Problem 7 - Technical Mathematics with Calculus

Question

Multiply each approximate number and retain the proper number of digits in your answer. $$93.9 \times 0.0055908$$

Step-by-Step Solution

Answer

The product of 93.9 and 0.0055908 rounded to three significant figures is 0.525.

Step 1: Identify the Significant Figures

The first number, 93.9, has three significant figures because there are three digits known with certainty (one of which is after the decimal). The second number, 0.0055908, has six significant figures. The zeros to the left are not significant; they are merely placeholders.

Step 2: Perform the Multiplication

Multiply the two numbers ignoring the significant figures for now.\[93.9 \times 0.0055908 = 0.52509132\]

Step 3: Round the Result to Proper Number of Significant Figures

The result should have the same number of significant figures as the number with the fewest significant figures in the original problem. Since 93.9 has three significant figures and that's the least, round the result to three significant figures.\[0.52509132 \approx 0.525\]

Key Concepts

Significant Figures Rules

When it comes to quantitative science, the accuracy and precision of measurements are paramount, and that's where significant figures come into play. Significant figures are the digits in a number that carry meaning contributing to its precision, and they include all non-zero numbers, zeros between non-zero digits, and trailing zeros in a decimal number. For instance, in the number 93.9, all three digits are significant: '9' and '3' as non-zero digits, and the final '9' as a trailing zero in a decimal number.

The rules for significant figures in multiplication dictate that the product must not have more significant figures than the least precise number used in the calculation. This ensures that the result reflects the precision of the least accurate measurement. For example, when multiplying 93.9 with 0.0055908, because 93.9 has three significant figures and that's the lowest count between the two, the result is rounded to three significant figures, giving an answer of approximately 0.525.

Rounding Significant Figures

Rounding to the proper number of significant figures is essential in maintaining the integrity of measurements throughout calculations. When rounding, if the digit to the right of the last significant figure is five or higher, you increase the last significant figure by one. If it's lower than five, you leave the last significant figure as it is.

Consider our multiplication example: \(93.9 \times 0.0055908 = 0.52509132\). Rounding to three significant figures, we start at the first non-zero digit and count three figures to the right. The third figure is '5', and the next number, '0', does not require us to round up. Thus, we retain the '5' and arrive at \(0.525\), ensuring the final answer appropriately communicates its precision. It's worth noting that only the numbers in the final answer are rounded; intermediate numbers retain more significant figures to avoid rounding error accumulation.

Multiplication of Approximate Numbers

Working with approximate numbers can introduce uncertainty in calculations. In multiplication, this uncertainty must be reflected in the final product. That’s why the result's number of significant figures is based on the input with the smallest number of significant figures.

In this context, approximate numbers with many significant figures usually represent a higher precision. Yet, in the process of multiplication, it's essential not to give the impression that the product is more precise than the least precise number implies. Thus, the result after multiplying should represent the inherent precision limits of the initial measurements. This consciousness in precision during calculations helps maintain consistent and reliable handling of significant figures throughout the scientific process.