Question

How many digits after the decimal point should be reported when the calculation \((10,434-9.3344)\) is performed?

Short answer

There should be 3 digits after the decimal point reported when the calculation is performed, and the rounded answer is \(1.100\).

Step-by-step solution

Perform the subtraction

Subtract the numbers as given: \(10.434 - 9.3344 = 1.0996\)

Identify least number of digits after the decimal point

Determine the least number of digits after the decimal point in the original numbers: 10.434 has 3 digits after the decimal point, while 9.3344 has 4 digits; hence, the least number among them is 3.

Round the result to the appropriate number of digits after the decimal point

Since we need 3 digits after the decimal point, we round the result accordingly: \(1.0996 \approx 1.100\) (notice that we round up because the next digit, 9, is greater than or equal to 5). So there should be 3 digits after the decimal point reported when the calculation is performed, and the rounded answer is 1.100.

Key concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used in science and engineering to make calculations with such numbers more manageable. In this notation, a number is written as the product of two parts: a coefficient that is greater than or equal to 1 and less than 10, and a power of 10. For example, the number 0.000123 can be written in scientific notation as \(1.23 \times 10^{-4}\).

The use of scientific notation is ubiquitous in chemistry, where chemists often deal with quantities like the Avogadro number, \(6.022 \times 10^{23}\), or extremely small measurements such as the mass of an electron \(9.109 \times 10^{-31}\) kg. Mastering scientific notation is crucial for students, as it ensures accuracy and simplifies the analysis and communication of information.

Decimal Places

The term 'decimal places' refers to the number of digits to the right of the decimal point in a number. It is a critical concept in chemistry and other sciences, where precise measurements are necessary. The number of decimal places has a direct impact on the precision of the calculations. For instance, when reporting the result of the calculation \((10,434 - 9.3344)\), the number of decimal places is determined by the original numbers involved in the computation.

Ruling Decimal Places in Calculations

When adding or subtracting numbers, the result should be reported with the same number of decimal places as the number with the least decimal places from those being added or subtracted. In the provided exercise, \(10.434\) has three decimal places, and \(9.3344\) has four, so the result is rounded to three decimal places, yielding \(1.100\). This rule helps ensure that the reported result is not misleadingly precise.

Arithmetic in Chemistry

Arithmetic in chemistry involves applying mathematical operations such as addition, subtraction, multiplication, and division to chemical data. This often includes working with measurements and must accommodate the significant figures and decimal places associated with them.

Understanding significant figures and correct rounding practices is crucial because they reflect the precision and limitations of the experimental data. It’s important not to overstate the precision of calculated results, as done correctly in the subtraction exercise where the answer, \(1.0996\), was rounded to three decimal places based on the least precise measurement, resulting in \(1.100\). In more complex chemistry calculations, understanding how to propagate uncertainty through these calculations ensures accurate and credible results, which are fundamental to scientific integrity and progress.