Problem 57 - Chemistry

Question

The diameter of the Earth at the equator is \(7926.381 \mathrm{mi}\). Round off this quantity to four significant figures; then to two significant figures. Express the answers in scientific notation.

Step-by-Step Solution

Answer

Four significant figures: \(7.926 \times 10^3\); Two significant figures: \(7.9 \times 10^3\).

Step 1: Understanding Significant Figures

Significant figures are the digits in a number that carry meaningful information about its precision. For example, in the number 7926.381, all of the digits are significant because they are not zero.

Step 2: Rounding to Four Significant Figures

To round the number 7926.381 to four significant figures, we look at the first four digits: 7926. Since the number immediately following is a 3 (which is less than 5), we do not round up the fourth digit. Thus, 7926.381 rounded to four significant figures is 7926.

Step 3: Converting Four Significant Figures to Scientific Notation

The number 7926 in scientific notation is written as follows: \(7.926 \times 10^3\). The process involves moving the decimal point three places to the left to produce a coefficient between 1 and 10, multiplying by \(10^3\) because the decimal point was moved three places.

Step 4: Rounding to Two Significant Figures

To round the number 7926.381 to two significant figures, we focus on the first two digits: 79. The third digit after these two significant digits is a 2, which means we do not round the second significant digit up. Therefore, 7926 rounded to two significant figures is 7900.

Step 5: Converting Two Significant Figures to Scientific Notation

The number 7900 in scientific notation is \(7.9 \times 10^3\). Again, we move the decimal three places to the left, resulting in \(7.9\), and multiply by \(10^3\) since the decimal was moved three places.

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 commonly used in scientific and educational mathematics to simplify numbers and make them easier to read, compare, and use in calculations. This method uses powers of ten. In scientific notation, a number is typically written as the product of two parts:

• A decimal number between 1 and 10, known as the coefficient.
• A power of ten, which indicates how many places the decimal point was moved.

For example, the number 7926 can be written in scientific notation as \(7.926 \times 10^3\). Here, the decimal point is moved three places to the left, giving our coefficient \(7.926\), followed by \(10^3\) because we moved the decimal three places. This format makes it easier to handle extremely large or small numbers across different types of exercises, such as ones involving Earth's diameter or tiny measurements in scientific experiments.

Rounding Numbers

Rounding numbers involves reducing the number of significant figures in a number while trying to keep it as close to the original value as possible. It's crucial for simplifying numbers, especially when precision beyond a certain point isn't necessary. Here’s the general guideline for rounding:

• Identify the digit up to which you want to round.
• Look at the next digit to decide whether to round up or leave it as is.
• If the next digit is 5 or more, round up. If it is less than 5, do not round up.

In our exercise, we were asked to round 7926.381 to four significant figures, which resulted in 7926 (since the next digit is 3, less than 5) and then to two significant figures, resulting in 7900 (since the next digit is 2, less than 5).
This concept of rounding is not only critical in math exercises but also in real-life scenarios like financial estimations and scientific calculations, where an exact number might not be necessary.

Educational Mathematics

Educational mathematics covers a wide range of topics, including significant figures, rounding numbers, and scientific notation. These concepts are fundamental learning blocks in any math curriculum, serving critical roles in helping students understand precision, accuracy, and simpler ways of expressing complex numbers.
Understanding these concepts enables students to tackle more sophisticated mathematical problems involving physics, chemistry, and even everyday calculations. Here are some of the benefits:

• Precision and Accuracy: Helps in understanding the proper level of precision needed in calculations.
• Simplification: Allows for easier manipulation and understanding of both very large and very small numbers.
• Universal Application: Gives students tools that are applicable in various scientific and real-world contexts.

Educators often emphasize the importance of mastering these topics as they create a strong foundation for advanced mathematics and sciences. By using examples like the Earth's diameter, students can relate these abstract concepts to real-life phenomena, making learning both engaging and practical.