Problem 110 - Chemistry Matter and Change

Question

Round each figure to three significant figures. a. 0.003210 \(\mathrm{g}\) \(\quad\) d. 25.38 \(\mathrm{L}\) b. 3.8754 \(\mathrm{kg}\) \(\quad\) e. 0.08763 \(\mathrm{cm}\) c. \(219,034 \mathrm{m}\) \(\quad\) f. 0.003109 \(\mathrm{mg}\)

Step-by-Step Solution

Answer

a. 0.00321 g; b. 3.88 kg; c. 219,000 m; d. 25.4 L; e. 0.0876 cm; f. 0.00311 mg.

Step 1: Understand Significant Figures

Significant figures in a number are all the non-zero digits, any zeros between them, and any trailing zeros in the decimal portion. For example, in 0.003210, the digits 3, 2, 1, and 0 are significant, meaning the number has 4 significant figures.

Step 2: Round 0.003210 g

Identify the first 3 significant figures in 0.003210, which are 3, 2, and 1. The digit after the third significant figure is 0, which does not affect the rounding. Thus, the rounded number is 0.00321 g.

Step 3: Round 3.8754 kg

Identify the first 3 significant figures in 3.8754, which are 3, 8, and 7. Check the fourth digit, which is 5, indicating we round the third digit up. Hence, 3.8754 rounds to 3.88 kg.

Step 4: Round 219,034 m

Identify the first 3 significant figures in 219,034, which are 2, 1, and 9. The fourth digit is 0, thus rounding doesn't affect the third digit. So, it rounds to 219,000 m.

Step 5: Round 25.38 L

Identify the first 3 significant figures in 25.38, which are 2, 5, and 3. Since the fourth digit is 8, we round up the third digit. Therefore, it rounds to 25.4 L.

Step 6: Round 0.08763 cm

Identify the first 3 significant figures in 0.08763, which are 8, 7, and 6. The fourth digit is 3, so the third significant figure remains unchanged. Consequently, it rounds to 0.0876 cm.

Step 7: Round 0.003109 mg

Identify the first 3 significant figures in 0.003109, which are 3, 1, and 0. The digit after the third significant digit is 9, so 0 is rounded up. Therefore, it rounds to 0.00311 mg.

Key Concepts

Rounding Numbers

Rounding numbers is a mathematical process used to simplify numbers, often making them easier to work with or understand. This is particularly useful when dealing with lengthy numbers or when looking for a number that is close to the true value but less precise. Rounding typically follows a basic set of rules:

• If the digit to be dropped is less than 5, the last remaining digit stays as is.
• If this digit is 5 or more, you increase the last remaining digit by 1.

In real-world applications, rounding is used in financial transactions, scientific measurements, and daily calculations. For instance, if you have a number like 3.8754, and you need to round it to three significant figures, you would end up with 3.88. This process ensures clarity and consistency, particularly in scenarios where excessive detail is unnecessary.

Mathematical Concepts

Mathematical concepts form the foundation of understanding and handling numbers. It's the logic behind the numbers and the operations we perform on them. In this context, we focus on significant figures — a critical concept in precision and accuracy. Significant figures represent all the meaningful digits in a number. Non-zero numbers are always considered significant. Zeros, on the other hand, need a little more attention:

• Zeros sandwiched between significant digits are significant.
• Trailing zeros in a decimal number are considered significant because they indicate precision.
• Leading zeros serve only as placeholders and are not significant.

Consider a number like 0.003210; the significant figures are 3, 2, 1, and 0. Using mathematical principles helps maintain consistency and precision in calculations, providing a robust framework for interpreting measurements and numerical data.

Precision in Measurement

Precision in measurement refers to the detail and exactness of the measurement. It shows how finely a measurement is made and how closely individual measurements are to each other. Precision is crucial in fields like engineering, science, and even finance, where exact values can significantly influence outcomes. Precision is reflected through the number of significant figures in a measurement. For example, in the number 219,034, choosing three significant figures results in 219,000. This indicates high precision in the original measurement but is simplified while maintaining the measurement's essence. Having a good grasp of precision helps in improving the reliability of data. It ensures that measurements are neither over-exaggerated nor understated, offering a clearer picture of the quantity being examined. In practice, this might mean recording measurements with appropriate instruments or ensuring numerical data is reported with just the right amount of detail.