Problem 26 - Chemistry: Principles and Reactions

Question

A listing of a house for sale states that there are 5 bedrooms, \(4000 \mathrm{ft}^{2}\) of living area, and a living room with dimensions \(17 \times 18.5 \mathrm{ft}\). How many significant figures are there in each quantity? (Your answer may include the words ambiguous and exact.)

Step-by-Step Solution

Answer

Based on the given information, determine the number of significant figures for each of the following quantities: the number of bedrooms, the living area in square feet, and the living room dimensions in feet. Answer: 1. Number of bedrooms: 1 significant figure. 2. Living area: 1 significant figure. 3. Living room dimensions: 2 significant figures for 17 ft and 3 significant figures for 18.5 ft.

Step 1: Determine significance of the number of bedrooms

There are 5 bedrooms mentioned in the listing. Since this is an integer, it would typically be considered to have an exact value, as it's not a measurement with potential variability. Number of bedrooms has 1 significant figure.

Step 2: Determine significance of the living area

The living area is given as \(4000 \mathrm{ft}^{2}\). Since there are no decimals, we count the number of non-zero significant digits. In this case, there is only one significant figure, which is the 4 in 4000. However, this number might be ambiguous because it could have been rounded down, and the exact value might contain more significant figures. Without further information, we cannot assume additional significant figures.

Step 3: Determine significance of living room dimensions

The dimensions of the living room are given as \(17 \times 18.5 \mathrm{ft}\). In this case, both numbers are significant since they include a decimal point, and we count all the digits. The dimension 17 has an exact value, so it would have 2 significant figures. The other dimension, 18.5, has a decimal, so every digit is significant, giving it 3 significant figures in total. To summarize: 1. Number of bedrooms: 1 significant figure (exact). 2. Living area: 1 significant figure (ambiguous). 3. Living room dimensions: 2 significant figures for 17 (exact) and 3 significant figures for 18.5.

Key Concepts

Exact Numbers

Exact numbers are values that are known with complete certainty. Such numbers arise from counting actual items or are defined values within calculations. For instance, when we say there are 5 bedrooms, this number is exact. It is a count of individual, whole items – bedrooms, which don’t change based on measurement technique or instrument. Thus, these numbers do not involve any estimation or uncertainty.
In scientific notation and computations, exact numbers can be treated as having an infinite number of significant figures. This means they do not limit the precision of a calculation that includes measured values. Here are a couple of points to consider:

• Exact numbers arise from definitions, such as 12 inches in a foot.
• They also come from direct counts, like counting 5 apples.

It's important to note that exact numbers stand out because they are not derived from measuring something with a tool, hence reducing the potential for errors.

Measurement Precision

Measurement precision refers to how finely a measurement is expressed or the detail of the measurement. It reflects the smallest unit of measurement that can be reliably identified or measured. Precision plays a crucial role in determining significant figures of a number because it indicates how much detail or confidence there is in the value provided.
For example, when we consider the living room dimension of 18.5 ft, this number has three significant figures, showing that it's known quite precisely and hasn't been rounded in a way that loses detail. The presence of the decimal point in '18.5' further emphasizes this precision. Measuring to one decimal place means that slight changes in measurement, potentially missed if rounding to whole numbers, are captured.

• More significant digits imply a more precise measurement with less uncertainty.
• Measurement tools with finer resolution increase the precision.

This means that when precision is high, you see fewer "jumps" between measured values, and measurements are often more reliable for closer comparisons.

Rounding Ambiguity

Rounding ambiguity occurs when it's unclear how many significant figures a rounded number should represent. This happens when details about the rounding criteria are missing. In some instances, numbers appear without decimal points, like the living area of the house given as 4000 ft², leaving us uncertain whether it's exactly 4000 or a rounded approximation.
To handle such numbers, we often assume they have fewer significant figures unless there's additional context that clarifies precision. It's also important to be cautious as a number like 4000 could have been rounded from 3995 to 4004, thus making assumptions about its precision risky.

• Clarification of rounding intent can prevent misinterpretations.
• Communication about the level of precision can often help reduce rounding ambiguity.

This detail matters in scientific calculations, where results depend on exact numbers for accuracy. Misinterpreting significance due to rounding ambiguity can skew results and lead to significant errors in data interpretation.