Question

In which of the following pairs do both numbers contain the same number of significant figures? a. \(2.0500 \mathrm{~m}\) and \(0.0205 \mathrm{~m}\) b. \(600.0 \mathrm{~K}\) and \(60 \mathrm{~K}\) c. \(0.00075 \mathrm{~s}\) and \(75000 \mathrm{~s}\) d. \(6.240 \mathrm{~L}\) and \(6.240 \times 10^{-2} \mathrm{~L}\)

Short answer

Pairs (c) and (d) have the same number of significant figures.

Step-by-step solution

- Count significant figures in (a)

In the pair (a), count the significant figures in each number:- For 2.0500 m: It has 5 significant figures.- For 0.0205 m: It has 3 significant figures.

- Count significant figures in (b)

In the pair (b), count the significant figures in each number:- For 600.0 K: It has 4 significant figures.- For 60 K: It has 1 significant figure.

- Count significant figures in (c)

In the pair (c), count the significant figures in each number:- For 0.00075 s: It has 2 significant figures.- For 75000 s: It has 2 significant figures.

- Count significant figures in (d)

In the pair (d), count the significant figures in each number:- For 6.240 L: It has 4 significant figures.- For 6.240 × 10^{-2} L: It also has 4 significant figures.

- Compare and conclude

Compare the significant figures in each pair:- Pair (a) has different significant figures (5 and 3).- Pair (b) has different significant figures (4 and 1).- Pair (c) has the same significant figures (2 and 2).- Pair (d) has the same significant figures (4 and 4).Pairs (c) and (d) both have the same number of significant figures in each number.

Key concepts

Measurement Accuracy

Measurement accuracy describes how close a measured value is to the true value. For example, if you are measuring the length of a table and you get 2 meters, but the actual length is 2.05 meters, the accuracy of your measurement is how close that 2 meters figure is to 2.05 meters.
Several factors can affect measurement accuracy:

• Quality of the measuring instrument.
• Skill of the person taking the measurement.
• Environmental conditions, such as temperature.

To ensure high measurement accuracy, it's important to use calibrated instruments, proper techniques, and account for any possible errors.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It's commonly used in science and engineering to make these numbers easier to work with.
A number in scientific notation is written as the product of two numbers: a coefficient and a power of 10. For example, the number 6,240 can be written as 6.240 × 10^3.
Here are some key points:

• Coefficient: A number between 1 and 10.
• Exponent: Indicates how many times to multiply the coefficient by 10.

Using scientific notation helps maintain precision by clearly showing significant figures. For example, 6.240 × 10^-2 maintains all four significant figures from 6.240.

Precision in Measurements

Precision refers to how consistent repeated measurements are with each other. It is different from accuracy, which is about how close a measurement is to the true value.
Consider the following:

• If you measure the same object multiple times and get nearly the same result each time, your measurements are precise.
• Precision is often indicated by the number of significant figures; more significant figures usually mean a more precise measurement.

For example, in the original exercise, the number 2.0500 meters has five significant figures, which indicates a high level of precision compared to 0.0205 meters which has three significant figures.

Counting Significant Figures

Counting significant figures in a number helps determine its precision and how reliable it is. Here are the guidelines for counting significant figures:

• All non-zero digits are significant (e.g., 123 has three significant figures).
• Any zeros between significant digits are significant (e.g., 1002 has four significant figures).
• Leading zeros are not significant (e.g., 0.0025 has two significant figures).
• Trailing zeros in a decimal number are significant (e.g., 12.300 has five significant figures).

In the original exercise, for example, 6.240 and 6.240 × 10^-2 both have four significant figures, retaining their precision regardless of their actual value.