Problem 152 - Introductory Chemistry Atoms First

Question

Express \(23,000,000\) in scientific notation having: (a) Two significant figures (b) Three significant figures (c) Five significant figures (d) Six significant figures (e) Eight significant figures

Step-by-Step Solution

Answer

(a) \(2.3 \times 10^7\) (b) \(2.30 \times 10^7\) (c) \(23.000 \times 10^6\) (d) \(23.0000 \times 10^6\) (e) \(23.000000 \times 10^6\)

Step 1: (a) Two significant figures

: To write \(23,000,000\) with two significant figures, we round it to the nearest ten million, which is \(23,000,000\). In scientific notation, this becomes: \(2.3 \times 10^7\)

Step 2: (b) Three significant figures

: To write \(23,000,000\) with three significant figures, we round it to the nearest million, which also remains \(23,000,000\). In scientific notation, this becomes: \(2.30 \times 10^7\)

Step 3: (c) Five significant figures

: To write \(23,000,000\) with five significant figures, we round it to the nearest ten thousand. In this case, there are no ten thousands, so the number remains unchanged. In scientific notation, this becomes: \(23.000 \times 10^6\)

Step 4: (d) Six significant figures

: To write \(23,000,000\) with six significant figures, we round it to the nearest thousand. Again, there are no thousands, so the number remains unchanged. In scientific notation, this becomes: \(23.0000 \times 10^6\)

Step 5: (e) Eight significant figures

: To write \(23,000,000\) with eight significant figures, we need more decimal places than the original number has. In this case, we'll simply add more trailing zeros after the decimal point, without rounding. In scientific notation, this becomes: \(23.000000 \times 10^6\)

Key Concepts

Significant Figures

Significant figures are crucial in scientific notation as they indicate the precision of a measurement or calculation. These are the digits that carry meaningful information about the number's accuracy. For example, the number 23000000 can be written with different levels of precision:

• With two significant figures: 23
• With three significant figures: 23.0
• With more significant figures such as five, six, or eight, trailing zeroes are included to convey the precision needed.

A digit is considered significant if it contributes to the number's precision, especially when it appears in a place where rounding could affect the calculation's accuracy. In scientific contexts, representing a large number with the right number of significant figures ensures the calculations remain consistent and precise.

Rounding Numbers

Rounding numbers is the process of simplifying a number while maintaining its value as close as possible to the original. This is crucial when converting large numbers into scientific notation or when a specific number of significant figures are requested.

• For instance, rounding 23000000 to two significant figures might involve rounding to the nearest ten million, keeping only meaningful digits for your specific context.
• Even seemingly "non-existent" numbers like zeros can be crucial when they appear after a decimal in scientific notation.

Rounding helps simplify numbers without losing important information, which is why ensuring the correct amount of rounding depending on significant figures is so important in scientific calculations.

Scientific Notation in Chemistry

In chemistry, scientific notation is especially useful for expressing very large or very small numbers efficiently. Chemicals and their properties often exist in extreme quantities where regular notation becomes cumbersome.

• With scientific notation, a number's place value and significant figures are clearly conveyed by showing a coefficient (like 2.3) multiplied by a power of ten, such as \(2.3 \times 10^7\).
• Scientific notation helps chemists to convey measurements and calculations succinctly, ensuring their precision through the use of significant figures.

Using scientific notation makes interpreting and calculating with large numbers in chemistry much more manageable, maintaining clarity and retaining the necessary level of precision.

Mathematical Representation

Mathematical representation of numbers involves writing them in forms that are clear, precise, and appropriate for their context, such as scientific notation. Mathematical representations facilitate easier calculation and comparison of values.

• In scientific notation, a number is typically represented as a product of a single-digit number and a power of ten.
• This representation highlights all the significant figures in a number while keeping things concise and easy to interpret.

Mathematical representation through scientific notation is not only about simplifying numbers but also about making sure that the value retains its accuracy and can be quickly utilized in further mathematical operations or real-world applications.

Notation Format for Large Numbers

The notation format for large numbers, particularly with scientific notation, is designed to simplify writing and understanding while preserving significant information.

• Instead of writing cumbersome numbers like 23000000 in full, we use a concise format like \(2.3 \times 10^7\) which represents the same number succinctly.
• This format is helpful in various fields beyond chemistry, including physics and engineering, where large numbers are commonplace.
• Utilizing scientific notation ensures that large numbers are written and read with ease while ensuring that their scale and precision are clear and consistent.

Through a compact representation, large values become more approachable and easier to work with, illustrating the importance of using scientific notation in practical applications.