Problem 59 - General Chemistry

Question

The circumference of the earth at the equator is \(40,000 \mathrm{~km}\). This value is precise to two significant figures. Write this in scientific notation to express correctly the number of significant figures.

Step-by-Step Solution

Answer

\( 4.0 \times 10^4 \) retains two significant figures.

Step 1: Understand the Problem

We need to express the number 40,000 km, which has two significant figures, into scientific notation, ensuring that those two figures are retained.

Step 2: Identify Significant Figures

In the number 40,000, the '4' is a significant figure and one trailing zero must be included to retain two significant figures.

Step 3: Convert to Scientific Notation

Transform the number 40,000 into scientific notation by placing the decimal point after the first significant digit and counting the number of places the decimal has moved.

Step 4: Write in Scientific Notation

The number 40,000 can be written as \( 4.0 \times 10^4 \) to ensure that it has two significant figures. Here, '4.0' retains the two significant figures.

Key Concepts

Significant Figures

Significant figures are crucial when representing scientific measurements as they convey the precision of a number. In any measurement, significant figures include all the digits that are known accurately, plus one that is estimated. Consider the number 40,000 in our exercise. Here, the significant figures are essential for conveying that the figure is correct to a certain precision. By stating only two significant figures, we emphasize that there may be some uncertainty beyond this point.

For 40,000, the '4' is known accurately, and to express two significant figures, one trailing zero must also be considered significant. This helps maintain the measurement's precision. When numbers are converted to scientific notation, significant figures are preserved through careful placement of digits before and after the decimal point.

• The first non-zero digit is always significant.
• Trailing zeros in a whole number with no decimal shown may or may not be significant.

Maintaining significant figures ensures accuracy and reliability in scientific reporting.

Physics Education

Physics education often involves understanding how to accurately represent and interpret numerical data. This includes learning how to use scientific notation and significant figures, which are fundamental in quantifying physical phenomena. In physics, measurements must be as precise as possible to develop theories and conduct experiments.

For students, it is important to learn these concepts early to develop accuracy in interpreting and understanding data. Physics education covers various topics where scientific notation helps compress very large or very small numbers into a more manageable form. It simplifies calculations and enhances the understanding of data scales.

• Simulation of real-world problems often requires precise numerical expressions.
• Accurate representation of data helps in error estimation and experimental validation.

Recognizing significant figures and employing scientific notation properly equips students to apply these skills in complex scenarios.

Mathematical Notation

Mathematical notation provides a concise and standardized method for representing numbers and operations. Scientific notation is a form of mathematical notation that is particularly useful in dealing with very large or very small numbers, often seen in scientific fields.

It involves shifting the decimal point in a number to create a new number between 1 and 10, then multiplying by a power of ten. For example, the exercise about the Earth's circumference involves converting 40,000 into scientific notation as follows:

• Move the decimal 4 places to the left, creating 4.0.
• Indicate this movement with a power of ten: \(4.0 \times 10^{4} \).

This approach not only simplifies the number's format but ensures that its precision level is clear. Among other symbols, powers of ten are used extensively in mathematical notation, making it easier to communicate complex ideas succinctly. As students learn to manipulate and interpret these notations, they become better equipped to handle the intricacies of scientific data.