Perform each calculation to the correct number of significant figures. (a) \(1459.3+9.77+4.32\) (b) \(0.004+0.09879\) (c) \(432+7.3-28.523\) (d) \(2.4+1.777\)

Grasping the method for adding significant figures is crucial for maintaining precision in scientific calculations. When you are adding numbers together, like in example (a) with the numbers 1459.3, 9.77, and 4.32, the first step is always to align the decimal places of all the numbers you are adding. This means writing them so that all of the decimal points are lined up vertically.

Once they are aligned, you can add the numbers together as you would normally, but there's a catch that's often overlooked—the answer must be rounded off to the same number of decimal places as the number with the fewest decimal places. In example (a), that would be 2 decimal places. This method ensures that your final result reflects the appropriate level of precision.

Rounding is a technique used to reduce a number to a certain number of significant figures for precision. A standard rule for rounding is: if the digit to the right of your last significant figure is greater than or equal to 5, you increase the last significant figure by 1; if it's less than 5, you leave it as is.

In our calculations, such as in step 2 for problem (a), when the sum of 1459.3, 9.77, and 4.32 gives us 1473.39, we only need two decimal places. Since the third decimal place is a 9, we round up, getting 1473.39. Following proper rounding rules is vital to ensure the correct level of precision.

Precision in chemistry calculations directly affects the validity and reproducibility of results. It's essential to understand that the precision of a measured or calculated quantity is limited by the least precise measurement. For example, when you're performing a calculation with several terms, as in step 5 with problem (c) dealing with the terms 432, 7.3, and -28.523, you must pay attention to the number with the least decimal places (7.3, which has one decimal place). After performing the calculation, the result must be rounded off to maintain that level of precision (410.777 becomes 410.8). This approach helps to preserve the integrity of your data.

Decimal place alignment is imperative when adding or subtracting in scientific calculations. The numbers must be written so their decimal points are in a straight line, and when the sum or difference is determined, it should have the same number of decimal places as the number in the equation with the least. A brief example of this is in step 6 for problem (d): when adding 2.4 (which has 1 decimal place) to 1.777, we align the decimal points and perform the addition to get 4.177, but we round it to 4.2 to maintain the correct number of decimal places. Careful alignment of decimals is key to ensuring accurate calculations.