Problem 47 - Chemistry: The Central Science in SI Units, Global Edition

Question

Round each of the following numbers to three significant figures and express the result in standard exponential notation: \((\mathbf{a}) 2048732.23(\mathbf{b}) 0.000292945(\mathbf{c})-82454.09\) (d) \(942.057024(\mathbf{e})-0.00000324683 .\)

Step-by-Step Solution

Answer

The rounded numbers in standard exponential notation are: (a) \(2.05 \times 10^6\), (b) \(2.93 \times 10^{-4}\), (c) \(-8.25 \times 10^4\), (d) \(9.42 \times 10^2\), and (e) \(-3.25 \times 10^{-6}\).

Step 1: Part a: 2048732.23

First, identify the first three significant figures, which are 2, 0, and 4. Since the fourth significant digit (8) is greater than or equal to 5, round up the last significant digit. The rounded number is 2050000. Now, express the result in standard exponential notation: \(2.05 \times 10^6\).

Step 2: Part b: 0.000292945

Identify the first three significant figures (2, 9, and 2). Since the fourth significant digit (9) is greater than or equal to 5, round up the last significant digit. The rounded number is 0.000293. Express the result in standard exponential notation: \(2.93 \times 10^{-4}\).

Step 3: Part c: -82454.09

First, ignore the negative sign. Then find the first three significant figures, which are 8, 2, and 4. Since the fourth significant digit (5) is greater than or equal to 5, round up the last significant digit. The rounded number is 82500, and with the negative sign, it's -82500. Now, express the result in standard exponential notation: \(-8.25 \times 10^4\).

Step 4: Part d: 942.057024

First, identify the first three significant figures (9, 4, and 2). Since the fourth significant digit (0) is less than 5, do not round up the last significant digit. The rounded number is 942. Now, express the result in standard exponential notation: \(9.42 \times 10^2\).

Step 5: Part e: -0.00000324683

First, ignore the negative sign. Then find the first three significant figures (3, 2, and 4). Since the fourth significant digit (6) is greater than or equal to 5, round up the last significant digit. The rounded number is 0.00000325, and with the negative sign, it's -0.00000325. Now, express the result in standard exponential notation: \(-3.25 \times 10^{-6}\).

Key Concepts

Rounding Numbers

Rounding numbers is a crucial skill in mathematics, especially when dealing with significant figures, which reflect the precision of a number. Significant figures are important because they help us understand how accurate a measurement is.
When rounding to three significant figures, you follow these steps:

• Identify the first three significant digits in the number.
• Look at the fourth digit. If it’s 5 or greater, increase the third significant digit by one.
• If it's less than 5, keep the third digit as it is.

Consider the example of rounding 2048732.23 to three significant figures. The first three significant figures are 2, 0, and 4. Since the fourth digit (8) is greater than 5, we round up, resulting in 2050000. Rounding accurately helps to simplify and communicate numerical information effectively.

Exponential Notation

Exponential notation is a way of expressing numbers that makes them easier to read, especially when dealing with very large or very small quantities. This notation uses powers of 10 to convey how many places to move the decimal point.
For instance, the number 2050000 can be written in exponential notation as \(2.05 \times 10^6\). The base number, 2.05, is adjusted to fall between 1 and 10. The exponent, 6, indicates that the decimal point has moved six places to the right. This system is very useful in scientific work because it allows for clear and concise representation of figures without detailing all the zeros.
Exponential notation is integral to scientific notation, helping present data in a more manageable form.

Scientific Notation

Scientific notation is a specific form of exponential notation often used in scientific, engineering, and mathematical fields to handle numbers that are very large or very small. It simplifies complex calculations and makes them more readable.
To express a number in scientific notation, follow these steps:

• Move the decimal point in the number until only one non-zero digit remains on the left.
• Count the number of places the decimal has moved, which becomes the exponent of 10.
• If the original number is greater than 1, the exponent is positive. For numbers less than 1, the exponent is negative.

Looking at the number 0.000292945, the scientific notation would be \(2.93 \times 10^{-4}\). Here, the decimal is moved four places to the right, resulting in a negative exponent because the number is less than 1.
Scientific notation is essential for dealing efficiently with very large or very small numbers, ensuring precision and clarity in any calculations or data presentation.