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Chapter 1: Problem 16

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 to multiplying 26.2 by 16.43 is 430.466, as they consider the full value of the multiplication without rounding. On the other hand, a scientist would say the answer is 430, taking into account significant figures (3 in this case), which reflect the precision of the original measurements. This rounding prevents false precision and ensures that the answer stays true to the quality of the initial data.

Step by step solution

01

Multiply the numbers

To find the answer, we need to multiply 26.2 by 16.43. Use a calculator or perform the multiplication by hand: \(26.2 \times 16.43 = 430.466\)
02

A mathematician's answer

A mathematician would look at the full value of the multiplication, without any rounding or truncation of the decimals. Therefore, a mathematician would say the answer is: 430.466
03

A scientist's answer

A scientist would consider significant figures when giving an answer. Significant figures are the digits in a number that are meaningful and are used to indicate the precision of a measurement. In this case, we have two numbers: 26.2 has 3 significant figures (the trailing zero is considered significant because it follows a decimal point) 16.43 has 4 significant figures For multiplication and division, the rule is that the answer should have the same number of significant figures as the input number with the least significant figures. In this case, 26.2 has the least significant figures (3). Therefore, the answer should also have 3 significant figures.
04

Apply significant figures to the result

Now that we know the answer should have 3 significant figures, we need to round the multiplied result accordingly. In this case, we round the result of 430.466 to: 430 (3 significant figures)
05

Justifying the scientist's answer

The scientist's answer reflects the importance of the significant figures in real-world measurements because these figures indicate the precision and/or accuracy of the data. When we have a value with a limited number of significant figures, it means that its precision is limited. By rounding the answer, we prevent the introduction of false precision, ensuring that the answer stays true to the quality of the original measurements. In this specific case, since 26.2 has a limited precision, it's appropriate to limit the answer to the same number of significant figures, resulting in a rounded answer of 430.

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