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 calculate the exact product of \(26.2 \times 16.43\) as 430.486. However, a scientist would consider significant figures, providing the answer as 430. This is because in science, the context of measurements involves uncertainties and inaccuracies due to experimental limitations. By maintaining the answer with the same number of significant figures as the least precise number involved, we are essentially communicating the level of uncertainty in the data.

Step-by-step solution

Multiply the numbers

Firstly, let's multiply the two numbers to find the product: \(26.2 \times 16.43\). Use the normal multiplication process for decimal numbers, then count the total number of decimal places in both numbers (1 in 26.2 and 2 in 16.43) and place the decimal point in the product accordingly.

Mathematician's Answer

A mathematician would calculate the exact product and give the answer as follows: \(26.2 \times 16.43 = 430.486\) So, the product from a mathematician's perspective would be 430.486.

Determine the number of significant figures for each number

In the context of a scientist's perspective, we need to consider significant figures. Firstly, determine the number of significant figures in each number: - For 26.2, there are 3 significant figures. - For 16.43, there are 4 significant figures.

Find the lowest number of significant figures

According to the rule of multiplication for significant figures, the final answer should have a number of significant figures equal to the lowest number in the given multiplicands. In this case, 26.2 has the lowest number of significant figures, which is 3.

Round the product to the appropriate number of significant figures

Now, round the mathematician's answer 430.486 to the relevant number of significant figures (3): 430.486 rounded to 3 significant figures is 430.

Scientist's Answer and Justification

A scientist would provide the answer as 430. This is because in science, the context of measurements often involves uncertainties and inaccuracies due to experimental limitations. By maintaining the answer with the same number of significant figures as the least precise number involved, we are essentially communicating the level of uncertainty in the data. Thus, the scientist's answer is 430, and this approach takes into account the uncertainties and limitations of real-world data.

Key concepts

Scientific Notation

Understanding scientific notation first requires grasping its purpose. It is a way to write very large or very small numbers concisely. For example, the number 5,000 in scientific notation is written as \(5 \times 10^3\). Here, 5 is the significant figure, and \(10^3\) tells us how many places the decimal point has moved. This method allows for easier arithmetic operations and clearer data presentation.When dealing with scientific measurements, scientific notation is especially useful. It ensures precision and clarity, helping scientists to communicate findings effectively. Instead of writing long strings of numbers, they use scientific notation to maintain accuracy and make complex calculations more manageable.

Decimal Places

Decimal places are vital in mathematics and scientific measurements. They indicate how many digits appear after the decimal point in a number, such as in 26.2 or 16.43. When multiplying decimals, it's essential to consider the total number of decimal places. We align the numbers vertically without the decimal point and multiply them as if they are whole numbers. After multiplying, we place the decimal point in the product by counting the total decimal places from the multiplicands. For example, if you multiply two numbers with two and one decimal place, then your product should have three decimal places. Understanding decimal places ensures precise computations and results, especially in scientific contexts where precision is paramount.

Multiplication of Decimals

Multiplying decimals involves a few consistent steps. Initially, ignore the decimal points and multiply the numbers like whole numbers. Afterward, count the total number of decimal places in the factors. For example, when multiplying 26.2 (1 decimal place) by 16.43 (2 decimal places), ignore the decimal points and multiply 262 by 1643. After finding the product, count the total decimal places (3) and place the decimal correctly. The process ensures the accuracy of the product. This accuracy reflects how many measurements or data points you can trust in scientific calculations. The methodical approach of handling decimal places during multiplication helps maintain precision, vital for scientific experiments and data analysis.

Scientific Measurements

Scientific measurements are all about precision and accuracy. They often involve uncertainties due to instrument limitations or environmental factors. Thus, scientists use significant figures to convey the reliability of measurements. Significant figures provide a way to express numbers reflecting the precision of the measuring tool. For example, the number 26.2 has three significant figures, indicating it is precise to one decimal place. When multiplying measurements, the result must reflect the least precise measurement's number of significant figures. This rule maintains honest communication of data accuracy and reliability in scientific reporting. For example, by rounding 430.486 to three significant figures, we communicate the inherent uncertainty, maintaining transparency in scientific work.