Question

In which of the following pairs do both numbers contain the same number of significant figures? a. \(0.00575 \mathrm{~g}\) and \(5.75 \times 10^{-3} \mathrm{~g}\) b. \(0.0250 \mathrm{~m}\) and \(0.205 \mathrm{~m}\) c. \(150000 \mathrm{~s}\) and \(1.50 \times 10^{4} \mathrm{~s}\) d. \(3.8 \times 10^{-2} \mathrm{~L}\) and \(7.5 \times 10^{5} \mathrm{~L}\)

Short answer

Pairs a, b, and d have the same significant figures. Pair c does not.

Step-by-step solution

Count significant figures in 0.00575 g

Count all non-zero digits. The number 0.00575 has three significant figures (5, 7, and 5).

Count significant figures in 5.75 x 10^{-3} g

All digits in 5.75 are significant because it is in scientific notation. It has three significant figures (5, 7, and 5).

Compare significant figures in pair a

Both numbers, 0.00575 and 5.75 x 10^{-3}, have three significant figures.

Count significant figures in 0.0250 m

Count all non-zero digits and zeroes after the decimal point. The number 0.0250 has three significant figures (2, 5, and 0).

Count significant figures in 0.205 m

Count all non-zero digits and zeroes between them. The number 0.205 has three significant figures (2, 0, and 5).

Compare significant figures in pair b

Both numbers, 0.0250 and 0.205, have three significant figures.

Count significant figures in 150000 s

Count all non-zero digits. If there are no decimal points mentioned, only non-zero digits are significant. The number 150000 has two significant figures (1 and 5).

Count significant figures in 1.50 x 10^{4} s

All digits in 1.50 are significant because it is in scientific notation. It has three significant figures (1, 5, and 0).

Compare significant figures in pair c

The number 150000 has two significant figures, while 1.50 x 10^{4} has three significant figures, so they do not match.

Count significant figures in 3.8 x 10^{-2} L

All digits in 3.8 are significant because it is in scientific notation. It has two significant figures (3 and 8).

Count significant figures in 7.5 x 10^{5} L

All digits in 7.5 are significant because it is in scientific notation. It has two significant figures (7 and 5).

Compare significant figures in pair d

Both numbers, 3.8 x 10^{-2} and 7.5 x 10^{5}, have two significant figures.

Conclusion

Pairs a, b, and d have the same number of significant figures, while pair c does not.

Key concepts

scientific notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It is particularly useful in scientific calculations where precision is key. In scientific notation, numbers are written as a product of a coefficient (a number between 1 and 10) and a power of 10. For example, the number 5,750 can be written as \(5.75 \times 10^3\) in scientific notation. This makes it easier to handle and compare numbers, especially when they span many orders of magnitude. By converting numbers into scientific notation, you can more easily see the number of significant figures (which are crucial for precision) and avoid errors in calculations.

When you write numbers in scientific notation, make sure:

- The coefficient is a number between 1 and 10
- The exponent of 10 indicates how many times the base number is multiplied or divided by 10
- All non-zero digits in the coefficient are significant figures

For example, in the number \(5.75 \times 10^{-3}\), all three digits (5, 7, and 5) are significant figures.

counting significant figures

Counting significant figures is essential to ensure precision and accuracy in measurements. Significant figures include all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion. Here's how you can count them:

1. Non-zero digits: Always count
- Example: 123 has three significant figures
2. Captive zeros (zeros between non-zero digits): Always count
- Example: 1002 has four significant figures
3. Leading zeros (zeros before non-zero digits in decimal numbers): Do not count
- Example: 0.00575 has three significant figures, which are 5, 7, and 5
4. Trailing zeros (zeros at the end of a number): Count if they are after a decimal point
- Example: 100.0 has four significant figures (the zeros count because they are after the decimal point)

The step-by-step solution of the exercise uses these rules to count significant figures in each number. For example:
Pair (a): \(0.00575 \mathrm{~g}\) has three significant figures (5, 7, and 5), and \(5.75 \times 10^{-3} \mathrm{~g}\) also has three significant figures.
Pair (b): Both 0.0250 and 0.205 have three significant figures.
Pair (c): 150000 has two significant figures, whereas 1.50 \(\times 10^4\) has three. They don't match.
Pair (d): Both \(3.8 \times 10^{-2} \mathrm{~L}\) and \(7.5 \times 10^5 \mathrm{~L}\) have two significant figures.

precision in measurements

Precision in measurements refers to how close multiple measurements of the same quantity are to each other. More significant figures usually mean higher precision. However, it's essential to note that precision is not the same as accuracy, which measures how close a value is to the true or accepted value. Here's how precision matters:

- Consistent values: Multiple measurements should yield nearly the same value.
- More significant figures: More digits in a measurement point to higher precision.

For example, 1.50 \(\times 10^4 \) is a more precise representation than 150000 because it provides more detail with three significant figures. Similarly, 0.205 m is more precise than simply writing 0.2 m.

When comparing measurements, always look at the number of significant figures to gauge precision. This can be particularly crucial in scientific experiments where slight differences can greatly affect results. Understanding and using significant figures correctly ensures that measurements and calculations are as accurate and precise as possible.