Question

How many significant figures are in each of the following measured quantities? a. \(11.005 \mathrm{~g}\) b. \(0.00032 \mathrm{~m}\) c. \(36000000 \mathrm{~km}\) d. \(1.80 \times 10^{4} \mathrm{~kg}\) e. \(0.8250 \mathrm{~L}\) f. \(30.0^{\circ} \mathrm{C}\)

Short answer

a. 5, b. 2, c. 2, d. 3, e. 4, f. 3

Step-by-step solution

- Determine significant figures for 11.005 g

Count all the digits including zeros in the measurement. Since it is 11.005 g, all the digits are significant. So, there are 5 significant figures.

- Determine significant figures for 0.00032 m

Ignore the leading zeros and count the remaining digits. The number 0.00032 m has 2 significant figures.

- Determine significant figures for 36000000 km

In this number, if no additional information about precision is given, only the non-zero digits are significant. Thus, 36000000 km has 2 significant figures.

- Determine significant figures for 1.80 × 10^4 kg

The scientific notation makes it clear that we consider all digits shown. Therefore, 1.80 × 10^4 kg has 3 significant figures.

- Determine significant figures for 0.8250 L

Count all digits, including the trailing zero which indicates precision. Therefore, 0.8250 L has 4 significant figures.

- Determine significant figures for 30.0 °C

Count all digits, including the trailing zero which indicates precision. Therefore, 30.0 °C has 3 significant figures.

Key concepts

measurement precision

Measurement precision refers to how closely multiple measurements of the same quantity agree with each other. It is a crucial aspect in scientific experiments as it ensures the reliability of the data.
Precision is different from accuracy. While accuracy is about how close a measurement is to the true value, precision is about the consistency of repeated measurements.
In the context of significant figures, higher precision means more digits are used to express a measurement. For example, 0.8250 L has higher precision compared to 0.82 L because it indicates a more exact measurement with greater detail.
Understanding precision helps in interpreting data correctly and evaluating the quality of measurements.

scientific notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. This method makes it easier to handle and read these numbers.
A number in scientific notation is written as a product of a coefficient and a power of 10. For instance, 1.80 × 10^4 kg means 1.80 multiplied by 10 raised to the power of 4, which is 18,000 kg.
Scientific notation also clarifies the number of significant figures. In 1.80 × 10^4 kg, all the digits (1, 8, and 0) are significant, making it clear that there are 3 significant figures.

• It simplifies calculations involving very large or very small measurements.
• It reduces the risk of errors when transcribing or calculating with these numbers.

Knowing how to use scientific notation is essential for handling measurements in science and engineering.

leading zeros

Leading zeros are the zeros that precede all the non-zero digits in a number. These zeros are not considered significant because they only serve to place the decimal point.
For example, in the number 0.00032 m, the leading zeros (0.000) are not significant. Only the digits 3 and 2 are significant, making it a total of 2 significant figures.
Understanding leading zeros is important when determining the significant figures in a measured quantity. They help in distinguishing between small measurements and the actual precision of the measurement.

• Leading zeros only serve to set the decimal position.
• They are not counted as significant figures.

This concept helps avoid misinterpretation of measurement data, especially in cases with tiny values.

trailing zeros

Trailing zeros are the zeros that appear at the end of a number. Whether these zeros are considered significant depends on the context of the measurement.
Trailing zeros in a number without a decimal point are usually not considered significant unless indicated otherwise. For example, in 36000000 km, only the first two digits are significant unless a notation specifies the precision.
Conversely, trailing zeros in a number with a decimal point are significant. They indicate the precision of the measurement. In 0.8250 L, the trailing zero is significant, making it 4 significant figures.

• Zeros at the end of a number with a decimal point are significant.
• Zeros at the end without a decimal point need more context to determine their significance.

Recognizing when trailing zeros are significant helps in accurately interpreting and recording measurements.

significant digits

Significant digits (or significant figures) are the digits in a number that carry meaningful information about its precision. These digits include all non-zero numbers, any zeros between significant digits, and any trailing zeros in a decimal number.
For instance, in the measurement 11.005 g, all five digits are significant. This indicates a high precision for this measurement.
To determine significant digits correctly, it's important to follow these rules:

• All non-zero numbers are significant.
• Zeros between non-zero numbers are significant (e.g., 101).
• Leading zeros are not significant (e.g., 0.001).
• Trailing zeros are significant if there's a decimal point (e.g., 50.0).

Counting the correct number of significant digits ensures accurate representation and comparison of measurements. It avoids misleading precision and helps in scientific and mathematical calculations.