Problem 47 - Chemistry The Central Science

Question

Round each of the following numbers to four significant figures and express the result in standard exponential notation: (a) \(102.53070,\) (b) \(656.980,\) (c) 0.008543210 ,(d) \(0.000257870 (\mathbf{e})-0.0357202\)

Step-by-Step Solution

Answer

Rounded to four significant figures and expressed in standard exponential notation, the numbers are: (a) \(1.025\times10^2\), (b) \(6.57\times10^2\), (c) \(8.543\times10^{-3}\), (d) \(2.58\times10^{-4}\), and (e) \(-3.572\times10^{-2}\).

Step 1: Rounding to 4 significant figures (a)

First, we identify the first four significant figures in the number: 102.53070. The first four significant figures are 1, 0, 2, and 5. Then, we look at the digit after the 5, which is a 3. Since this is less than 5, we round down and the number becomes 102.5.

Step 2: Standard exponential notation (a)

Now, we convert 102.5 to standard exponential notation: \(1.025\times10^2\).

Step 1: Rounding to 4 significant figures (b)

First, we identify the first four significant figures in the number: 656.980. The first four significant figures are 6, 5, 6, and 9. Then, we look at the digit after the 9, which is an 8. Since this is greater than 5, we round up and the number becomes 657.0.

Step 2: Standard exponential notation (b)

Now, we convert 657.0 to standard exponential notation: \(6.57\times10^2\).

Step 1: Rounding to 4 significant figures (c)

First, we identify the first four significant figures in the number: 0.008543210. The first four significant figures are 8, 5, 4, and 3. Then, we look at the digit after the 3, which is a 2. Since it's less than 5, we round down and the number becomes 0.008543.

Step 2: Standard exponential notation (c)

Now, we convert 0.008543 to standard exponential notation: \(8.543\times10^{-3}\).

Step 1: Rounding to 4 significant figures (d)

First, we identify the first four significant figures in the number: 0.000257870. The first four significant figures are 2, 5, 7, and 8. Then, we look at the digit after the 8, which is a 7. Since it's greater than 5, we round up and the number becomes 0.000258.

Step 2: Standard exponential notation (d)

Now, we convert 0.000258 to standard exponential notation: \(2.58\times10^{-4}\).

Step 1: Rounding to 4 significant figures (e)

First, we identify the first four significant figures in the number: -0.0357202. The first four significant figures are 3, 5, 7, and 2. Then, we look at the digit after the 2, which is a 0. Since it's less than 5, we round down and the number becomes -0.03572.

Step 2: Standard exponential notation (e)

Now, we convert -0.03572 to standard exponential notation: \(-3.572\times10^{-2}\).

Key Concepts

Rounding Numbers

When rounding numbers, we determine the values that we keep, and we make adjustments based on subsequent digits. Rounding is crucial in ensuring precision without unnecessary detail. Here's how to do it effectively, especially when dealing with significant figures:

• Identify the number of significant figures you need to round the number to.
• Locate the digit at the position just after your significant figures. If this digit is 5 or more, you round up the last significant figure; otherwise, you round down.
• Apply this rule consistently across different types of numbers, be they integers or decimals.

For example, given the number 102.53070 and tasked with rounding to four significant figures, you identify 1, 0, 2, and 5 as those figures. Since the digit after 5 is a 3 (less than 5), you round down, resulting in 102.5. This technique ensures that calculations remain practical while retaining credibility.

Exponential Notation

Exponential notation offers a streamlined way of expressing extremely large or small numbers allowing for easier computation and comprehension. It consists of a base multiplied by a power of ten:

• The base is a decimal number, which is usually between 1 and 10.
• The exponent indicates how many places the decimal point must move to convert back to the standard format.

For example, converting 657.0 involves shifting the decimal two places to the right, which can be effectively expressed as \(6.57 \times 10^2\). Exponential notation not only simplifies number handling but also caters to maintaining accuracy in scientific calculations.

Scientific Notation

Scientific notation is a form of exponential notation particularly useful in fields like physics and chemistry to concisely represent very large or very small numbers. Here’s how it works:

• Identify and write down the significant figures of the number.
• Place a decimal after the first significant figure to form a new number (between 1 and 10).
• Calculate the exponent by counting the number of places you moved the decimal point from the original number's position.
• Combine the base and the exponent to express the number in scientific notation.

Consider converting the number 0.008543 to scientific notation. The decimal must shift three places to the right, giving us \(8.543 \times 10^{-3}\). This transformation allows for efficient communication and calculation of meaningful figures in science and engineering.