Question

Determine the number of significant figures in each of the following measurements: (a) \(0.006 \mathrm{~L},\) (b) \(0.0605 \mathrm{dm}\) (c) \(60.5 \mathrm{mg}\), (d) \(605.5 \mathrm{~cm}^{2}\) (e) \(9.60 \times 10^{3} \mathrm{~g}\) (f) \(6 \mathrm{~kg}\), (g) \(60 \mathrm{~m}\).

Short answer

(a) 1, (b) 3, (c) 3, (d) 4, (e) 3, (f) 1, (g) 1.

Step-by-step solution

Understanding Significant Figures

Significant figures are the digits in a number that contribute to its precision. This includes all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion.

Determine Significant Figures for 0.006 L

Identify all non-zero digits and zeros that contribute to the measurement precision. Here, the number is 0.006 L. The zeros before the 6 are not significant because they merely indicate the position of the decimal. There is only one significant figure: the digit 6.

Determine Significant Figures for 0.0605 dm

Identify all significant figures in 0.0605 dm. Zeros between non-zero digits are significant. The digits here are 6 and 5, and the zero between them is significant as well. This gives three significant figures: 6, 0, and 5.

Determine Significant Figures for 60.5 mg

In 60.5 mg, each number is significant. The digits 6, 0, and 5 all count towards the total. Therefore, this number has three significant figures.

Determine Significant Figures for 605.5 cm²

Each digit in 605.5 cm² is significant because there are no leading or trailing zeros. Thus, it has four significant figures.

Determine Significant Figures for 9.60 × 10³ g

In scientific notation, consider the digits in the coefficient only. Therefore, 9.60 has three significant figures (9, 6, and the zero, since it's trailing and after a decimal point).

Determine Significant Figures for 6 kg

The number 6 kg contains only one significant figure because there are no zeros or decimal points that would change this assessment.

Determine Significant Figures for 60 m

The number 60 m has ambiguous zeros because there is no decimal point. Generally, it is counted as one significant figure, unless context (like a decimal being present) suggests otherwise.

Key concepts

Precision in Measurements

Precision in measurements refers to the degree of refinement in a measurement. It indicates the closeness of several measurements together. Higher precision means more significant digits and a finer level of detail. Precision is not about true value; rather, it's about how detailed the measurements are.
For instance, measuring a desk as '1.75 meters' is more precise than just '2 meters'. Precision is improved with better instruments and can be influenced by human factors like the skill of the measurer.
Understanding precision helps in choosing the right tools and methods for measurements.

• Precision helps distinguish finely detailed results from broad estimates.
• It's crucial in scientific experiments where exactness is necessary.

Identifying Significant Figures

Identifying significant figures is a key skill in calculating the precision of any measurement. Significant figures not only include all certain digits but also the first uncertain digit in a measurement. They are the digits that carry meaning contributing to its accuracy.
Here's how to determine significant figures:

• All non-zero numbers (1-9) are significant.
• Zeros between non-zero digits are also significant, such as in 0.0605 (the zero counts, giving three significant figures).
• Leading zeros (zeros before non-zero digits) are not significant. They're just placeholders, as in 0.006 (one significant figure).
• Trailing zeros (zeros at the end) in a decimal portion are significant, such as in 9.60 (three significant figures because the zero is after a decimal and a number).

Identifying significant figures is essential in maintaining and expressing precision consistently.

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 key.
In scientific notation, numbers are written as the product of two factors: a coefficient and a power of 10. For example, 9.60 x 10³ grams.

• The coefficient (9.60) carries all the significant figures.
• The power of ten (10³) places the decimal to provide the correct magnitude.

By using scientific notation, you can neatly display significant figures and manage large or small numbers without losing precision. It ensures clarity and precision in data reporting.
This notation helps in simplifying the reading, calculating, and comparing of numbers, making it an essential tool in fields such as physics and chemistry.

Significant Digits

Significant digits are all the numbers in a measurement that indicate precision. Any digit that adds accuracy is considered significant. They show the reliability and trustworthiness of the measurement.
Here's a recap on dealing with significant digits:

• Figures 1-9 are always significant.
• Any zero between significant figures is also significant.
• Leading zeros do not count as significant.
• Trailing zeros are significant only if they come after a decimal point.

Consider 605.5 cm², it has four significant digits. This shows that each figure contributes to the measurement's accuracy. In ambiguous cases, such as 60 m where context isn't given, understand the convention that may imply one or two significant figures.
Significant digits are crucial for reporting accurate measurements and are fundamental in maintaining scientific precision.