Expressing Sums Without Parentheses And Calculating Results

In mathematics, especially when dealing with addition and subtraction of signed numbers, it's crucial to understand how to express these operations without parentheses. This not only simplifies the expressions but also makes calculations more straightforward. In this article, we will delve into the process of rewriting expressions involving signed numbers as sums without parentheses, rearranging terms to group like signs together, and then computing the final result. This approach is particularly useful in algebra and arithmetic, providing a solid foundation for more advanced mathematical concepts.

Before diving into the specifics, it's essential to grasp the concept of signed numbers. A signed number is simply a number that is either positive or negative. Positive numbers are greater than zero, while negative numbers are less than zero. The sign of a number indicates its direction relative to zero. For example, $+5$ represents a positive five, while $-5$ represents a negative five. Understanding signed numbers is foundational to performing operations such as addition, subtraction, multiplication, and division with these numbers.

The Significance of Parentheses

Parentheses in mathematical expressions serve the purpose of grouping terms and indicating the order of operations. They tell us which operations to perform first. When dealing with signed numbers, parentheses often enclose the numbers along with their signs to prevent ambiguity. However, for the sake of simplicity and calculation efficiency, it's beneficial to remove these parentheses and rewrite the expressions as simple sums.

Step-by-Step Process

The key to converting expressions with parentheses into sums without them lies in understanding the rules of signed number arithmetic. Here's a step-by-step process:

1. Identify the Operations: Look at the expression and identify all the addition and subtraction operations.
2. Rewrite Subtraction as Addition: Subtraction can be thought of as adding the negative of a number. For example, $a - b$ is the same as $a + (-b)$.
3. Remove Parentheses: Once all subtractions are rewritten as additions, you can remove the parentheses. The sign preceding the number remains with the number.

Let's illustrate this with an example:

$(+15) + (-27) - (-8) - (+12) + (+9)$

Following the steps:

1. Identify the Operations: We have addition and subtraction operations.
2. Rewrite Subtraction as Addition:
   • $-(-8)$ becomes $+8$ (since subtracting a negative is the same as adding a positive).
   • $-(+12)$ becomes $-12$ (since subtracting a positive is the same as adding a negative).

So the expression becomes:

$(+15) + (-27) + (+8) + (-12) + (+9)$

3. Remove Parentheses: Now, we can remove the parentheses:

$15 - 27 + 8 - 12 + 9$

Rearranging Terms

After removing parentheses, the next step is to rearrange the terms so that the positive numbers are grouped together and the negative numbers are grouped together. This makes the calculation process more manageable and less prone to errors. Rearranging terms is permissible due to the commutative property of addition, which states that the order in which numbers are added does not affect the sum.

In our example, $15 - 27 + 8 - 12 + 9$, we can rearrange the terms as follows:

$15 + 8 + 9 - 27 - 12$

Now, we have all the positive numbers together and all the negative numbers together.

Adding Positive Numbers

First, add all the positive numbers together:

$15 + 8 + 9 = 32$

Adding Negative Numbers

Next, add all the negative numbers together. Remember to keep the negative sign:

$-27 - 12 = -39$

Combining the Sums

Finally, combine the sum of the positive numbers and the sum of the negative numbers:

$32 + (-39) = 32 - 39 = -7$

So, the final result of the expression is $-7$.

Let's consider the expression:

(1) (+15) + (-27) - (-8) - (+12) + (+9)

Our goal is to rewrite this expression without parentheses, group like terms, and compute the result.

Step 1: Rewrite the Expression Without Parentheses

To start, we'll rewrite the subtraction operations as addition of negative numbers. This involves changing the subtraction of a negative number to the addition of its positive counterpart and the subtraction of a positive number to the addition of its negative counterpart.

$(+15) + (-27) - (-8) - (+12) + (+9)$

becomes

$15 + (-27) + 8 + (-12) + 9$

We've effectively removed the parentheses while maintaining the integrity of the mathematical operations. This step is crucial because it simplifies the expression and prepares it for the next phase.

Step 2: Group Like Terms Together

Now that we have an expression without parentheses, we can rearrange the terms to group the positive and negative numbers together. This rearrangement is based on the commutative property of addition, which states that the order in which numbers are added does not change the sum.

$15 + (-27) + 8 + (-12) + 9$

Rearrange to group like signs:

$15 + 8 + 9 + (-27) + (-12)$

This grouping strategy makes it easier to perform the calculations in a logical and organized manner, reducing the likelihood of errors.

Step 3: Perform the Addition

The next step involves adding the grouped numbers. We'll add the positive numbers first and then the negative numbers.

Adding Positive Numbers

$15 + 8 + 9$

Sum the positive numbers:

$15 + 8 = 23$

$23 + 9 = 32$

So, the sum of the positive numbers is $32$.

Adding Negative Numbers

Now, let's add the negative numbers:

$(-27) + (-12)$

Sum the negative numbers:

$-27 + (-12) = -39$

The sum of the negative numbers is $-39$.

Step 4: Combine the Sums

Finally, we combine the sum of the positive numbers and the sum of the negative numbers to get the final result.

Combine the sums:

$32 + (-39)$

This is the same as:

$32 - 39$

Perform the subtraction:

$32 - 39 = -7$

Therefore, the result of the expression $(+15) + (-27) - (-8) - (+12) + (+9)$ is $-7$.

Mastering the skill of rewriting expressions without parentheses and rearranging terms is essential for several reasons:

1. Simplifies Calculations: Removing parentheses and grouping like terms makes calculations easier and less prone to errors.
2. Foundation for Algebra: This skill is fundamental in algebra, where you often deal with complex expressions involving variables and signed numbers.
3. Problem-Solving: It enhances your problem-solving abilities by allowing you to break down complex problems into simpler steps.

When working with signed numbers and expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

1. Incorrectly Removing Parentheses: Make sure to correctly apply the rules of signed number arithmetic when removing parentheses.
2. Forgetting the Sign: Always keep track of the signs of the numbers, especially when rearranging terms.
3. Miscalculating Sums: Double-check your addition and subtraction to avoid errors.

To reinforce your understanding, try solving the following exercises:

1. $(-10) - (+5) + (-3) - (-7)$
2. $(+20) + (-15) - (-10) + (-5)$
3. $(-18) + (+12) - (+6) - (-9)$

In conclusion, rewriting expressions with signed numbers as sums without parentheses, grouping like terms, and performing the calculations is a crucial skill in mathematics. It simplifies complex problems, builds a foundation for algebra, and enhances your problem-solving abilities. By following the steps outlined in this article and practicing regularly, you can master this skill and excel in your mathematical endeavors. The ability to manipulate mathematical expressions efficiently and accurately is a cornerstone of mathematical proficiency, and mastering these techniques will undoubtedly benefit you in more advanced mathematical studies.