Question

Evaluate each of the following, and write the answer to the appropriate number of significant figures. a. \(149.2+0.034+2000.34\) b. \(1.0322 \times 10^{3}+4.34 \times 10^{3}\) c. \(4.03 \times 10^{-2}-2.044 \times 10^{-3}\) d. \(2.094 \times 10^{5}-1.073 \times 10^{6}\)

Short answer

= \(2149.574\)

Step-by-step solution

Calculate the sum of numbers with varying decimal positions

We'll first add the numbers in expression 'a' as they have varying decimal places. To do this, we'll align the decimal points and add the decimal parts together, followed by the non-decimal parts: \(149.2 + 0.034 + 2000.34\)

Key concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form, which is immensely useful in chemistry and other sciences. For instance, the number 0.0004 can be written as \(4 \times 10^{-4}\) in scientific notation. The structure of scientific notation is simple: a number between 1 and 10 is multiplied by 10 raised to an exponent. The exponent shows how many places the decimal point has moved.

When writing or interpreting scientific notation, remember two things:

• The exponent is positive if the number is large and the decimal point has moved to the left.
• The exponent is negative if the number is small and the decimal point has moved to the right.

Understanding scientific notation becomes crucial when working with significant figures because it allows you to easily track the precision of a number, regardless of its size.

Calculating Significant Figures

Calculating significant figures is essential for accurately conveying the precision of a measurement in chemistry. Significant figures, or 'sig figs', refer to the digits in a number that carry meaning contributing to its precision. This includes all non-zero numbers, any zeroes between significant digits, and any trailing zeros in the decimal portion of a number.

For example, the number 2050 has three significant figures, the 2, the 5, and the zero because it is between significant digits. However, the final zero isn't significant unless the number is written with a decimal point as 2050.0, which then has four significant figures. The rule of thumb is:

• Non-zero digits are always significant.
• Any zeros between two significant figures are significant.
• Leading zeros are never significant.
• Trailing zeros are significant only if the number contains a decimal point.

These rules help ensure that figures are calculated consistently.

Addition and Subtraction with Significant Figures

When performing addition and subtraction with significant figures, the rule of thumb is that the result should be reported with the same number of decimal places as the measurement with the fewest decimal places.

Consider the expression \(149.2 + 0.034 + 2000.34\). The number with the fewest decimal places is 149.2, which has one decimal place. Thus, when adding these numbers, the result should be rounded to one decimal place. The correct sum with proper significant figures would be \(2149.6\).

Similarly, subtracting numbers requires the same attention to decimal places. In the expression \(2.094 \times 10^{5} - 1.073 \times 10^{6}\), the number of significant decimal places is determined by the term with the fewest decimal places after the standardization of scientific notation. The result, therefore, must have the same level of precision in the decimal places, which influences the number of significant figures in the answer.