Problem 73 - Introductory Chemistry Atoms First

Question

Using scientific notation, write the measurement \(30 \mathrm{ft}\) as having an uncertainty of: (a) \(\pm 1 \mathrm{ft}\) (b) \(\pm 0.1 \mathrm{ft}\) (c) \(\pm 0.01 \mathrm{ft}\)

Step-by-Step Solution

Answer

(a) \(3 \times 10^1 \mathrm{ft} \pm 1 \mathrm{ft}\) (b) \(3 \times 10^1 \mathrm{ft} \pm 1 \times 10^{-1} \mathrm{ft}\) (c) \(3 \times 10^1 \mathrm{ft} \pm 1 \times 10^{-2} \mathrm{ft}\)

Step 1: Convert measurement into scientific notation

To convert the value \(30 \mathrm{ft}\) into scientific notation, we first identify \(a\) and \(b\) in the form \(a \times 10^b\), such that \(a\) has only one non-zero digit in front of the decimal and \(b\) is an integer. For the given value, \(a = 3\) and \(b = 1\), so the scientific notation for \(30 \mathrm{ft}\) is \(3 \times 10^1 \mathrm{ft}\). Now let's add uncertainty to this value. (a) For an uncertainty of \(\pm 1 \mathrm{ft}\):

Step 2: Include uncertainty

We can add the uncertainty to the measurement by expressing it in the same form as our scientific notation. Here, our uncertainty is \(1 \mathrm{ft}\). So, the scientific notation for the measurement with this uncertainty is: \(3 \times 10^1 \mathrm{ft} \pm 1 \mathrm{ft}\) (b) For an uncertainty of \(\pm 0.1 \mathrm{ft}\):

Step 3: Include uncertainty

We can add the uncertainty to the measurement by expressing it in the same form as our scientific notation. Here, our uncertainty is \(0.1 \mathrm{ft}\), which corresponds to \(1 \times 10^{-1} \mathrm{ft}\). So, the scientific notation for the measurement with this uncertainty is: \(3 \times 10^1 \mathrm{ft} \pm 1 \times 10^{-1} \mathrm{ft}\) (c) For an uncertainty of \(\pm 0.01 \mathrm{ft}\):

Step 4: Include uncertainty

We can add the uncertainty to the measurement by expressing it in the same form as our scientific notation. Here, our uncertainty is \(0.01 \mathrm{ft}\), which corresponds to \(1 \times 10^{-2} \mathrm{ft}\). So, the scientific notation for the measurement with this uncertainty is: \(3 \times 10^1 \mathrm{ft} \pm 1 \times 10^{-2} \mathrm{ft}\)

Key Concepts

Measurement Uncertainty

Understanding measurement uncertainty is crucial for students and professionals alike. It reflects the acknowledgment that no measurement can be exact and that there is always a margin of error. When we say that a length is \(30 \mathrm{ft}\), we imply that it is not precisely \(30 \mathrm{ft}\) but somewhere in a range around that value. This range is the uncertainty of the measurement.

For example, when we have an uncertainty of \(\pm 1 \mathrm{ft}\), this means the true value could be as low as \(29 \mathrm{ft}\) or as high as \(31 \mathrm{ft}\). This understanding is crucial in fields such as engineering, physics, and even finance, where precise measurements are fundamental.

Expressing Uncertainty

Expressing uncertainty is about communicating the range within which the actual measurement lies. This is typically shown as a \(\pm\) following the measured value, indicating the margin of error. For instance, \(30 \mathrm{ft} \pm 1 \mathrm{ft}\) conveys that the value could be one foot less or one foot more than 30 feet.

The choice of whether to express uncertainty in standard form or scientific notation usually depends on the precision required and the context of the measurement. In science, particularly, using scientific notation to express uncertainty helps maintain consistency of format, especially when dealing with very large or very small quantities, and it aligns with the precision of the instruments used to make the measurement.

Scientific Notation Conversion

Scientific notation is a concise way to express both very large and very small numbers. It simplifies calculations and comparisons between such numbers. Converting a number to scientific notation involves identifying a coefficient \(a\) that is greater than or equal to 1 and less than 10, and a power of 10 \(10^b\) that brings the coefficient to the original number's scale. For instance, \(30 \mathrm{ft}\) becomes \(3 \times 10^1 \mathrm{ft}\) in scientific notation.

When converting measurements with uncertainty into scientific notation, both the central value and the uncertainty should be expressed in scientific notation. This is seen in how \(30 \mathrm{ft} \pm 0.01 \mathrm{ft}\) is correctly written as \(3 \times 10^1 \mathrm{ft} \pm 1 \times 10^{-2} \mathrm{ft}\). This forms a consistent method to represent data with precision and is essential for effectively communicating measurements in science and engineering.