Question

How many significant figures do the following measured quantities have? (a) \(0.0230 \mathrm{~g}\) (d) \(614.00 \mathrm{mg}\) (b) \(105.303 \mathrm{~m}\) (e) \(10 \mathrm{~L}\) (c) \(0.007 \mathrm{~kg}\)

Short answer

a) 3 significant figures, b) 6 significant figures, c) 1 significant figure, d) 5 significant figures, e) 1 significant figure.

Step-by-step solution

Identifying Significant Figures in 0.0230 g

In the measurement 0.0230 grams, the zeros after the decimal point but before a non-zero digit are not significant. However, the zero after the '3' is considered significant because it denotes the precision of the measurement. Therefore, there are three significant figures in this measurement.

Identifying Significant Figures in 614.00 mg

In the measurement 614.00 milligrams, all the numbers are significant. The zeros are significant because they are trailing zeros that come after a decimal point and after a non-zero digit. Therefore, there are five significant figures in this measurement.

Identifying Significant Figures in 105.303 m

In the measurement 105.303 meters, all the digits are significant because they are either non-zero digits or zeros between non-zero digits. Therefore, there are six significant figures in this measurement.

Identifying Significant Figures in 10 L

In the measurement 10 liters, without additional context or decimal places, we assume that the trailing zero is not significant. So, there is only one significant figure in this measurement.

Identifying Significant Figures in 0.007 kg

In the measurement 0.007 kilograms, the zeros that precede the first non-zero digit are not significant. The '7' is the only significant figure. Therefore, there is one significant figure in this measurement.

Key concepts

Understanding Measurement Precision

Measurement precision speaks to the level of detail and exactness in a measurement. It refers to how closely a measurement can be replicated by different measurements of the same quantity. Precision is directly tied to the concept of significant figures. Significant figures in a measurement include all the known digits plus one estimated digit, providing insight into the precision of that measurement.

Using the provided examples, when a measurement such as 0.0230 g is given, the zeroes indicate that the measurement is precise to the ten-thousandth of a gram. Higher precision is usually indicated by more significant figures, which means the measurement is more reliable. On the other hand, a value like 10 L, which lacks decimal points or additional digits, suggests lower precision and uncertainty in all digits except the first. Therefore, improving measurement precision may involve using more sensitive instruments or techniques that provide more significant figures in the recorded results.

Rules of Zeroes in Significant Figures

Zeroes play a crucial role in determining the number of significant figures and therefore impact the precision of a measurement. The rules for when zeroes are significant are dictated by their position in the number:

• Leading zeroes: These are zeroes that precede all non-zero digits. They are not significant because they merely indicate the position of the decimal point.
• Captive or sandwiched zeroes: Zeroes that fall between non-zero digits are always significant. They represent an actual measured or estimated quantity.
• Trailing zeroes in a decimal: These zeroes are significant if they come after a non-zero digit, as they indicate a measure of precision. Trailing zeroes in a whole number are only significant if a decimal point is expressly shown.

For instance, in 0.007 kg, the leading zeros are not counted as significant. However, in 614.00 mg, the trailing zeros after the decimal point represent significant figures because they reflect the precision of the measurement at the milligram level.

Utilizing Scientific Notation for Clarity

Scientific notation is a method of writing numbers that accommodates a wide range of values in a clear and concise manner, by expressing them as a product of a number between 1 and 10 and a power of 10. This can be particularly useful in measurements, as it can clearly indicate precision and make significant figures explicit. For example:

• A number like 0.007 kg can be written as \(7 \times 10^{-3}\) kg, indicating one significant figure (the 7).
• Similarly, a number like 614.00 mg can be represented as \(6.1400 \times 10^2\) mg, which includes all five digits as significant figures.

The use of scientific notation is especially helpful when dealing with very large or small numbers, as it simplifies calculations and clearly conveys the number of significant figures. Scientific notation thus reinforces the precision communicated through significant figures and is a tool for both scientific calculation and the effective communication of data.