Question

Three quantities, the results of measurements, are to be added. They are 2.0600,3.163 , and 1.12 . What is their sum to the correct mumber of significamt fiomes?

Short answer

Question: Add the following numbers while considering significant figures: 2.0600, 3.163, and 1.12. Answer: The sum of the three numbers, expressed to the correct number of significant figures, is 6.34.

Step-by-step solution

Identify significant figures and least precise number

First, we need to determine how many significant figures are in each of the measurements: - 2.0600: 5 significant figures - 3.163: 4 significant figures - 1.12: 3 significant figures The least precise number has 3 significant figures, so our final answer will have 3 significant figures as well.

Add the numbers

Now, we will add the three numbers together without rounding: Sum = 2.0600 + 3.163 + 1.12 = 6.3430

Round to the correct number of significant figures

The sum is currently given with 5 significant figures. However, we need to present the sum with the same number of decimal places as the least precise number, which has 3 significant figures: Sum = 6.34 (rounded to two decimal places) Therefore, the sum of the three numbers, expressed to the correct number of significant figures, is 6.34.

Key concepts

Measurement Accuracy

Measurement accuracy refers to how closely a measured value agrees with the true value of the quantity being measured. In physics, accuracy is pivotal because reliable results depend on how precise these measurements are. To understand measurement accuracy, consider a standard ruler. If you measure the length of an object with a ruler that has millimeter markings, you can be more accurate in your measurement compared to using a ruler with only centimeter markings.

When combining measurements with different degrees of accuracy, such as in the exercise involving the addition of 2.0600, 3.163, and 1.12, the accuracy of the result is limited by the least accurate measurement. This is why understanding the concept of significant figures is crucial in expressing your final answer with the appropriate accuracy.

Significant Figure Calculation

Significant figure calculation is the process of determining which digits in a number are meaningful in terms of contributing to its accuracy. In our exercise, the measurement 2.0600 has five significant figures, 3.163 has four, and 1.12 has three. The rule is that when adding or subtracting numbers, the result should be reported with no more decimal places than the least precise measurement.

In other words, the number of decimal places in the result reflects the number of decimal places in the measurement with the fewest decimal places. This is because the extra decimal places in more precise measurements cannot be justified if they are being added to a less precise measurement. The resultant sum therefore must be rounded off to maintain consistency across the precision levels.

Scientific Notation and Precision

Scientific notation is a method of writing numbers that accommodates values too large or too small to be conveniently written in standard decimal form. It is written as the product of a number between 1 and 10 and a power of 10. This method is particularly useful in physics for expressing quantities that span a vast range of scales.

For example, the sum of our measurements, 6.3430, could be expressed in scientific notation as \(6.3430 \times 10^0\). The precision of this number is given by the significant figures it contains. Precision refers to the degree of variation that occurs when multiple measurements of the same quantity are taken. It is related, but not equivalent, to accuracy — while accuracy indicates how close a measurement is to the true value, precision indicates consistency. When using scientific notation, it's important to only include significant figures in the coefficient (the number before the \(\times 10^n\)), thus communicating the precision of the measurement effectively.