Commutative Property Of Addition In Complex Expressions

Jul 11, 2025 by ADMIN 56 views

Which Expression Demonstrates the Commutative Property of Addition in Simplifying (-1 + i) + (21 + 5i)? A Comprehensive Guide

Introduction

Understanding the commutative property of addition is crucial for simplifying complex expressions in mathematics. This property allows us to change the order of addends without affecting the sum. In this article, we will delve into the commutative property, demonstrate its application in simplifying the expression (-1 + i) + (21 + 5i), and analyze the given options to identify the correct one. Whether you're a student grappling with complex numbers or a math enthusiast looking to refresh your knowledge, this guide will provide a clear and comprehensive understanding of the commutative property and its practical applications.

The question at hand asks us to identify which expression demonstrates the use of the commutative property of addition in the first step of simplifying the complex number expression (-1 + i) + (21 + 5i). The commutative property of addition states that for any two numbers, a and b, a + b = b + a. This property is fundamental in arithmetic and algebra, allowing us to rearrange terms in an addition without changing the result. Before diving into the options, let's break down the commutative property and its significance in simplifying expressions. Understanding this property will enable us to correctly identify the step that utilizes it in the given expression. We'll explore why this property is so important and how it fits into the broader context of mathematical operations.

Understanding the Commutative Property of Addition

The commutative property of addition is a basic principle in mathematics that states that the order in which numbers are added does not change the sum. Mathematically, this can be expressed as: a + b = b + a, where a and b can be any real or complex numbers. This property is fundamental because it simplifies calculations and algebraic manipulations. For example, 3 + 5 is the same as 5 + 3, both resulting in 8. While seemingly simple, this principle is the backbone of many algebraic techniques, especially when dealing with more complex expressions. The commutative property helps us rearrange terms to group like terms together, making simplification easier and more intuitive.

This property is not just limited to real numbers; it also applies to complex numbers, matrices, and other mathematical entities where addition is defined. When working with complex numbers, which have both a real and an imaginary part, the commutative property allows us to rearrange the terms so that the real parts and imaginary parts can be grouped separately. This is essential for simplifying expressions involving complex numbers, as it allows us to combine like terms more efficiently. For example, if we have an expression like (a + bi) + (c + di), we can use the commutative property to rearrange it as a + c + bi + di, making it easier to combine the real parts (a and c) and the imaginary parts (bi and di). This rearrangement is a crucial step in simplifying and solving equations involving complex numbers.

In the context of simplifying the given expression (-1 + i) + (21 + 5i), the commutative property allows us to rearrange the terms so that the real numbers (-1 and 21) and the imaginary numbers (i and 5i) are grouped together. This rearrangement makes it easier to combine like terms and arrive at the simplified form. Without the commutative property, simplifying such expressions would be significantly more complicated. It’s a foundational tool that streamlines the process and helps avoid errors. Therefore, recognizing and applying the commutative property correctly is crucial for efficient and accurate simplification of mathematical expressions.

Analyzing the Expression (-1 + i) + (21 + 5i)

The given expression is (-1 + i) + (21 + 5i). To simplify this expression using the commutative property of addition, the initial step involves rearranging the terms to group the real and imaginary parts together. The real parts are -1 and 21, while the imaginary parts are i and 5i. The commutative property allows us to change the order of these terms without altering the final result. The goal is to rearrange the expression to a form that allows us to combine the real numbers and the imaginary numbers separately, making the simplification process more straightforward. This is a standard technique when dealing with complex numbers, and it relies heavily on the flexibility provided by the commutative property.

Applying the commutative property, we can rearrange the expression as (-1 + 21) + (i + 5i). This rearrangement makes it clear that we are adding the real parts (-1 and 21) together and the imaginary parts (i and 5i) together. This step is a direct application of the property a + b = b + a, where a and b can be real or complex numbers. By grouping like terms, we set the stage for the next step in simplification, which involves combining these terms. Understanding this step is crucial because it demonstrates how the commutative property is not just a theoretical concept but a practical tool for simplifying complex expressions.

Once the terms are grouped, the next step is to perform the addition. Adding the real parts, -1 + 21 equals 20. Adding the imaginary parts, i + 5i equals 6i. Therefore, the simplified expression becomes 20 + 6i. This process highlights the efficiency of using the commutative property to rearrange and simplify expressions. By identifying and applying this property correctly, we can break down complex problems into more manageable steps. This approach is not only useful for complex numbers but also for other algebraic expressions where combining like terms is necessary for simplification.

Evaluating the Given Options

Now, let's evaluate the given options to determine which one demonstrates the use of the commutative property of addition in the first step of simplifying the expression (-1 + i) + (21 + 5i).

Option A: (-1 + i) + (21 + 5i) + 0

This option introduces the addition of 0. While adding 0 does not change the value of the expression (due to the identity property of addition), it does not demonstrate the commutative property. The commutative property involves changing the order of addends, not adding an additional term. Therefore, Option A is not the correct answer.
Option B: -1 + (i + 21) + 5i

This option rearranges the parentheses but does not explicitly demonstrate the commutative property in the first step. The associative property is being applied here, which states that the way numbers are grouped in addition does not change the sum (i.e., (a + b) + c = a + (b + c)). While associative and related, it is not the commutative property, which is what we're looking for. Therefore, Option B is not the correct answer.
Option C: (-1 + 21) + (i + 5i)

This option correctly demonstrates the commutative property of addition. It rearranges the terms in the original expression to group the real numbers (-1 and 21) and the imaginary numbers (i and 5i) together. This is a direct application of the commutative property, making it the correct answer. By rearranging the terms, we can clearly see how the real parts and imaginary parts are being combined separately, which is a crucial step in simplifying complex number expressions.
Option D: -(1 - i) + (21 + 5i)

This option changes the sign of the first complex number, which is not an application of the commutative property. It introduces a different operation (changing the sign) rather than rearranging terms. Therefore, Option D is not the correct answer.

Conclusion

In conclusion, the correct answer is Option C: (-1 + 21) + (i + 5i). This option demonstrates the use of the commutative property of addition in the first step of simplifying the expression (-1 + i) + (21 + 5i) by rearranging the terms to group the real and imaginary parts together. Understanding and applying the commutative property is essential for efficiently simplifying mathematical expressions, especially those involving complex numbers. This property allows us to rearrange terms without changing the sum, making it easier to combine like terms and arrive at the simplified form.

By mastering fundamental properties like the commutative property, students and math enthusiasts can approach complex problems with greater confidence and accuracy. The ability to recognize and apply these properties correctly is a cornerstone of mathematical proficiency. This article has provided a detailed explanation of the commutative property, its application in simplifying complex expressions, and a step-by-step analysis of the given options. With this knowledge, you are well-equipped to tackle similar problems and deepen your understanding of mathematical principles.