Question

How many significant figures are in each of the following? (a) \(0.890 \mathrm{~cm}\) (b) \(210^{\circ} \mathrm{C}\) (c) \(2.189 \times 10^{6} \mathrm{nm}\) (d) \(2.54 \mathrm{~cm}=1 \mathrm{in}\) (e) \(140.00 \mathrm{~g}\)

Short answer

Question: Determine the number of significant figures in each of the following numbers: (a) \(0.890 \mathrm{~cm}\) (b) \(210^{\circ} \mathrm{C}\) (c) \(2.189 \times 10^{6} \mathrm{nm}\) (d) \(2.54 \mathrm{~cm}=1 \mathrm{in}\) (e) \(140.00 \mathrm{~g}\) Answer: (a) 3 significant figures (b) 2 significant figures (c) 4 significant figures (d) 3 significant figures (e) 5 significant figures

Step-by-step solution

(a) \(0.890 \mathrm{~cm}\)

There are no leading zeros in this number. It has three digits: a nonzero digit (8), a zero between nonzero digits (0), and a nonzero digit (9). All of these are significant, so the number of significant figures is 3.

(b) \(210^{\circ} \mathrm{C}\)

There are two nonzero digits (2 and 1), and one trailing zero (0). The trailing zero is not after a decimal point, so it is not significant. Thus, the number of significant figures is 2.

(c) \(2.189 \times 10^{6} \mathrm{nm}\)

In scientific notation, we only need to look at the number before the multiplication symbol (\(\times\)). The number has four digits: three nonzero digits (2, 1, and 9) and one zero between nonzero digits (8). All of these are significant, so the number of significant figures is 4.

(d) \(2.54 \mathrm{~cm}=1 \mathrm{in}\)

We only need to find the number of significant figures in the number \(2.54 \mathrm{~cm}\). It has three digits: two nonzero digits (2 and 5) and one trailing zero after a decimal point (4). All of these are significant, so the number of significant figures is 3.

(e) \(140.00 \mathrm{~g}\)

This number has five digits: two nonzero digits (1 and 4) and three trailing zeros after a decimal point (0, 0, and 0). All of these are significant, so the number of significant figures is 5.

Key concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used by scientists, mathematicians, and engineers to make calculations easier and to clearly indicate the precision of measurements.

In scientific notation, a number is written as a product of two factors: a decimal number between 1 and 10, and a power of 10. For example, the number 2,189,000 can be written in scientific notation as \(2.189 \times 10^{6}\). The first part of this expression, \(2.189\), is called the 'significant' and it determines the number of significant figures in the entire number. In the step-by-step solution provided, for section (c), the significant is \(2.189\) and so the number of significant figures is four.

When converting numbers into scientific notation, always ensure that the decimal point is placed after the first non-zero digit and before adjusting the exponent of 10 to match the shift in the decimal place. This process preservers the original number's precision and ensures clarity in communication within scientific contexts.

Measurement Precision

Measurement precision denotes how detailed and exact a measurement is. Precision is intricately related to the idea of significant figures, which are the digits in a number that carry meaning contributing to its precision. In the context of our step-by-step solutions, precision indicates how accurately the measurements are reported.

For example, in section (e) \(140.00 \mathrm{~g}\), the number has five significant figures, two non-zero numbers, and three zeros following a decimal point. Those additional zeros imply a greater level of precision in the measurement; it suggests that the weight was measured to the nearest hundredth of a gram.

The precision of a measurement can depend on the instrument used for the measurement and the method of relaying that measurement. For instance, a scale able to measure to the nearest thousandth of a gram would provide more precise measurements than one that only measures to the nearest gram. Similarly, more significant figures usually mean a more precise measurement when the numbers are reported.

Decimal Places

Decimal places are the number of digits to the right of the decimal point in a number. They play a crucial role in the precision of a measurement and are key in determining the number of significant figures a number has.

In the exercise, look at section (a) \(0.890 \mathrm{~cm}\). Although the number starts with a zero, which is not significant, the number still has three significant figures because there are exactly three digits after the decimal place that are known with certainty (8, 9, and the final 0). The presence of these numbers after the decimal point implies a high degree of measurement accuracy.

Adding more decimal places usually means a higher precision unless they are leading zeros, which do not count as significant figures. For instance, the number 0.0089 only has two significant figures because the first two zeros do not provide specific information about the quantity measured. Decimal places are an intuitive way to communicate precision in measurements and calculations.