Question

Calculate to the correct number of significant figures. MISSED THIS? Read Section 1.7; Watch KCVs \(1.6,1.7,\) IWEs 1.5,1.6 a. \(43.7-2.341\) b. \(17.6+2.838+2.3+110.77\) c. \(19.6+58.33-4.974\) d. \(5.99-5.572\)

Short answer

a. 41.4, b. 133.5, c. 73.0, d. 0.42

Step-by-step solution

Calculating a. (43.7 - 2.341)

Perform the subtraction: 43.7 (three significant figures) - 2.341 (four significant figures). Because we must match the least number of decimal places in the original numbers, the final answer should have only one decimal place. Calculate the result and round appropriately.

Calculating b. (17.6 + 2.838 + 2.3 + 110.77)

Perform the addition: 17.6 (three significant figures, one decimal place) + 2.838 (four significant figures, three decimal places) + 2.3 (two significant figures, one decimal place) + 110.77 (five significant figures, two decimal places). The answer should match the least number of decimal places in the original numbers, which is one decimal place. Add the numbers and round to the nearest tenth.

Calculating c. (19.6 + 58.33 - 4.974)

Perform the addition and subtraction: 19.6 (three significant figures, one decimal place) + 58.33 (four significant figures, two decimal places) - 4.974 (four significant figures, three decimal places). The result should be rounded to match the least number of decimal places in the original numbers, which is one decimal place. Calculate the combined result and round appropriately.

Calculating d. (5.99 - 5.572)

Perform the subtraction: 5.99 (three significant figures, two decimal places) - 5.572 (four significant figures, three decimal places). The result must be rounded to the least number of decimal places in the original numbers, which is two decimal places. Calculate the result and round to the nearest hundredth.

Key concepts

Scientific Notation

When dealing with very large or very small numbers, it's often convenient to use scientific notation. This method simplifies numbers by expressing them as a product of two factors: a coefficient and a power of ten. For example, the distance of the Earth from the Sun, approximately 149,600,000 kilometers, is more concisely written in scientific notation as \(1.496 \times 10^8\) kilometers.

The coefficient should be a number greater than or equal to 1 and less than 10, and it captures the significant figures of the original number. When moving from standard to scientific notation, the decimal point is shifted such that only significant figures remain, and the power of ten shows how many places the decimal point moved. By representing numbers in this format, it becomes easier to manage calculations, especially when using a calculator, and to showcase the precision of measurements in terms of significant figures.

Rounding Numbers

Rounding numbers is a fundamental concept in mathematics, used to make numbers easier to work with or to reflect the precision of the data. In essence, when you round a number, you are finding the closest number to it with a specified level of precision, such as the nearest ten, one, or tenth. When rounding to a specific decimal place, look at the digit to the right of the desired place. If this digit is 5 or greater, you increase the target digit by one; if it's less than 5, the target digit remains the same.

For example, rounding \(0.7745\) to three significant figures results in \(0.775\), because the digit in the fourth place is 5, prompting an increase in the third place. It's vital to round off only as a final step in calculations to avoid accumulating rounding errors.

Significant Figures Rules

Significant figures are the digits that carry meaning in the precision of a number, and there are specific rules for identifying them in different types of numbers. To determine the number of significant figures:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.
• Trailing zeros in a decimal number are significant.
• Trailing zeros in a whole number may or may not be significant, depending on whether a decimal is present.

When performing calculations, the number of significant figures in the final answer should reflect the least precise measurement used in the calculation. In addition, when adding or subtracting, the result should be rounded to the least number of decimal places of any number in the operation, while multiplication and division results should reflect the least number of significant figures from the numbers involved. These rules ensure that the precision of the result isn't overstated, maintaining the integrity of the data.