Question

Round the number on the left to the number of significant figures indicated by the example in the first row. (Use scientific notation as needed to avoid ambiguity.) \begin{tabular}{lccc} & Rounded to 4 Significant Figures & Rounded to 2 Significant Figures & Rounded to 1 Significant \\ \(94.52118\) & \(94.52\) & 95 & Figure \\ \(105.4545\) & & & \\ \(0.455981\) & & & \\ \(0.009999991\) & & & \\ \hline \end{tabular}

Short answer

\(105.4545\) rounded to 4 s.f. is \(1.055 \times 10^2\), to 2 s.f. is \(1.1 \times 10^2\), to 1 s.f. is \(1 \times 10^2\). \(0.455981\) rounded to 4 s.f. is \(4.560 \times 10^{-1}\), to 2 s.f. is \(4.6 \times 10^{-1}\), to 1 s.f. is \(5 \times 10^{-1}\). \(0.009999991\) rounded to 4 s.f. is \(1.000 \times 10^{-2}\), to 2 s.f. is \(1.0 \times 10^{-2}\), to 1 s.f. is \(1 \times 10^{-2}\).

Step-by-step solution

Round 105.4545 to 4 Significant Figures

Firstly, identify the first 4 significant figures in 105.4545, which are 1054. Now, look at the digit immediately after the 4th significant figure which is 5. Since 5 is equal to or greater than 5, we round the last significant figure up to 5 to get 1055. In scientific notation, this becomes: \[105.4545 \approx 1.055 \times 10^2\].

Round 105.4545 to 2 Significant Figures

Identify the first 2 significant figures in 105.4545, which are 10. The third digit is 5, meaning we round up the second digit. Therefore, \[105.4545 \approx 1.1 \times 10^2\].

Round 105.4545 to 1 Significant Figure

The first significant figure is 1. The next digit (0) is less than 5, hence we do not round up the first significant figure. The result is: \[105.4545 \approx 1 \times 10^2\].

Round 0.455981 to 4 Significant Figures

The first four significant figures are 4559. The digit following 9 is 8, so we round up to get 4560. In scientific notation, \[0.455981 \approx 4.560 \times 10^{-1}\].

Round 0.455981 to 2 Significant Figures

The first two significant figures are 45. Since the number immediately following is 5, we round up to get 46. The representation in scientific notation is: \[0.455981 \approx 4.6 \times 10^{-1}\].

Round 0.455981 to 1 Significant Figure

The first significant figure is 4. The next digit (5) is equal to 5, meaning we round up the first significant figure to 5. Therefore, the rounded number is: \[0.455981 \approx 5 \times 10^{-1}\].

Round 0.009999991 to 4 Significant Figures

Ignore the leading zeros and identify the four significant figures which are 9999. The following digit is 9 so we round up to get 1000. Hence, the number in scientific notation is: \[0.009999991 \approx 1.000 \times 10^{-2}\].

Round 0.009999991 to 2 Significant Figures

The first two significant figures are 99. Since the next digit is 9, we round up the second figure to 0 and increment the leading 9, resulting in: \[0.009999991 \approx 1.0 \times 10^{-2}\].

Round 0.009999991 to 1 Significant Figure

The first digit, which is a 9, is followed by another 9, therefore we round up the first figure to 1. The number becomes: \[0.009999991 \approx 1 \times 10^{-2}\].

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 used extensively in the sciences and engineering to manage the wide range of values that occur in these fields. For instance, the speed of light in a vacuum, a fundamental constant, is approximately 299,792,458 meters per second, which is more succinctly expressed as \(2.99792458 \times 10^8\) m/s in scientific notation.

In the context of the exercise where numbers need to be rounded to significant figures, scientific notation allows for clear indication of precision. It ensures that the number of significant digits is easily visible and not obscured by zeros, which could be mistaken for significant figures themselves. For example, \(105.4545\) rounded to four significant figures is \(1.055 \times 10^2\), clearly showing that the first four digits are the significant ones.

Accuracy in Chemistry

In chemistry, accuracy refers to how close a measured or calculated value is to the true value. In the laboratory, precise measurements are crucial. This accuracy often depends on the significant figures that are reported. As seen in the given exercise, when the value \(0.009999991\) is rounded to four significant figures as \(1.000 \times 10^{-2}\), it indicates a high accuracy to the fourth decimal place.

Chemists report measurements in significant figures to reflect the precision of their equipment and the reliability of their results. Reporting more significant figures than the measuring instrument can reliably produce is misleading and overstates the precision. It is thus essential to match the number of significant figures reported with the instrument's capability to ensure accurate communication of chemical measurements.

Numerical Precision

Numerical precision is the exactness of a number, which deals with the detail in the representation of the number. In this exercise, rounding involves determining which digits in a number are significant and should be retained for precision. The process of rounding to a specified number of significant figures can affect this precision. Rounding \(105.4545\) to two significant figures gives \(1.1 \times 10^2\), which is less precise than rounding it to four significant figures as \(1.055 \times 10^2\).

Numerical precision is important in fields that rely heavily on accurate calculations, such as science, engineering, and finance. A loss of precision can lead to significant errors in calculations and results. Therefore, it's essential to balance the need for precision with the practical limitations of measuring instruments and the contexts in which these numbers will be used.