text{Lower Width} = 5 - 0.5 = 4.5 \text{ cm} \\ \text{Upper Width} = 5 + 0.5 = 5.5 \text{ cm}$$
     
     

3. Calculate Bounds of Area:

   • Lower Bound of Area: $$\text{Lower Area} = \text{Lower Length} \times \text{Lower Width} = 7.5 \times 4.5 = 33.75 \text{ cm}^2$$
   • Upper Bound of Area: $$\text{Upper Area} = \text{Upper Length} \times \text{Upper Width} = 8.5 \times 5.5 = 46.75 \text{ cm}^2$$

Sp the area is between $33.75 cm^2 \quad \text{and} \quad 46.75cm^2$