Question

Consider the addition of 15.4 to 28. 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

A mathematician would add the numbers directly and say the answer is 43.4. A scientist, considering significant figures, would round the answer to the least precise measurement (in this case, no decimal places) and say the answer is 43. The scientist's answer accounts for the uncertainty and precision of the measurements, acknowledging that there is inherent uncertainty in the data.

Step-by-step solution

Find the sum of the numbers - Mathematician's answer

Add the two numbers: \(15.4 + 28 = 43.4\) So, a mathematician would say the answer is 43.4.

Estimate the sum with significant figures - Scientist's Answer

Here, the given numbers are 15.4 and 28. The first number has one decimal place, while the second number has no decimal places. A scientist would likely consider the significance of the decimal place in each number and round their answer accordingly. When adding numbers, the result should have the same number of decimal places as the least precise measurement. In this case, that would be the number 28, since it has no decimal places.

Round the sum to the appropriate number of significant figures

After calculating the sum of 43.4, we should round it to the nearest whole number since the least precise measurement has no decimal places. In this case, the sum rounded to the nearest whole number is: \(43.4 \approx 43\) So, a scientist would say the answer is 43.

Explain the reasoning behind the scientist's answer

A scientist's answer is different from a mathematician's answer because scientists want to account for the uncertainty and precision of the measurements given. In this case, only one number has a decimal place (15.4) while the other number has none (28). The scientist would use the concept of significant figures to ensure the result is expressed with the appropriate level of precision. In this context, using the number 43 as the sum reflects the level of precision in the original measurements and acknowledges that there is uncertainty in the data.

Key concepts

Precision in Measurement

Precision in measurement refers to the detail and exactness of a numerical value that describes a quantity. It denotes how specific a value is considering the limitations of the measuring tools used. In scientific contexts, every measurement comes with a degree of uncertainty. This means:

• Measurements are never perfectly exact but have estimated digits.
• The degree of exactness can significantly impact results in scientific computations.

In the example from the exercise, when adding 15.4 and 28, the precision is determined by the number of decimal places. The number 15.4 has one decimal place, indicating a more precise measurement than 28, which has none. Thus, scientists prefer maintaining results that reflect the least precise measurement used in the calculation to avoid indicating precision that doesn’t exist. This ensures that any findings or conclusions drawn are supported by the reliability of the data provided.

Rounding Numbers

Rounding numbers is a crucial part of expressing the calculated sums in a way that accurately presents the precision of the measurements used. When rounding, the result should reflect the measurement with the least number of decimal places or significant figures, depending on the context.

• When adding numbers, the sum should have the same number of decimal places as the number with the least decimal places.
• When performing other operations, like multiplication or division, the answer should have the same number of significant figures as the measurement with the fewest significant figures.

During the rounding process, numbers are adjusted to the nearest value that can express the precision accurately. In the exercise, 43.4 was rounded down to 43 because we needed to match the precision level of 28. This method ensures that the result does not suggest any superfluous accuracy.

Mathematical vs. Scientific Notation

Mathematical notation and scientific notation serve different purposes in handling numbers. Mathematical notation is often straightforward, presenting numbers as they are, like calculations with decimal points and whole numbers. However, scientific notation is used to express very large or very small numbers in a simplified form, emphasizing their significant figures.

• Mathematical notation: Simple representation, ensuring accurate calculations without altering the original precision.
• Scientific notation: Expresses numbers as a product of a number between 1 and 10 and a power of 10, such as \( 4.34 \times 10^1 \) for 43.4, allowing easier interpretation and notation of precision.

In the context of our exercise, while a mathematician would simply compute the sum and provide it directly (e.g., 43.4), a scientist would alter the result to reflect with an appropriate level of precision, potentially using scientific notation to communicate this effectively, especially when dealing with vast ranges of values.