Question

How many significant figures are there in each of the following? (a) \(0.136 \mathrm{~m}\) (b) \(0.0001050 \mathrm{~g}\) (c) \(2.700 \times 10^{3} \mathrm{nm}\) (d) \(6 \times 10^{-4} \mathrm{~L}\) (e) \(56003 \mathrm{~cm}^{3}\)

Short answer

(a) \(0.136 \mathrm{~m}\) (b) \(0.0001050 \mathrm{~g}\) (c) \(2.700 \times 10^{3} \mathrm{nm}\) (d) \(6 \times 10^{-4} \mathrm{~L}\) (e) \(56003 \mathrm{~cm}^{3}\) Answer: (a) 3 significant figures (b) 3 significant figures (c) 4 significant figures (d) 1 significant figure (e) 5 significant figures

Step-by-step solution

Recall the rules for significant figures

There are a few rules for identifying significant figures in a number: 1. Non-zero digits are always significant. 2. Any zeroes between significant digits (captive zeroes) are significant. 3. Leading zeroes are not significant. These are zeros that precede non-zero digits. 4. Trailing zeroes in a decimal number (after a decimal point) are significant. Now, let's apply these rules to each value.

Identify the significant figures in (a)

For (a) \(0.136 \mathrm{~m}\), we have: 1. The non-zero digits 1, 3, and 6 are significant. 2. There are no captive zeroes. 3. The leading zero before the decimal point is not significant. 4. There are no trailing zeroes. Thus, there are 3 significant figures in (a).

Identify the significant figures in (b)

For (b) \(0.0001050 \mathrm{~g}\), we have: 1. The non-zero digits 1 and 5 are significant. 2. There are no captive zeroes. 3. The leading zeroes before the decimal point and before 1 are not significant. 4. The trailing zero after 5 is significant since it is after the decimal point. Thus, there are 3 significant figures in (b).

Identify the significant figures in (c)

For (c) \(2.700 \times 10^{3} \mathrm{nm}\), we have: 1. The non-zero digit 2 is significant. 2. There are no captive zeroes between non-zero digits. 3. There are no leading zeroes. 4. The zeroes after the decimal point (and after the 2) are significant since they are trailing zeroes in a decimal number. Thus, there are 4 significant figures in (c).

Identify the significant figures in (d)

For (d) \(6 \times 10^{-4} \mathrm{~L}\), we have: 1. The non-zero digit 6 is significant. 2. There are no captive zeroes. 3. There are no leading zeroes. 4. There are no trailing zeroes. Thus, there is 1 significant figure in (d).

Identify the significant figures in (e)

For (e) \(56003 \mathrm{~cm}^{3}\), we have: 1. The non-zero digits 5, 6, and 3 are significant. 2. The zero between the 6 and 3 is a captive zero, so it is significant. 3. There are no leading zeroes. 4. There are no trailing zeroes in this whole number. Thus, there are 5 significant figures in (e).

Key concepts

Measurement Precision

When we talk about measurement precision in scientific terms, we are referring to the level of detail that a measurement conveys. It's important to distinguish between 'precision' and 'accuracy'. Precision is about the consistency of measurements or the smallest increment an instrument can detect, whereas accuracy is how close a measurement is to the true value.

In the context of significant figures, precision is reflected in the number of digits reported. More significant figures imply greater precision, as each digit represents a finer level of measurement. For example, a scale that reports weight as 20.00 grams is more precise than one that reports it only as 20 grams.

Remember that a common source of confusion is trailing zeros. For instance, in measurement (c) from the exercise, the recorded value is given as \(2.700 \times 10^{3}\) nanometers (nm), indicating that the measurement is precise to the tenths place—even though the last two digits are zeros, they still count towards the precision of the measurement.

Scientific Notation

Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It is often used in science and engineering to make calculations easier to perform and to clearly indicate the number of significant figures.

A number in scientific notation is expressed as the product of a number (between 1 and 10) and a power of ten. For example, \(2.700 \times 10^{3}\) means that the decimal point in 2.700 is moved 3 places to the right, giving 2700, which indicates that the original number has four significant figures. Likewise, in the exercise (d) \(6 \times 10^{-4}\) L, the notation implies a single significant figure, because the coefficient, 6, has only one non-zero digit.

Decimal Places

Decimal places are used to indicate the precision of a decimal number. They count the number of digits after the decimal point. For example, in a measurement of length that is 12.345 meters, there are three decimal places, reflecting a certain degree of precision regarding the hundredths of a meter.

When discussing significant figures, decimal places play a critical role in determining which zeroes are significant. Trailing zeroes, or zeroes at the end of a number after the decimal point, are considered significant because they define the decimal place of the final digit. Such as in measurement (b) from the exercise, the zeros after the 5 are significant because the measurement has been recorded to that level of precision: \(0.0001050\) g has three significant figures.