Commutative Property Of Addition How To Simplify Complex Expressions

by ADMIN 69 views

Which option accurately illustrates the application of the commutative property of addition as the initial step in simplifying the expression (-1+i)+(21+5i)?

In the realm of mathematics, particularly when dealing with complex numbers, understanding fundamental properties like the commutative property of addition is crucial for simplifying expressions effectively. This article delves into the application of the commutative property within the context of complex number addition, using the expression (-1 + i) + (21 + 5i) as a practical example. We will dissect the expression, identify the role of the commutative property, and elucidate how it streamlines the simplification process.

Understanding the Commutative Property

At its core, the commutative property of addition states that the order in which numbers are added does not affect the sum. Mathematically, this is expressed as a + b = b + a, where a and b can be any real or complex numbers. This seemingly simple property is a cornerstone of arithmetic and algebra, allowing us to rearrange terms in an expression to facilitate easier calculations. In the context of complex numbers, this property becomes especially handy as it allows us to group real and imaginary parts together, making addition more intuitive.

When simplifying expressions involving complex numbers, the commutative property provides the flexibility to rearrange terms. This flexibility is particularly useful when you want to group together the real parts and the imaginary parts of complex numbers. For example, in the expression (-1 + i) + (21 + 5i), we have two complex numbers being added. Each complex number consists of a real part and an imaginary part. The commutative property allows us to rearrange the terms so that the real parts (-1 and 21) are together and the imaginary parts (i and 5i) are together. This rearrangement makes it easier to perform the addition because we can add the real parts and the imaginary parts separately.

By applying the commutative property, we transform the original expression into a more manageable form. This transformation is not just about changing the order of the terms; it's about setting the stage for the next steps in the simplification process. Once the real and imaginary parts are grouped together, we can then apply the associative property and the distributive property (if necessary) to further simplify the expression. This step-by-step approach, starting with the commutative property, ensures that we handle complex number addition accurately and efficiently. Understanding the commutative property is therefore not just about knowing a rule; it's about understanding how to use this rule to make complex mathematical operations simpler and more understandable.

Dissecting the Expression: (-1 + i) + (21 + 5i)

The given expression, (-1 + i) + (21 + 5i), involves the addition of two complex numbers. To simplify this expression, we aim to combine the real parts and the imaginary parts separately. The commutative property of addition plays a crucial role in this process. Let's break down the expression:

  1. Identifying the Components: The expression consists of two complex numbers: (-1 + i) and (21 + 5i). Each complex number has a real part (-1 and 21, respectively) and an imaginary part (i and 5i, respectively).
  2. The Challenge: Directly adding the numbers in the given order can be cumbersome. We need a way to group the real and imaginary parts together to facilitate addition.
  3. The Solution: Commutative Property: This is where the commutative property comes into play. By applying this property, we can rearrange the terms in the expression without changing its value. This allows us to group the real numbers (-1 and 21) and the imaginary numbers (i and 5i) together.

The initial expression presents a challenge because it mixes real and imaginary components within parentheses. This mixing makes it difficult to see the like terms that can be combined. The commutative property is the key to unlocking a more straightforward approach. By rearranging the terms, we can bring the real numbers together and the imaginary numbers together, creating a clearer path to simplification. This rearrangement is not just a cosmetic change; it's a strategic move that makes the subsequent steps of addition much easier and more intuitive. Understanding how to apply the commutative property in this context is crucial for mastering complex number arithmetic.

Therefore, by recognizing the structure of the expression and the role of the commutative property, we set the stage for a more efficient simplification process. The next step involves actually applying the property to rearrange the terms, which we will explore in detail in the following sections. This initial analysis highlights the importance of understanding fundamental mathematical properties and how they can be used to tackle complex problems in a systematic way.

Applying the Commutative Property: A Step-by-Step Demonstration

Now, let's demonstrate how the commutative property is applied to simplify the expression (-1 + i) + (21 + 5i). This involves rearranging the terms to group the real parts and the imaginary parts together.

  1. Original Expression: (-1 + i) + (21 + 5i)
  2. Applying Commutative Property: Our goal is to rearrange the terms so that the real numbers (-1 and 21) are together, and the imaginary numbers (i and 5i) are together. We can rewrite the expression as -1 + i + 21 + 5i.
  3. Rearranging Terms: Now, we apply the commutative property to rearrange the terms: -1 + 21 + i + 5i.

This rearrangement is the key step in utilizing the commutative property. We've effectively changed the order of addition without altering the value of the expression. This allows us to clearly see the components that can be combined, setting the stage for the next phase of simplification.

The transformation from the original expression to the rearranged expression highlights the power of the commutative property. It's not just about changing the order of numbers; it's about strategically positioning them to make the subsequent arithmetic operations easier. In this case, by bringing the real numbers and the imaginary numbers together, we've created a visually clearer structure that facilitates the addition process. This step is crucial for anyone learning to work with complex numbers, as it demonstrates how a simple property can significantly streamline the simplification of complex expressions. The ability to recognize and apply the commutative property in this way is a fundamental skill in algebra and beyond.

