Question

Round each number to four significant figures. \(\begin{array}{ll}{\text { a. } 431,801 \mathrm{kg}} & {\text { d. } 0.004384010 \mathrm{cm}} \\ {\text { b. } 10,235.0 \mathrm{mg}} & {\text { e. } 0.00078100 \mathrm{mL}} \\ {\text { c. } 1.0348 \mathrm{m}} & {\text { f. } 0.0098641 \mathrm{cg}}\end{array}\)

Short answer

The rounded values to four significant figures are: a. 431,800 kg b. 10,240 mg c. 1.035 m d. 0.004384 cm e. 0.0007810 mL f. 0.009865 cg

Step-by-step solution

Rounding value a: 431,801 kg

To round the number 431,801 kg to four significant figures, we will look at the fifth digit, which is 1. Since 1 is less than 5, we keep the fourth digit as is and get 431,800 kg.

Rounding value b: 10,235.0 mg

To round the number 10,235.0 mg to four significant figures, we look at the fifth digit, which is 5. Since 5 is equal to 5, we round up the fourth digit and get 10,240 mg.

Rounding value c: 1.0348 m

To round the number 1.0348 m to four significant figures, we look at the fifth digit, which is 8. Since 8 is greater than 5, we round up the fourth digit and get 1.035 m.

Rounding value d: 0.004384010 cm

To round the number 0.004384010 cm to four significant figures, we look at the fifth digit, which is 4. Since 4 is less than 5, we keep the fourth digit as is and get 0.004384 cm.

Rounding value e: 0.00078100 mL

To round the number 0.00078100 mL to four significant figures, we look at the fifth digit, which is 1. Since 1 is less than 5, we keep the fourth digit as is and get 0.0007810 mL.

Rounding value f: 0.0098641 cg

To round the number 0.0098641 cg to four significant figures, we look at the fifth digit, which is 6. Since 6 is greater than 5, we round up the fourth digit and get 0.009865 cg.

Key concepts

Understanding Significant Figures

Significant figures are essential in representing the precision of a number. They indicate which digits in a measurement are reliable and meaningful. When you're dealing with measurements, it's important to convey not just the number itself, but also the confidence and resolution of the measurement. Significant figures help convey that. Here's a simple breakdown of what counts as significant:

• All non-zero digits (1, 2, 3, etc.) are significant.
• Zeros between non-zero digits are significant. For example, in 101.5, all the digits are significant, including the zero.
• Leading zeros (those appearing before all non-zero digits) are not significant. In 0.004, only the 4 is significant.
• Trailing zeros in a number with a decimal point are significant. In 2.300, all four digits are significant.
• Trailing zeros in a number without a decimal point are not significant. For instance, in 1500, only the 1 and 5 are significant unless specified otherwise (e.g., 1500.).

Being aware of these rules helps ensure measurements are as precise and transparent as possible. When rounding, these guidelines will determine which digits are retained and which are discarded.

Rounding Techniques for Accuracy

Rounding is a fundamental technique used to simplify numbers while preserving their significance and intent. It's especially crucial in scientific and financial calculations where precision matters. Here’s a concise guide on rounding numbers based on significant figures:

• If the digit to be dropped is less than 5, simply truncate after the required digit (e.g., rounding 6.342 to 2 significant figures results in 6.3).
• If the next digit is 5 or higher, increase the last retained significant digit by 1 (e.g., round 3.678 to 2 significant figures to get 3.7).
• When rounding a sequence of repeated calculations, apply rounding only at the final stage, not at each step, to avoid accumulating rounding errors.
• Round numbers consistently using the above rules to ensure that round-off errors do not bias your results in a particular direction.

These techniques ensure that you round numbers accurately using significant figures, ensuring clarity and consistency in every calculation.

Mastering Decimal Places

Decimal places are closely related to significant figures and are crucial when it comes to finite measurement accuracy. They determine how many digits should appear after the decimal point, reflecting the precision of the number.

• Decimal places can usually be counted straightforwardly by noting all digits to the right of the decimal point. For instance, in 3.141, there are three decimal places.
• In scientific contexts, controlling decimal places helps balance the level of precision with the ease of interpretation, preventing overly lengthy representations while maintaining significant precision.
• The choice of decimal places in a calculation often depends on the context, such as the type of calculation or the specific requirements of a task or assignment.
• In general, aim to match the decimal place precision used within the constraints of significant figures, which will ensure accuracy without cluttering numbers with unnecessary zeros.

By understanding and monitoring decimal places, you maintain a high level of precision in calculations, facilitating more reliable results.