Question

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

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

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. \(26.2 \times 16.43 = 430.246\)

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

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.

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.

Key concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It is particularly useful in science, where dealing with extremely big or tiny quantities is common. In scientific notation, numbers are written as the product of a coefficient and a power of ten. For example, the number 430, which we obtained in our multiplication exercise, could further be expressed in scientific notation as \(4.3 \times 10^2\).

This method of representation is helpful for several reasons:

• It simplifies calculations by concentrating on significant digits along with the exponent.
• Makes it easier to compare the scale of numbers, since the exponent tells you how many zeros follow.
• Improves readability and clarity, especially when dealing with scientific data that involves many zeros.

For students or professionals working with measurements and scientific data, mastering scientific notation is essential because it keeps track of significant figures and computational accuracy across calculations.

Rounding Rules

Rounding rules are crucial when dealing with significant figures in scientific calculations. Precision in representation is important, but so is understanding when and how much to round off.

When you round numbers, follow these basic rules:

• If the digit to be dropped is less than 5, the last retained digit remains unchanged.
  Example: Rounding 430.2 to one decimal place gives 430.
• If the digit to be dropped is 5 or greater, you increase the last retained digit by one.
  Example: Rounding 430.246 to three significant figures results in 430.

These rounding rules are applied after performing your calculations. The purpose is to ensure that the final result is reported to a precision level that matches the least accurate measurement used. This consistency helps prevent overstating the precision of the result.

Precision in Measurements

Precision in measurements indicates the reproducibility or repeatability of a series of measurements. It's about how closely multiple measurements of the same object or quantity are to each other.

In scientific practices, every decimal or non-decimal number carries meaning related to precision:

• The number 26.2 in our exercise has three significant figures, meaning its precision is to the tenths decimal place.
• The number 16.43, with four significant figures, is more precise, going down to the hundredths decimal place.
• The outcome of rounding after multiplication must reflect the least precise measurement, which keeps the final result's validity in line with source uncertainties.

Acknowledging measurement precision helps in realistic data analysis. It's vital for assessing the quality and reliability of scientific data and understanding the inherent limitations of experimental instruments.

Scientific Calculations

Scientific calculations frequently involve multiple steps, integrating measurements and mathematical operations. Each step must respect the rules of significant figures and rounding to maintain overall accuracy.

When performing calculations like multiplication:

• Determine the number of significant figures in each number before starting.
• Complete the math operation using exact values first without rounding.
• Apply rounding rules at the end, aligning significant figures with the input that has the least precision.

By sticking to this approach, scientists ensure that inconsistencies are minimized, and the results obtained truly reflect the data's accuracy. Scientific calculations not only provide results but do so considering the limitations in measurements, ensuring interpretations are grounded in reality.
Exploring these concepts helps in understanding how scientific theories and conclusions rely heavily on rigorously quantified calculations.