Question

Round each number to four significant figures. \(\begin{array}{ll}{\text { a. } 84,791 \mathrm{kg}} & {\text { c. } 256.75 \mathrm{cm}} \\ {\text { b. } 38.5432 \mathrm{g}} & {\text { d. } 4.9356 \mathrm{m}}\end{array}\)

Short answer

When rounded to four significant figures: a. 84,791 kg becomes \(84,800 kg\) b. 38.5432 g becomes \(38.54 g\) c. 256.75 cm becomes \(256.8 cm\) d. 4.9356 m becomes \(4.936 m\)

Step-by-step solution

a. Rounding 84,791 kg

To round the number 84,791 kg to four significant figures, we first identify the fifth digit, which is 9. Since 9 is greater than or equal to 5, we will round up the fourth significant figure. The rounded number will be: 84,800 kg

b. Rounding 38.5432 g

To round the number 38.5432 g to four significant figures, we first identify the fifth digit, which is 3. Since 3 is less than 5, we will leave the fourth significant figure as it is. The rounded number will be: 38.54 g

c. Rounding 256.75 cm

To round the number 256.75 cm to four significant figures, we first identify the fifth digit, which is 5. Since 5 is greater than or equal to 5, we will round up the fourth significant figure. The rounded number will be: 256.8 cm

d. Rounding 4.9356 m

To round the number 4.9356 m to four significant figures, we first identify the fifth digit, which is 5. Since 5 is greater than or equal to 5, we will round up the fourth significant figure. The rounded number will be: 4.936 m

Key concepts

Rounding Numbers

Rounding numbers is a key skill in both mathematics and science. It helps to simplify numbers while maintaining a level of precision necessary for calculations and interpretations. Rounding numbers involves changing a number to a simpler or cleaner version, usually to a specified number of significant figures or decimal places.

To round numbers accurately:

• Identify which digit you need to round to, generally based on the number of significant figures required. For example, if you need four significant figures, you focus on the fourth digit.
• Look at the digit immediately following it to determine whether to round up or leave it as is.
• If this next digit is 5 or greater, increase the significant digit by one. If it's less than 5, then leave the significant figure as it is.

Through rounding, we streamline numbers like 84,791 kg down to 84,800 kg with four significant figures. By doing this consistently, calculations become more straightforward without losing necessary detail.

Precision in Measurements

Precision in measurements refers to how consistently you can obtain the same measurement results under unchanged conditions. It's vital in science and engineering where measurements form the basis of data interpretation and decision-making.

Precision is not the same as accuracy:

• Accuracy describes how close a measured value is to the true value.
• Precision refers to the repeatability and consistency of a set of measurements.

When taking measurements:

• Use tools and instruments capable of providing the necessary level of detail.
• Note the number of significant figures. They reflect the precision of the measurement.
• Understand that more significant figures usually indicate a higher precision.

Being precise and understanding significant figures ensures that your measurements are consistent and reliable over time, enabling you to draw valid conclusions from your experiments and research.

Chemistry Calculations

Chemistry calculations often involve a level of complexity due to the need for precision. Calculations can vary from finding molar concentrations to balancing chemical equations, all demanding careful consideration of significant figures.

Some essential tips for handling chemistry calculations include:

• Always maintain the correct number of significant figures throughout your calculations to ensure precision.
• Begin by identifying the least precise measurement or data point. Use its significant figures as a guideline for the precision of your final answer.
• Apply rounding consistently as per the rules for significant figures, particularly when performing multiple steps in calculations.

By adhering to these practices and rounding accurately—such as simplifying 256.75 cm to 256.8 cm while maintaining four significant figures—you can ensure your chemistry results are both reliable and meaningful. Such diligence improves the integrity of your findings and supports scientific rigour.