This step-by-step demonstration underscores the practical application of the commutative property in the context of complex number addition. By rearranging the terms, we've made the expression more amenable to simplification, paving the way for the final steps of combining like terms and arriving at the simplified form. The next section will delve into these final steps, showcasing how the initial application of the commutative property leads to a straightforward and efficient solution.

Simplifying Further: Combining Like Terms

After applying the commutative property and rearranging the terms to get -1 + 21 + i + 5i, the next step is to combine the like terms. This involves adding the real parts together and the imaginary parts together.

  1. Real Parts: -1 + 21 = 20
  2. Imaginary Parts: i + 5i = 6i
  3. Combining Results: Now, we combine the results from the real and imaginary parts: 20 + 6i

This final step yields the simplified form of the original expression. By adding the real parts and the imaginary parts separately, we arrive at a complex number in standard form (a + bi), where a is the real part and b is the imaginary part. In this case, the simplified expression is 20 + 6i.

The process of combining like terms is a fundamental algebraic technique that is greatly facilitated by the prior application of the commutative property. By grouping the real and imaginary parts, we made it straightforward to perform the addition. This highlights the interconnectedness of mathematical properties and techniques; each step builds upon the previous one to achieve the final simplified result. The commutative property laid the groundwork, and combining like terms brought us to the solution.

The ability to simplify complex expressions is a crucial skill in mathematics, with applications ranging from basic algebra to advanced calculus and beyond. This example demonstrates how a clear understanding of fundamental properties, such as the commutative property, can make complex tasks more manageable. By breaking down the problem into smaller steps and applying the appropriate properties, we can systematically arrive at the solution. This approach not only leads to the correct answer but also enhances our understanding of the underlying mathematical principles.

Identifying the Correct Expression

Now that we've demonstrated the application of the commutative property in simplifying the expression (-1 + i) + (21 + 5i), let's examine the given options to identify the one that correctly demonstrates the use of this property in the first step:

A. (-1 + i) + (21 + 5i) + 0 B. -1 + (i + 21) + 5i C. (-1 + 21) + (i + 5i) D. -(1 - i) + (21 + 5i)

To determine the correct option, we need to identify the expression that shows the rearrangement of terms using the commutative property as the initial step.

  • Option A: (-1 + i) + (21 + 5i) + 0. This option introduces the addition of 0, which is an application of the identity property of addition, not the commutative property. Therefore, this is not the correct answer.
  • Option B: -1 + (i + 21) + 5i. This option uses the associative property of addition by regrouping terms within parentheses. While the associative property is related to addition, it's not the commutative property we're looking for. Thus, this option is also incorrect.
  • Option C: (-1 + 21) + (i + 5i). This option correctly demonstrates the commutative property in the first step. It rearranges the terms to group the real parts (-1 and 21) and the imaginary parts (i and 5i) together. This is precisely what we did in our step-by-step demonstration, making this the correct answer.
  • Option D: -(1 - i) + (21 + 5i). This option involves distributing a negative sign, which is a different algebraic manipulation. It doesn't directly demonstrate the commutative property of addition. Therefore, this option is incorrect as well.

Therefore, the correct expression that demonstrates the use of the commutative property of addition in the first step of simplifying the expression (-1 + i) + (21 + 5i) is Option C: (-1 + 21) + (i + 5i). This option clearly shows the rearrangement of terms, grouping the real and imaginary parts together, which is the essence of applying the commutative property in this context.

Conclusion: Mastering the Commutative Property

In conclusion, understanding and applying the commutative property of addition is fundamental to simplifying expressions, especially those involving complex numbers. This property allows us to rearrange terms without changing the value of the expression, making it easier to combine like terms and arrive at a simplified form.

In the context of the expression (-1 + i) + (21 + 5i), the commutative property enables us to group the real parts (-1 and 21) and the imaginary parts (i and 5i) together. This rearrangement transforms the expression into a more manageable form, facilitating the subsequent steps of addition.

By correctly identifying Option C: (-1 + 21) + (i + 5i) as the expression that demonstrates the commutative property in the first step, we reinforce the importance of recognizing and applying fundamental mathematical principles. This skill is not only crucial for simplifying complex number expressions but also for a wide range of algebraic manipulations.

Mastering the commutative property, along with other fundamental properties, is essential for building a strong foundation in mathematics. It empowers us to approach complex problems with confidence and efficiency, making the journey through algebra and beyond smoother and more rewarding. This article has provided a comprehensive guide to understanding and applying the commutative property in the context of complex number addition, equipping you with the knowledge and skills to tackle similar problems with ease. By grasping these core concepts, you can unlock a deeper understanding of mathematics and its applications.