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Chapter 1: Problem 18

Consider multiplying \(26.2\) by \(16.43 .\) What would a mathematician say the answer is? What would a scientist say? Justify the scientist's answer, not merely citing the rule, but explaining it.

Short Answer

Expert verified
A mathematician would say the answer is 430.246, which is the exact product of 26.2 and 16.43. A scientist, on the other hand, would say the answer is 430, rounded to 3 significant figures based on the least precise measurement (26.2 has 3 significant figures). This is because significant figures provide an estimate of the precision and accuracy of measurements, and rounding helps scientists maintain consistency in their calculations and acknowledges the limitations of measurements or instruments used.

Step by step solution

01

Multiply the given numbers.

Multiply the given numbers 26.2 and 16.43. This would be the mathematician's answer as it provides an exact product. \(26.2 \times 16.43 = 430.246\)
02

Identify significant figures in the given numbers.

To provide the scientist's answer, we need to round off the above product to an appropriate significant figure count. First, let's identify the significant figures in each given number: - In 26.2, there are 3 significant figures - In 16.43, there are 4 significant figures
03

Round off the product to the least significant figure count.

In scientific calculations, we round off the product to the least number of significant figures in any of the given numbers. In our case, the least count of significant figures is 3. Therefore, we should round off the product obtained in Step 1 to 3 significant figures. We have the product: 430.246 Round it off to 3 significant figures, and we get 430.
04

Justify the scientist's answer.

The scientist's answer is rounded off to 3 significant figures because the concept of significant figures is important in scientific calculations. Significant figures provide an estimate of the precision and accuracy of measurements. When multiplying two numbers with different significant figures, the result should reflect the least precise measurement (i.e., the measurement with the least number of significant figures). This is because, in a chain of calculations, the accuracy of the final result cannot be more precise than the least accurate measurement involved. In our case, "26.2" has 3 significant figures, which indicates that the measurement is precise up to one decimal place. To maintain this level of precision consistently in the final product, we round off our answer to the same number of significant figures. Thus, the scientist would say the answer is 430, while the mathematician would say it is 430.246. This rounding helps scientists maintain the consistency of precision in their calculations and acknowledges the limitations of measurements or instruments used for such measurements.

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Most popular questions from this chapter

A rule of thumb in designing experiments is to avoid using a result that is the small difference between two large measured quantities. In terms of uncertainties in measurement, why is this good advice?

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