Question

Determine the number of significant figures in each measurement $\begin{array}{ll}\text{ a. }0.049450\mathrm{s}& \text{ c. }3.1587\times {10}^{-4}\mathrm{g}\\ \text{ b. }0.000482\mathrm{mL}& \text{ d. }0.0084\mathrm{mL}\end{array}$

Short answer

a. 0.049450 s has 5 significant figures. b. 0.000482 mL has 3 significant figures. c. $3.1587\times {10}^{-4}$ g has 5 significant figures. d. 0.0084 mL has 2 significant figures.

Step-by-step solution

a. 0.049450 s

There are two leading zeros (0.0) that are not significant. The following 5 digits (49450) are all non-zero and significant, so there are a total of 5 significant figures in this measurement.

b. 0.000482 mL

There are three leading zeros (0.000) that are not significant. The next 3 digits (482) are all non-zero and significant, so there are a total of 3 significant figures in this measurement.

c. 3.1587 × 10⁻⁴ g

The scientific notation here implies that all digits (3.1587) are significant. There are no zeros that would be considered non-significant. So, there are 5 significant figures in this measurement.

d. 0.0084 mL

There are two leading zeros (0.00) that are not significant. The next 2 digits (84) are non-zero and significant, so there are a total of 2 significant figures in this measurement.

Key concepts

Understanding Measurement and Significant Figures

Measurement provides the quantitative description that is central to science and engineering. In measurement, significant figures play a crucial role because they tell us about the precision of the measured value. The number of significant figures in a measurement reflects the certainty of that measurement, from the instrument used and the process applied.

Significant figures include all the digits that are known precisely plus one last digit that is estimated. To determine the number of significant figures:

• Count all non-zero digits as significant.
• Zeros between non-zero digits are significant.
• Leading zeros, which precede all of the non-zero digits, are not significant.
• Trailing zeros in a number with a decimal point are significant.

By understanding these rules, one can express a measurement accurately, providing clear information about its reliability.

The Power of Scientific Notation

Scientific notation is a method to express very large or very small numbers conveniently. It is particularly useful in scientific calculations where precision is needed, and it reduces the potential for error when dealing with many zeros.

In scientific notation, numbers are written in the form of $m\times {10}^n$, where:

• $m$ is a number greater than or equal to 1 and less than 10.
• $n$ is an integer.

For example, 3.1587 × 10⁻⁴ is written in scientific notation, where 3.1587 is the base ($m$) and -4 is the exponent ($n$).

Using scientific notation helps to maintain the correct number of significant figures. When numbers are expressed this way, all digits in the base $m$ (except for non-significant zeros if there were any) are significant, as they directly show the precision of the measurement.

Deciphering Non-significant Zeros

Understanding non-significant zeros is crucial for determining the precision of a measurement. Non-significant zeros do not contribute to the accuracy of a measurement, but they help in positioning the decimal point.

There are primarily two types of non-significant zeros:

• Leading Zeros: These are zeros that appear before the first non-zero digit (e.g., 0.000482 mL). Leading zeros are never significant. They merely act as placeholders and help in expressing the number correctly in a readable format.
• Captive Zeros: Although these zeros are placed between non-zero digits, they are considered significant because they contribute to the specificity of the number (e.g., the zeros in 1.05 are significant).

It’s essential to identify and exclude leading zeros when counting significant figures to ensure that the measurement is not overestimated in terms of its precision.