Question

Do each calculation without your calculator and give the answers to the correct number of significant figures. a. \(1.76 \times 10^{-3} / 8.0 \times 10^{2}\) b. \(1.87 \times 10^{-2}+2 \times 10^{-4}-3.0 \times 10^{-3}\) c. \(\left[\left(1.36 \times 10^{5}\right)(0.000322) / 0.082\right](129.2)\)

Short answer

a. The result for (a) is rounded to 2 significant figures, the least number in the original numbers. b. For (b) the answer is kept to the least number of decimal places in the problem, after reconciling the exponents. c. The result for (c) is rounded to the least number of significant figures amongst the factors within the brackets.

Step-by-step solution

Identify Significant Figures

Determine the number of significant figures in each given number. For a., 1.76 has 3 significant figures, and 8.0 has 2 significant figures. For b., 1.87 has 3 significant figures, 2 has 1 significant figure, and 3.0 has 2 significant figures. For c., 1.36 has 3 significant figures, 0.000322 has 3 decimal significant figures, 0.082 has 2 decimal significant figures, and 129.2 has 4 significant figures.

Perform the Calculation for a.

To divide the numbers, calculate the coefficients (1.76 by 8.0) and then subtract the exponents of 10 (-3 - 2). The calculation yields (1.76 / 8.0) * 10^(-5). When performing the division, keep the result to the least number of significant figures in the coefficients.

Perform the Calculation for b.

Sum the numbers ensuring that they're all in the same exponential form for consistency and comparison, adjusting where necessary. Align according to the least significant decimal place when adding or subtracting, and the result should have the same number of decimal places as the term with the most decimal places involved in the operation.

Perform the Calculation for c.

For the calculation within the brackets, multiply 1.36 * 10^5 by 0.000322, then divide by 0.082. The multiplication and division should be done in sequence while keeping track of significant figures. Afterward, multiply the result from the brackets by 129.2.

Round the Final Answers

Round the final answers from steps 2, 3, and 4 to the correct number of significant figures determined from the respective least number of significant figures of the numbers involved in the calculations for each part a, b, and c.

Key concepts

Significant Figures Rules

Understanding the rules of significant figures is crucial for precision in scientific calculations. Significant figures are the digits in a number that carry meaning contributing to its precision. This includes all non-zero digits, zeros between non-zero digits, and trailing zeros in a decimal. To accurately identify them, remember these key points:

• All non-zero numbers (1-9) are always significant.
• Any zeros between non-zero digits are significant.
• Leading zeros (zeros before non-zero numbers) are not significant; they merely indicate the position of the decimal point.
• Trailing zeros (zeros after a decimal point) are significant only if the number has a decimal point present.
• In a whole number with a clear decimal point, trailing zeros are significant.

When you're adding or subtracting with significant figures, the result should have the same number of decimal places as the measurement with the least decimal places. During multiplication and division, the result should have the same number of significant figures as the measurement with the least significant figures. Correctly applying these rules ensures your final answer has appropriate precision.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used in science, engineering, and mathematics to simplify numbers and computations. In scientific notation, a number is written as the product of two factors:

• A coefficient that is greater than or equal to 1 but less than 10.
• 10 raised to a power (the exponent), which indicates the number of places the decimal point has been moved.

For example, the number 0.00076 in scientific notation is expressed as \(7.6 \times 10^{-4}\). This notation eliminates ambiguity by clearly showing the significant figures in a number. When performing calculations with numbers in scientific notation, it’s crucial to handle the coefficients separately from the exponents. Combining or separating the powers of ten appropriately can simplify the arithmetic operations.

Arithmetic with Significant Figures

Performing arithmetic with significant figures requires mindfulness to maintain the correct level of precision. Here's a quick guide to help you

Addition and Subtraction

When adding or subtracting, align the numbers by their decimal points and ensure that the result has the same number of decimal places as the least precise measurement.

Multiplication and Division

For multiplication and division, it is the number of significant figures that matters, not the decimal places. The final result should be reported with no more significant figures than the least precise measurement used in the calculation.

By keeping these rules in mind, you can accurately perform arithmetic operations while respecting the precision of the initial measurements. This methodology maintains consistency and reduces errors in calculations.

Rounding Significant Figures

Rounding to the correct number of significant figures is the final step to ensure that your result appropriately represents the data's accuracy. The convention is straightforward:

• If the digit to be removed is less than 5, simply drop it along with all the following digits.
• If the digit to be removed is 5 or greater, increase the last remaining digit by one.

Round off the results from calculations only at the very end to prevent rounding errors from affecting the accuracy. When confronted with sequential operations, carry through as many significant figures as possible until the final step. Remember that calculators do not automatically observe the rules of significant figures, so it is up to you to manually round off your results appropriately.