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

The mathematician's answer to the addition of \(15.4\) and \(28\) would be \(43.4\), while the scientist's answer would be \(43\). The scientist's answer considers the significant figures involved in the measurements, acknowledging that the answer can't have more precision than the least precise measurement. This approach takes into account the accuracy and uncertainty of the measurements.

Step-by-step solution

Adding the numbers

To begin, we will add the two numbers together: \(15.4 + 28 = 43.4\)

Understanding the mathematician's answer

A mathematician would simply report the answer we calculated in Step 1, which is: \(43.4\)

Understanding significant figures

To find the scientist's answer, we need to understand significant figures. Significant figures represent the number of known digits in a measurement, with the idea that the more significant figures, the more accurate a measurement is. In this problem, we have the following numbers of significant figures: - 15.4 has three significant figures - 28 has two significant figures When performing mathematical operations, we need to follow the rules for significant figures in order to maintain the accuracy of the measurements.

Applying the significant figures rules in addition

In addition, the answer should have the same number of decimal places as the least precise measurement (the one with the least number of decimal places). In this case, the least precise measurement is 28, which has no decimal places. Therefore, the scientist's answer should also have no decimal places.

Calculate the scientist's answer

To find the scientist's answer, we'll round the mathematician's answer to the nearest whole number: \(43.4 \approx 43\)

Explaining the scientist's answer

The scientist's answer is different from the mathematician's answer because the scientist is considering the precision of the measurements. By limiting the answer to the same precision as the least precise measurement (in this case, 28), we're acknowledging that the answer can't have more precision than the least precise measurement involved. This approach best represents the accuracy and uncertainty of the measurement. So, the mathematician's answer is \(43.4\), while the scientist's answer is \(43\).

Key concepts

Measurement Accuracy

Measurement accuracy is crucial when dealing with scientific data. It refers to how close a measured value is to the true value or the accepted standard of measurement. In our everyday calculations and scientific experiments, accuracy dictates the reliable representation of our results. In the given exercise, accuracy is ensured by considering significant figures. When adding numbers like in the exercise, you must ensure the final answer reflects the precision of the original measurements. This is because each measurement carries inherent uncertainty. In our case, this means limiting the number of decimal places in the answer to match the least precise measurement. The number 28, which does not have any decimal places, is less precise compared to 15.4, which has one decimal place. Therefore, the result should be rounded according to this least precise value, giving us an accurate representation of the measurements used.

Scientific Notation

Scientific notation is a way to express numbers that accommodate values either too large or too small for standard decimal notation. It's essential for simplifying the reading and calculations of such numbers, making it widely used in scientific and engineering fields. While scientific notation wasn't explicitly utilized in this particular exercise, understanding it helps when the numbers involved are outside the "comfortable" range.Scientific notation expresses numbers as a product of a factor (ranging from 1 to less than 10) and a power of ten. For example:

• The number 2,800 can be written as \(2.8 \times 10^3\).
• 0.0154 becomes \(1.54 \times 10^{-2}\).

Even though our exercise used straightforward numbers, knowing how to write them in scientific notation can be crucial when handling vast data sets. This ensures consistency in measurement accuracy and significant figure application.

Significant Figures Rules

Significant figures are all about identifying which numbers in a measurement contribute to its precision. They give scientists a necessary protocol for when and how results should be rounded to maintain a reasonable level of accuracy. Some key rules for significant figures include:

• All non-zero numbers are significant (e.g., 123 has three significant figures).
• Zeros located between significant numbers are significant (e.g., 1002 has four significant figures).
• Leading zeros are not significant (e.g., 0.0034 has two significant figures).
• Trailing zeros are significant if they come after a decimal point (e.g., 2.3400 has five significant figures).

In operations like addition or subtraction, the total result should match the decimal placement of the least precise measurement, as seen in our exercise where 15.4 (one decimal) and 28 (zero decimals) were added. The result, 43.4, was then rounded to 43 to reflect the least decimal placement, 28, preserving the measurement's precision.