Problem 30 - Technical Mathematics with Calculus

Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$\begin{aligned} &809-463+744\\\ &758-964+508 \end{aligned}$$

Step-by-Step Solution

Answer

The results of the computations are \(1090\) and \(302\) respectively.

Step 1: Perform the First Calculation

Begin with the first series of operations: \(809 - 463 + 744\). First, subtract 463 from 809, which equals 346. Then add 744 to this result: \(346 + 744 = 1090\).

Step 2: Perform the Second Calculation

Now, for the second series: \(758 - 964 + 508\). Subtract 964 from 758, which gives a negative result: \(758 - 964 = -206\). Then, add 508 to this result: \(-206 + 508 = 302\).

Step 3: Round the Results (if necessary)

In this case, there is no need to round since the original numbers are whole numbers and the operations result in whole numbers.

Key Concepts

Combined Operations

When working with mathematical problems, it's common to encounter scenarios that require the use of combined operations. These involve performing a series of calculations following a specific order, typically enforced by the rules of arithmetic. For instance, in the exercise given, the operations involve subtraction followed by addition.

It’s essential to execute operations in the correct sequence to avoid errors. This is where the BODMAS/BIDMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) rule comes into play. However, in our problem, as there are no brackets or powers, we start with subtraction and then proceed to addition.

Exploring the Steps:

1. Identify the sequence of operations required.
2. Perform each operation in the correct order.
3. Keep an eye on the signs of the numbers, especially when dealing with negatives.

Careful computation is key to avoid slipping up with combined operations. In our example, we ensure that the subtraction is done first, followed by the addition while paying attention to the signs to ensure accuracy.

Approximate Numbers

In many real-world scenarios, working with exact numbers isn't always possible. This is where approximate numbers come into play. Approximate numbers are not exact; they are close to the true value but rounded or estimated. When dealing with these, it's important to keep in mind that any operations performed may also yield approximate results.

Considering Approximations:

1. Be aware that approximate numbers can lead to approximate results when used in calculations.
2. Use approximate numbers responsibly by clearly indicating they are estimates.
3. Understand that operations with approximate numbers can affect precision and accuracy.

While the numbers in our provided exercise are not inherently approximate, they might represent real-world measurements that have been rounded. Hence, the results will also be treated as approximations of the actual values.

Significant Digits

Significant digits (or significant figures) are crucial when we wish to represent the precision of a number. They include all digits that contribute to the measurement precision, from the first non-zero digit to the last accurately known digit.

Using significant digits correctly allows us to convey exactly how precise our numbers are. When performing operations like addition or subtraction, the result should not have more significant digits than the least precise number involved in the computation.

Minding Significant Digits:

1. Identify the least precise number in a set of data.
2. After performing operations, ensure that your result does not exceed this level of precision.
3. Round off your final answer to reflect the correct number of significant digits.

In our textbook exercise, since we're dealing with whole numbers, we don't have to adjust for significant digits. However, if we had started with approximate numbers, rounding off to the appropriate number of significant digits in our final answer would be imperative to maintain the integrity of the data.